                                                                        WEIL’S QUADRATIC FORM
                                                                       VIA THE SCREW FUNCTION

                                                                                  MASATOSHI SUZUKI


                                                  Abstract. We establish a unified framework for understanding the results on the
                                                  Weil quadratic form obtained by Yoshida (1992), Bombieri (2001, 2003), Connes–
                                                  Consani (2023), and Connes–Consani–Moscovici (2025+) from the perspective of the
                                                  screw function introduced in Suzuki (2023). An advantage of the approach via the
                                                  screw function is that it provides a method to study the Weil quadratic form, which is
                                                  originally defined in terms of distributions, by means of continuous functions. Based




arXiv:2606.09096v1 [math.NT] 8 Jun 2026
                                                  on this framework, we formulate a conjecture stating that a self-adjoint operator whose
                                                  eigenvalues are the imaginary parts of the nontrivial zeros of the Riemann zeta function
                                                  can be obtained as the limit, as a → ∞, of self-adjoint operators arising from nonlocal
                                                  realizations of the first-order differential operator on the finite interval [−a, a]. All
                                                  these results are obtained without assuming the Riemann Hypothesis. This conjecture
                                                  may be compared with the limit formula for the Riemann zeta function expressed in
                                                  terms of zeta-regularized products proposed by Connes, Consani, and Moscovici, and
                                                  it sheds new light on the spectral-theoretic interpretation of the nontrivial zeros of the
                                                  Riemann zeta function.




                                                                                   1. Introduction
                                          1.1. Historical Background and Motivation. For test functions f , the Weil func-
                                          tional f 7→ W (f ) is defined by
                                                         Z ∞                             ∞                   ∞
                                                                                        X    Λ(n)           X    Λ(n)
                                               W (f ) :=       f (x)(ex/2 + e−x/2 )dx −      √ f (log n) −        √ f (− log n)
                                                          −∞                            n=1
                                                                                               n            n=1
                                                                                                                    n
                                                                                Z ∞n                                o ex/2 dx
                                                        − (log 4π + C0 )f (0) −        f (x) + f (−x) − 2e−x/2 f (0) x        ,
                                                                                   0                                  e − e−x
                                          where Λ(n) denotes the von Mangoldt function, defined as Λ(n) = log p if n = pk with
                                          k ∈ Z>0 , and Λ(n) = 0 otherwise, and C0 is the Euler–Mascheroni constant. Although
                                          the precise class of test functions will be specified as needed later, our primary object
                                          of interest is the symmetric (or Hermitian) quadratic form
                                                                QW (v1 , v2 ) := W (v1 ∗ ve2 ),     QW (v) := QW (v, v),
                                          where                             Z ∞
                                                         (v1 ∗ v2 )(x) :=         v1 (y)v2 (x − y) dy     and ve(x) := v(−x).
                                                                             −∞
                                          The Riemann Hypothesis (RH) states that all nontrivial zeros of the Riemann zeta
                                          function ζ(s) lie on the critical line ℜ(s) = 1/2. A fundamental result due to Weil [15]
                                          states that RH is equivalent to the condition that
                                                                           QW (v) ≥ 0      for all v ∈ Cc∞ (R),
                                          a property known as Weil’s positivity criterion for RH (although Weil did not formulate
                                          the criterion in terms of compactly supported smooth test functions; see [15, 16] and
                                          [13, Section 3.2]). In the commentary to his Collected Works, Weil emphasized the
                                             Date: Version of June 9, 2026.
                                             2020 Mathematics Subject Classification. 11M26 42A82 46E22 47B25 .
                                             Key words and phrases. Riemann Hypothesis; Weil quadratic form; screw function; de Branges
                                          spaces; Hilbert–Pólya operator .
                                                                                              1
2                                        M. SUZUKI


distributional nature of this criterion and suggested that it deserved further investigation
by analysts. Despite this invitation, the functional W and the quadratic form QW have
not been extensively studied, with notable exceptions such as the works of Yoshida [17],
Bombieri [1, 2], Connes and Consani [3], and Connes, Consani, and Moscovici [4].
   The study of the localization of the Weil quadratic form QW was pioneered by Yoshida
[17], who established that RH is equivalent to the positive definiteness of QW on the
space
                     Cc∞ (−a, a) := {v ∈ Cc∞ (R) | supp(v) ⊂ [−a, a]}
for every a > 0. In [17, Proposition 1], Yoshida further demonstrated that the condition
QW (v) > 0 for all non-zero odd functions v ∈ Cc∞ (R) implies RH, while a similar condi-
tion for all non-zero even functions implies RH except for possible real zeros. Crucially,
he initiated the variational study of QW by considering the infimum of the Rayleigh
quotient QW (v)/∥v∥2L2 on the localized space
    K(a) := {v | v(x) = f (x)1[−a,a] (x) for some f ∈ C ∞ (R) which has period 2a},
proving that it is positive for sufficiently small a > 0 [17, Lemma 2] or for appropriate
subspaces KN (a) of finite codimension [17, Lemma 3]. The non-degeneracy of QW on the
                [ (,→ L2 (−a, a)) was shown to be equivalent to RH in [17, Theorem
“completion” K(a)
                           [ does not denote the Fourier transform.
2]. Note that the hat in K(a)
   Building on this foundation, Bombieri [1, 2] addressed the minimization of the Rayleigh
quotient on the Sobolev space H01 (−a, a) [1, Problem 1] (and [2, Problem B]) and on
L2 (E) for a finite union of intervals E [1, Problem 2] (and [2, Problem A]). While the
existence of a minimizer for the lower bound on L2 (E) was established in [1, Theorem 3]
(and [2, Theorem 4.3]), the subsequent argument concerning the continuity of this lower
bound with respect to a remains an analytically delicate issue. Specifically, although
it was claimed in [1, Theorem 5] that the lower bound varies continuously when the
minimization is restricted to the subspaces of either even or odd functions, this asser-
tion calls for a more careful analysis. In view of Yoshida’s result, such continuity would
essentially reproduce the equivalence between the failure of RH and the existence of a
degenerating element v ∈ D(QW ) satisfying QW (v) = 0, offering a simplification of the
original proof that relied on explicit formulas. See the discussion following Theorem 1.3
for further details.
   More recently, a rigorous operator-theoretic refinement has been provided by Connes
and Consani [3] and Connes, Consani, and Moscovici [4]. They established that the
localized closed symmetric form
                                    a
                                   QW := QW |L2 (−a,a)
is lower bounded and lower semicontinuous [3, Section 2], leading to the construction of
a canonical, densely defined self-adjoint operator Aa on L2 (−a, a) such that
                                    a
                                   QW (v) = ⟨Aa v, v⟩L2                                (1.1)
for v ∈ D(Aa ) ⊂ D(QW   a ) ⊂ L2 (−a, a) [4, Section 3] (here we reformulate their results,

originally stated on [1/λ, λ], in terms of the variable a = log λ), where the bracket is the
standard L2 inner product
                                              Z ∞
                               ⟨v1 , v2 ⟩L2 =     v1 (x)v2 (x)dx.
                                            −∞
This operator Aa , which can be viewed as a refinement of Bombieri’s Lagrangian ([1,
Lemma 1], [2, Sections 5–6]), possesses a discrete lower bounded spectrum [4, Theorem
3.6] and a corresponding ground state [4, Corollary 3.7]. Furthermore, it was shown in
[4, Theorem 5.10] that, when QW  a is restricted to a certain finite-dimensional subspace
     2
of L (−a, a), the Fourier transform of the eigenfunction corresponding to the lowest
eigenvalue possesses only real zeros.
                   WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                               3


   First, we aim to unify these various results through the framework of the screw func-
tion associated with ζ(s), which was introduced in [13]. This approach not only provides
transparent re-derivations of previous results, but also leads to more refined results pre-
sented below. A key advantage of this screw function approach is that it enables us
to understand Weil’s quadratic form through continuous (and well-behaved) functions,
thereby avoiding the delicate distribution-theoretic aspects of Weil’s original formula-
tion. The connection between the Weil quadratic form and the screw function, already
emphasized in [13, Proposition 3.1], was expected to provide new insight into the study
of the Weil quadratic form. Indeed, [13] reinterpreted some of the results by Yoshida [17],
Bombieri [1, 2], and Connes–Consani [3] from the viewpoint of screw functions. How-
ever, these earlier works had not yet been placed within a fully unified framework. Since
then, the theoretical foundations of the screw-function approach to zeta functions have
been substantially developed.
   In particular, [14] established a completely new picture of the Hilbert space obtained
as the completion of Cc∞ (R) with respect to QW under RH. This was made possible
by the favorable analytic properties of the screw function as an integral kernel, such
as positivity and smoothness. Our main objective is to understand what new insights
about QW can be obtained by localizing the global framework of [14] to the interval
[−a, a], thereby returning to Yoshida’s original viewpoint. This line of investigation was
further motivated by recent results in [4, Theorem 5.10, Sections 7–8]. Let
                              ξ(s) := s(s − 1)π −s/2 Γ(s/2)ζ(s)
be the Riemann ξ-function, where Γ(s) is the usual gamma function. Let va be the
eigenfunction corresponding to the lowest eigenvalue of Aa (denoted by ξλ with a = log λ
in their notation). Denote by
                                         Z ∞
                                fb(z) :=     f (x) eizx dx
                                            −∞
the Fourier transform of f . Motivated by the Hilbert–Pólya operator constructed in [14,
Section 6], we next formulate an analogue of the conjectural limiting formula discussed
in [4, Section 7],
                               lim ca vba (z) = ξ(1/2 + iz).                        (1.2)
                                a→∞
This line of investigation led to a unified perspective on the earlier works mentioned
above and ultimately to Corollary 1.6, which constitutes the main conjectural statement
of the present paper.
1.2. Main Results. Before presenting the main results, we emphasize that none of
them depends on RH. The first step in the study of the Weil quadratic form via screw
functions is to obtain an explicit description of the self-adjoint operator Aa in (1.1).
While the existence of Aa was established abstractly in [4, Section 3] via the theory of
quadratic forms, the use of screw functions provides a constructive and explicit formula
for Aa .
  Let g(t) be a continuous real-valued even function on R, defined by
                                     X Λ(n)
     g(t) = −4(et/2 + e−t/2 − 2) +          √ (|t| − log n)
                                              n
                                      n≤exp(|t|)                                          (1.3)
                |t|                     1                                          
             − (ψ(1/4) − log π) −          Φ(1, 2, 1/4) − e−|t|/2 Φ(e−2|t| , 2, 1/4) ,
                 2                      4
where ψ(s) is the digamma function and Φ(z, s, a) = ∞                    −s n
                                                          P
                                                            n=0 (n + a) z is the Hurwitz–
Lerch zeta function. A function g on R satisfying g(t) = g(−t) is called a screw function
on the real line if the kernel g(t − u) − g(t) − g(−u) + g(0) is nonnegative for all t, u ∈ R.
As shown in [13, Theorem 1.2], the above function g is a screw function in the sense of
4                                        M. SUZUKI


Krein–Langer [8, Section 5] if and only if RH holds. For this reason, we shall refer to it
as the screw function associated with ζ(s). We define the integral operator G by
                                       Z ∞
                           (Gu)(x) :=       g(x − y)u(y) dy.                        (1.4)
                                          −∞
The integral on the right-hand side converges absolutely for u ∈ Cc∞ (R), since g is
continuous. However, as noted immediately after Theorem 1.6 in [13], g(t) does not
tend to zero as |t| → ∞, and hence the integral does not necessarily converge for a
general u ∈ L2 (R). Within the theory of screw functions, it would be more natural
to consider the integral operator G e with the kernel g(t − u) − g(t) − g(−u) + g(0) (cf.
Section 8); nevertheless, for applications to QW , the simpler operator G is sufficient.
    Throughout this paper, for each a > 0, we view L2 (−a, a) as a closed subspace of
L (R) by extending functions to be zero outside (−a, a). Let L20 (−a,Ra) denote the closed
  2
                                                                      a
subspace of L2 (−a, a) consisting of functions u satisfying ub(0) = −a u(x)dx = 0. We
denote by Pa the orthogonal projection from L2 (R) onto L20 (−a, a) (see (2.1)), and define
                         Ga := Pa GPa : L20 (−a, a) → L20 (−a, a).                    (1.5)
The self-adjointness of the operator Ga follows immediately from the fact that g is real-
valued and even. By [13, Proposition 3.1], the integral operator Ga associated with the
screw function is directly related to Weil’s quadratic form QW via differentiation of test
functions. Let D = i d/dx denote the differential operator on L2 (−a, a) with Dirichlet
boundary conditions, and let D∗ be its adjoint. We then define
            Ba := D∗ Ga D : L2 (−a, a) → L2 (−a, a),        D(Ba ) = H01 (−a, a).     (1.6)
This operator is symmetric but fails to be self-adjoint. Then, the operator Aa in (1.1)
can be characterized as follows.
Theorem 1.1. For each a > 0, the self-adjoint operator Aa defined by (1.1) is the
Friedrichs extension of the symmetric operator Ba defined by (1.6).
   This result provides a concrete operator-theoretic realization of the spectral interpre-
tation of QWa established in [4] through the use of screw functions, thereby clarifying the

connection between this spectral interpretation and the classical variational approach of
Yoshida [17].
   Moreover, as shown in [4, Theorem 3.6], for each a > 0, the spectrum of the self-adjoint
operator Aa is bounded from below and discrete, with +∞ as its only accumulation
point. In particular, the largest lower bound of the spectrum of Aa , denoted by λa , is an
eigenvalue. Therefore, there exists a nonzero function v ∈ D(Aa ) attaining the infimum
                                                    a (v)
                                                   QW
                                λa =       inf a ) ∥v∥2
                                                          .                           (1.7)
                                       0̸=v∈D(QW
                                                      L2
Furthermore, the domain D(Aa ) is strictly larger than D(Ba ) = H01 (−a, a) and contains
functions such as constants. As will be shown later (Lemma 3.1), we have
                        a
                       QW (v) = ⟨Ba v, v⟩L2      for   v ∈ H01 (−a, a).               (1.8)
Since the form-norm closure of Cc∞ (−a, a) (⊂ H01 (−a, a)) with respect to QW a contains
                                a
the core of the quadratic form QW as in the proof of Theorem 1.1, we obtain the following
result.
Corollary 1.2. For each a > 0, any function attaining the infimum in (1.7) belongs
to the closure of Cc∞ (−a, a) with respect to the form norm associated with QW
                                                                             a . In

particular, λa is the infimum of the Rayleigh quotient
                                       ⟨Ba v, v⟩L2
                                         ∥v∥2L2
over Cc∞ (−a, a).
                   WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                               5


  This is analogous to [1, Problem 2] (and [2, Problem A]) and [4, Theorem 3.6], but
here the class over which the infimum is taken is made explicit. For the relation between
the Rayleigh quotient in [1, Problem 1] (and [2, Problem B]) and the operator Ga , see
Section 8.4.
   One expects that the eigenvalues of the self-adjoint operator Aa vary continuously in
a. However, because Aa is unbounded and its domain is difficult to describe explicitly,
it is not straightforward to prove such continuity. Indeed, in [1, Theorem 5] (and [2,
Theorem 4.4]), analogues of the infimum in (1.7) are considered on the spaces of even
and odd functions, and their continuity with respect to a is asserted, but the details of
the proof are not fully provided there.
   In contrast, by exploiting the asymptotic expansion (2.2) of the screw function near
the origin, which involves a logarithmic singularity, one can prove the continuity of the
lowest eigenvalue λa without imposing any parity restriction.
Theorem 1.3. The lowest eigenvalue λa is continuous in a.
   Since the continuity of λa can be established without assuming RH, Theorem 1.3
immediately yields, as a corollary, another proof of Yoshida’s result [17] that RH is
equivalent to the nondegeneracy of QW  a for every a > 0. Indeed, the failure of RH is

equivalent to the existence of some a > 0 for which λa < 0. Since λa > 0 for sufficiently
                                                                 a must be degenerate
small a > 0 [17, Lemma 2], it follows that if RH is false, then QW
for some value of a by continuity of λa .
   We now turn to the eigenfunction corresponding to the lowest eigenvalue, following
the approach of [4]. Roughly speaking, [4, Theorem 5.10] (see also [4, Section 7]) shows
that if the lowest eigenvalue λa is simple and its eigenfunction va ∈ D(Aa ) is even,
then all zeros of the Fourier transform vba (z) (corresponding to ξbλ (z) in [4]) are real.
More precisely, they study restrictions of QW a to suitable finite-dimensional subspaces

of L2 (−a, a), obtaining the corresponding reality result at each finite-dimensional level.
In our framework, however, the simplicity of λa and the evenness of the correspond-
ing eigenfunction follow naturally for sufficiently small a by once again exploiting the
asymptotic expansion (2.2) of the screw function.
Theorem 1.4. For sufficiently small a > 0, the lowest eigenvalue λa is positive, simple,
and satisfies
                            1
                    λa = log + µ1 − log(2π) + ψ(2) − 1 + O(a)
                            a
as a → 0+, for some constant µ1 > 0. Furthermore, the corresponding eigenfunction is
even.
   Let us return to the eigenfunction va corresponding to the lowest eigenvalue. The
fact that all zeros of its Fourier transform vba (z) are real follows from the property that
it serves as the characteristic function (given by a zeta-regularized product) of a self-
adjoint perturbation of the differential operator d/dx with periodic boundary conditions
[4, Theorem 5.10]. However, since such a differential operator cannot be defined directly
as an operator on L2 (−a, a), the authors of [4] instead consider restrictions of QW      a to

finite-dimensional subspaces V . Consequently, what is actually established there is not
directly the reality of the zeros of vba (z) itself, but rather the reality of the zeros of the
Fourier transform of a function minimizing the Rayleigh quotient of QW        a | .
                                                                                 V

  Instead of considering perturbations of the differential operator d/dx with periodic
boundary conditions, we construct an analogue of vba (z) by considering self-adjoint ex-
tensions of the minimal operator Da in a Hilbert space different from the usual L2 space.
Choose λ such that λa > λ, and define Ta = Ta,λ := Aa − λI. Since Ta is positive,
                          ∥v∥2Ta := ⟨Ta v, v⟩L2 = QW
                                                   a
                                                     (v) − λ∥v∥2L2                       (1.9)
6                                         M. SUZUKI


defines a norm on Cc∞ (−a, a). Let H(Ta ) denote the corresponding completion. Then
Cc∞ (−a, a) embeds injectively into H(Ta ). We therefore consider the differential operator
Da := i d/dx on H(Ta ) with domain
                                    D(Da ) := Cc∞ (−a, a).                             (1.10)
It turns out that Da is a symmetric operator with deficiency indices (1, 1) (Lemmas 6.1
and 6.2). Hence Da admits a family of self-adjoint extensions D a,θ parametrized by
θ ∈ [0, 2π). A function arising from the boundary form of the adjoint operator Da∗
provides a counterpart to vba (z) in [4], as follows.
Theorem 1.5. Let a > 0 and θ ∈ [0, 2π). Let v± (a, x) be eigenfunctions of the ad-
joint operator Da∗ corresponding respectively to the eigenvalues ±i, normalized so that
∥v+ (a, ·)∥Ta = ∥v− (a, ·)∥Ta . Then the function
                              Z a                                Z a
      W (a, θ; z) := (z − i)      v+ (a, x)eizx dx + eiθ (z + i)     v− (a, x)eizx dx (1.11)
                             −a                               −a

is entire in z. The eigenvalues of the self-adjoint operator D a,θ are precisely the zeros
of W (a, θ; z). Furthermore, all zeros of W (a, θ; z) are real.
    The equation W (a, θ; z) = 0 is equivalent to
                            Z a                       Z a            
            iθ                           izx                         izx
           e = − (z − i)          v+ (x)e dx      (z + i)     v− (x)e dx .
                               −a                             −a

This is analogous to the equation eiθ = exp(iza)/ exp(−iza), which characterizes the
eigenvalues of the self-adjoint extension ∂θ of the minimal operator ∂ = i d/dx on
L2 (−a, a) through the boundary values of its eigenfunctions.
   Let us compare Theorem 1.5 with [4, Theorem 5.10]. First, the two results are parallel
in that both establish the reality of the zeros of a certain function through its relation to
the self-adjointness of a differential operator. However, whereas the result in [4] requires
strong assumptions, such as the simplicity of λa and the evenness of the corresponding
eigenfunction, our Theorem 1.5 has the advantage of being proved unconditionally. In
fact, its proof does not require detailed information on the arithmetic terms appearing
in the Weil form QW   a . It relies essentially only on the fact that the prime contribution
      a
to QW involves only finitely many primes for fixed a > 0. Moreover, they obtain
a determinant representation of vba (z) in terms of a zeta-regularized determinant (by
restricting to finite-dimensional subspaces), thereby interpreting vba (z) as a characteristic
function. By contrast, our expression (1.11) arises from a Lagrangian condition.
   A further difference appears in the approach to RH, as discussed in [4, Sections 7 and
8]. While their function vba (z) is expected to approximate ξ(1/2 − iz) in the limit a → ∞
through the conjectural formula (1.2), our function W (a, θ; z) is expected, as explained
below, to approximate the reciprocal of its logarithmic derivative.
Corollary 1.6. If one can choose θ = θ(a) and ϕ(a, z) (̸= ∞ for any a > 0 and z ∈ C)
such that
                                                      ξ(1/2 − iz)
                        lim eϕ(a,z) W (a, θ; z) = z 2 ′           ,            (1.12)
                       a→∞                           ξ (1/2 − iz)
holds uniformly on every compact subset K ⊂ C, then RH holds.
  A direct computation of the eigenfunctions shows that every eigenvalue of D a,θ has
multiplicity one. The right-hand side of (1.12) is consistent with this simplicity of
the spectrum. The exponential factor exp(ϕ(a, z)) is included to allow for a possible
normalization in the limit process. It is plausible that no such correction is actually
needed, but we do not pursue this question here.
  The reason why the limit formula (1.12) is expected to hold will be discussed in
Section 7. Briefly speaking, it is motivated by the determination of the structure of
                   WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                            7


H(A∞ ) := H(T∞,0 ) under RH in [14, Theorem 1.1], together with the construction of
the Hilbert–Pólya operator described in [14, Section 6].
   Under RH, one has Aa > 0, so that λ = 0 may be chosen in Ta for every a > 0. The
limit formula (1.12) is formulated with the situation Ta = Aa (i.e. λ = 0) in mind, and
indeed Section 7 proceeds under the assumption Aa > 0. Nevertheless, when Aa > 0,
both H(Ta ) and H(Aa ) are defined and are isomorphic as Hilbert spaces. Hence the zeros
of W (a, θ; z) are expected to be independent of the choice of λ used in the definition of
Ta . However, in any attempt to prove the limit formula (1.12), control of λ (< λa ) will
likely play an important role. Such control is expected to require a detailed analysis
of the arithmetic contribution coming from the prime terms in QW    a , which lies beyond

the arguments used in the proof of Theorem 1.5, where only the finiteness of the prime
contribution for fixed a plays a role.
   Since the reality of the zeros of W (a, θ; z) for each a > 0 has already been established
in Theorem 1.5, it would suffice to prove (1.12) as an identity of complex functions in
order to deduce RH. As explained in Section 8, the eigenfunctions v± (a, x) are solutions
of Fredholm integral equations of the first kind associated with the integral operator
Ga whose kernel is the continuous function g(x − y). Consequently, the conjectural
limit formula (1.12) in Corollary 1.6 can be formulated as a purely analytic statement,
independently of H(Ta ).
   The organization of this paper is as follows. In Section 2, we study the asymptotic
expansion of the screw function g near the origin and derive from it several formulas for
the quadratic form ⟨Ba v, v⟩. The latter formulas will be used repeatedly in the proofs
of the main results. Using the results of Section 2, we prove Theorem 1.1 in Section 3
and Theorem 1.3 in Section 4. In both proofs, the compactness of the embedding
D(QW a ) → L2 (−a, a) plays an essential role. Theorem 1.4 is proved in Section 5. In

addition to the results of Section 2, the proof relies on the theory of Dirichlet forms. In
Section 6, we prove Theorem 1.5 using von Neumann’s theory of self-adjoint extensions.
Section 7 explains the motivation for the limit formula in Corollary 1.6 by relating it to
the results of [14]. The main tool there is the theory of de Branges spaces. Finally, in
                                                           a (v) = ⟨A v, v⟩
Section 8, we relate the theory of the quadratic forms QW              a    L2 developed in
this paper to the theory of the quadratic forms ⟨Ga u, u⟩L2 . While the former belongs to
the theory of distributions and is analytically more delicate, the latter has the advantage
of being treatable within the theory of integral operators with continuous kernels.

                                    2. Preliminaries
2.1. Precise definition of operators. Let
                         L20 (−a, a) = {u ∈ L2 (−a, a) | u
                                                         b(0) = 0},
as in Section 1. Then D = i d/dx denotes the differentiation operator on L2 (−a, a)
with Dirichlet boundary conditions v(±a) = 0, whose domain and range are D(D) =
H01 (−a, a) and R(D) = L20 (−a, a), respectively. Let D∗ denote the adjoint of D with
respect to the standard L2 inner product. While D∗ acts as the same differential op-
erator i d/dx, its domain is D(D∗ ) = H 1 (−a, a) and its range is R(D∗ ) = L2 (−a, a).
Consequently, D is not self-adjoint. Define the projection Pa : L2 (R) → L20 (−a, a) by
                                                           Z a
                                       1                 1
                   (Pa u)(x) := u(x) − u b(0) = u(x) −         u(y) dy               (2.1)
                                      2a                2a −a
for x ∈ (−a, a), and (Pa u)(x) = 0 otherwise. Then,
                                           Z a
                                         1
              (Pa Ku)(x) = (Ku)(x) −           (Ku)(t) dt,    x ∈ (−a, a).
                                        2a −a
This defines the operator Ga in (1.5). The kernel function g is continuous, and its
first derivative is piecewise continuous with only a discrete set of discontinuities at
8                                           M. SUZUKI


which finite one-sided limits exist. It follows that for any u ∈ L20 (−a, a), we have
Ga u ∈ D(D∗ ) = H 1 (−a, a). Consequently, the operator Ba on L2 (−a, a) with domain
D(Ba ) := H01 (−a, a) is well-defined by (1.6). Since Ga is self-adjoint, the relation
⟨Ba v, w⟩L2 = ⟨v, Ba w⟩L2 holds for all v, w ∈ D(Ba ). Thus, Ba is a symmetric operator,
but it is not self-adjoint. Indeed, the domain of the adjoint operator, D(Ba∗ ), consists
of all w such that the functional v 7→ ⟨Ba v, w⟩ is L2 -continuous, which also includes
functions w ∈ H 1 (−a, a) that do not vanish on the boundary. That is, D(Ba ) ⊊ D(Ba∗ ).
2.2. Expansion near the origin. We derive an asymptotic formula for the screw
function g in a neighborhood of the origin. The resulting asymptotic formula will be
used repeatedly throughout the paper. [6, 9.554] with z = e−2t , m = 2, and v = 1/4
yields
              e−t/2 Φ(e−2t , 2, 1/4) = 2t(log(2t) + ψ(1/4) − ψ(2))
                                                          ∞
                                                          X                     (−2t)n
                                         + ζ(2, 1/4) +          ζ(2 − n, 1/4)
                                                                                  n!
                                                          n=2
for t → 0+, where ζ(s, a) denotes the Hurwitz zeta function. By definition (1.3),
        g(t) = −4(et/2 + e−t/2 − 2) − (|t|/2)(ψ(1/4) − log π) − (1/4)(F (0) − F (t))
                   X Λ(n)
               +          √ (|t| − log n),
                            n
                 n≤exp(|t|)

where
                                                                   ∞
                                                                   X                     (−2|t|)n
     F (t) := 2|t|(log(2|t|) + ψ(1/4) − ψ(2)) + ζ(2, 1/4) +              ζ(2 − n, 1/4)            .
                                                                                           n!
                                                                   n=2
We have ψ(2) = 1 − C0 [6, (8.365), (8.366)] and ψ(1/4) = −(π/2) − 3 log 2 − C0 [6,
(8.366)], where C0 denotes the Euler–Mascheroni constant. It follows from the defining
series that Φ(1, 2, 1/4) = ζ(2, 1/4). Using also ζ(0, 1/4) = 1/4 [6, (9.523)], we obtain
                       1                    X Λ(n)
              g(t) = |t| log |t| + A|t| +          √ (|t| − log n) + r(t)             (2.2)
                       2                              n
                                             n≤exp(|t|)

in a neighbourhood of t = 0, where r is an even C 2 -function satisfying r(t) = O(t2 ), and
                 1                     1
             A = (log(2π) − ψ(2)) = (log(2π) + C0 − 1) = 0.707546 . . . .
                 2                     2
The sum involving the von Mangoldt function in (2.2) is understood to be zero if |t| <
(1/2) log 2.
2.3. Quadratic form formulas. For the quadratic form
                       QBa (v) := ⟨Ba v, v⟩L2 ,   v ∈ D(Ba ) = H01 (−a, a)
we compute the quadratic forms corresponding to each term on the right-hand side of
(2.2). First, for the first term on the right-hand side, we have
                  Z aZ a
                           |x − y|v ′ (y)v ′ (x)dxdy = −2∥v∥2L2 , v ∈ D(Ba ).
                  −a    −a
Next, for the second term on the right-hand side, if we define
                1 a a |v(x) − v(y)|2            1 a
                  Z Z                             Z
      La (v) :=                         dx dy −        log(a2 − x2 )|v(x)|2 dx                        (2.3)
                4 −a −a      |x − y|            2 −a
for v ∈ D(Ba ) (to ensure the convergence of the second term), then we obtain
     Z aZ a
            |x − y| log |x − y|v ′ (y)v ′ (x)dxdy = 2La (v) − 2∥v∥2L2 , v ∈ D(Ba ).                   (2.4)
      −a   −a
                    WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                              9


We verify this relation. Let I(v) denote the left-hand side. Applying integration by
parts with respect to y, we have
        Z a                                Z a
                                ′
            |x − y| log |x − y|v (y)dy = −     sgn(y − x)(log |x − y| + 1)v(y)dy.
          −a                                      −a
Substituting this into I(v), we introduce an ϵ-cutoff to handle the singularity of the
kernel rigorously:
                      Z a         Z                                      !
         I(v) = − lim     v ′ (x)       sgn(y − x)(log |x − y| + 1)v(y)dy dx.
                    ϵ→0 −a              |y−x|>ϵ

Let Fϵ (x) denote the inner integral inside the brackets. Differentiating it by Leibniz’s
rule, we obtain
                                                         Z
                ′                                                  v(y)
              Fϵ (x) = −(log ϵ + 1)v(x − ϵ) + v(x + ϵ) −                  dy.
                                                          |y−x|>ϵ |x − y|
Using this, we integrate by parts with respect to x. Taking into account the boundary
conditions v(±a) = 0 and the fact that v(x ± ϵ) → v(x) as ϵ → 0, we find
                         Z a"                       Z                  #
                                                            v(x)v(y)
            I(v) = − lim      2(log ϵ + 1)|v(x)|2 +                  dy dx.
                     ϵ→0 −a                          |y−x|>ϵ |x − y|

Here, we use the identity 2Re(v(x)v(y)) = |v(x)|2 + |v(y)|2 − |v(x) − v(y)|2 to convert the
product into a difference of squares (the imaginary part vanishes due to the symmetry
of the double integral). Evaluating the integral concerning |v(x)|2 yields
                  Z
                                        1
                                             dy = −2 log ϵ + log(a2 − x2 ).
                    y∈(−a,a),|y−x|>ϵ |x − y|

Substituting this and simplifying the coefficient of |v(x)|2 , the contribution from the first
term is 2 log ϵ + 2, while the contribution from the second term is −2 log ϵ + log(a2 − x2 ).
Consequently, log ϵ is cancelled out, and we obtain
                  Z a
                                                       1 a a |v(x) − v(y)|2
                                                        Z Z
                             2          2     2
                                                
       I(v) = −        |v(x)| 2 + log(a − x ) dx +                               dxdy.
                    −a                                 2 −a −a        |x − y|
Rearranging the terms yields (2.4).
                                                      −1/2 Λ(n) |t| − log n and calculating the
                                      P                                    
  Furthermore, setting g0 (t) =          n≤exp(|t|) n
corresponding integral,
   Z aZ a
          g0 (x − y) v ′ (y)v ′ (x) dx dy
    −a −a
                                                                                            !
          X Λ(n) Z a−log n                                     Z a
   =−            √                    v(x + log n)v(x) dx +              v(x − log n)v(x) dx .
                   n       −a                                   −a+log n
        n≤exp(2a)

Computing QBa (v) by using the asymptotic expansion (2.2), we get
               QBa (v) = La (v) − (2A + 1)∥v∥2L2
                              X Λ(n) Z a−log n
                         −           √           v(x + log n)v(x) dx
                                       n    −a
                             n≤exp(2a)
                                                      Z a                           !    (2.5)
                                                  +               v(x − log n)v(x) dx
                                                       −a+log n
                             Z aZ a
                         −              r′′ (x − y) v(y)v(x)dxdy
                              −a   −a
for v ∈ H01 (−a, a).
10                                         M. SUZUKI


2.4. Formulas in Fourier transforms. Expanding the numerator of
                                        |v(x) − v(y)|2
                       ZZ
                     1
                                                       dx dy.
                     4 (−a,a)2 ,|x−y|>ϵ     |x − y|

as |v(x)|2 + |v(y)|2 − 2 Re(v(x)v(y)) and using symmetry, it equals
                                               !
   1 a
     Z            Z                                    ZZ
               2                        1            1                    v(x)v(y)
         |v(x)|                              dy dx −                               dx dy.
   2 −a             (−a,a)\(x−ϵ,x+ϵ) |x − y|         2    (−a,a) , |x−y|>ϵ |x − y|
                                                                2


Since
                 Z x−ϵ               Z a
                           1                1
                              dy +              dy = log(a2 − x2 ) − 2 log ϵ,
                   −a     x−y         x+ϵ y − x
we obtain
                      "        Z a                  ZZ                                  #
                                             1                            v(x)v(y)
        La (v) = lim − log ϵ    |v(x)|2 dx −                                       dx dy .
                 ϵ→0         −a              2           (−a,a)2 , |x−y|>ϵ |x − y|

The second term may be rewritten, by Plancherel’s theorem, as the quadratic form
associated with the convolution kernel Kϵ (x) = |x|−1 1{|x|>ϵ} (x). Its Fourier transform
is
                                                   Z ∞
                                      e−izx
                                Z
                                                         cos u
                       Kϵ (z) =
                       c                    dx = 2             du.
                                 |x|>ϵ |x|          ϵ|z|   u
Using the asymptotic expansion of the cosine integral [6, (8.230)], we obtain
              Z ∞
                    cos u
                          du = − log ϵ − log |z| − C0 + o(1)    (ϵ → 0).
               ϵ|z|   u
Substituting this into the Fourier representation yields
                                     Z ∞
                                  1
                                                                 v (z)|2 dz
                                                               
                    La (v) = lim            −2 log ϵ − Kcϵ (z) |b
                             ϵ→0 4π −∞
                                 Z ∞                                                         (2.6)
                              1
                                                      v (z)|2 dz.
                                                    
                           =           log |z| + C0 |b
                             2π −∞
Using the Fourier transform vb of v (defined as the zero-extension of v outside (−a, a))
                    Z ∞                         Z ∞
                                              1
                         v(x ∓ τ )v(x)dx =            v (z)|2 e±izτ dz,
                                                     |b
                      −∞                     2π  −∞

which yields
               Z a−log n                           Z a
                           v(x + log n)v(x) dx +               v(x − log n)v(x) dx
                 −a                                 −a+log n
                                                     Z ∞
                                                   1
                                               =               v (z)|2 cos(z log n)dz.
                                                              |b
                                                   π     −∞

Furthermore, noting that ∥v∥2L2 = (2π)−1 ∥b   v ∥2L2 , we obtain from (2.5) that
                           Z ∞
                        1
             QBa (v) =          (log |z| − log(2π)) |b   v (z)|2 dz
                       2π −∞
                                                                          
                             Z ∞
                           1                      X       Λ(n)
                       −          |v̂(z)|2                √ 2 cos(z log n) dz              (2.7)
                          2π −∞                              n
                                              n≤exp(2a)
                             Z ∞
                           1
                       +                   v (z)|2 dz
                                  rb′′ (z)|b
                          2π −∞ a
                    WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                                 11


for v ∈ H01 (−a, a). To interpret the term rba′′ (z), we decompose r(t) as r(t) = r0 (t)+r1 (t),
where
                                                                 ∞
                                                               1X               (−2|t|)n
         r0 (t) = −4(et/2 + e−t/2 − 2)      and     r1 (t) =      ζ(2 − n, 1/4)          .
                                                               4                  n!
                                                                n=2

Since rb′′ (z) cannot be defined directly, we define rba′′ (z) by treating these two parts
separately. For r0 , we take the Fourier transform of r0,a ′′ (x) := r ′′ (x)1
                                                                      0        (−2a,2a) (x), whereas
                                                               d (z)+ rb′′ (z). As shown below,
for r1 we compute rb′′ (z) explicitly. We then set rb′′ (z) := r′′
                      1                                 a            0,a    1
all resulting terms except those arising from the polar factor s(s − 1) coincide with the
explicit formula.
   Using ζ(2 − n,Pa)  = −Bn−1 (a)/(n − 1) [10, (25.11.14)] and the generating series
                    ∞
ye /(e − 1) = n=0 Bn (a)y n /n! [6, (9.621)], we have
   ay   y


                                                  e−|t|/2   1
                                   r1′′ (t) =             −     .
                                                1 − e−2|t| 2|t|
Hence, the Fourier transform is given by
               Z ∞                 !              Z ∞              !
     ′′                e−t/2     1                       e−u/4   1
    r1 (z) = 2
    b                         −      cos(zt) dt =              −     cos(zu/2) du
                0    1 − e−2t 2t                   0    1 − e−u u
                  "Z                             !                           #
                          e−u/4 cos(zu/2) e−u
                      R                                 Z R −u
                                                            e − cos(zu/2)
           = lim                          −        du +                    du .
             ϵ→+0   ϵ        1 − e−u          u          ϵ        u
              R→∞

Differentiating [6, (8.341.3)] with respect to z, we obtain
                             Z ∞  −u
                                            e−su
                                                   
                                    e
                     ψ(s) =             −            du (Re(s) > 0).
                               0      u   1 − e−u
By taking the real part for s = 1/4 + iz/2, we find
               Z ∞                          !
                     e−u/4 cos(zu/2) e−u
                                                                 
                                                             1 iz
                                      −       du  = − Re  ψ    +
                0         1 − e−u         u                  4   2

for real z. On the other hand, we have
      Z ∞ −u         Z ∞
            e            cos(zu/2)
                du −               du = E1 (ϵ) + Ci(|z/2|ϵ) = log |z| − log 2 + O(ϵ)
       ϵ      u       ϵ      u
for real z by E1 (ϵ) = −C0 − log ϵ + O(ϵ) [10, (6.6.2)] and Ci(z) = C0 + log z + O(|z|) [10,
(6.6.6)]. Therefore,
                                    
                              1 iz
             ′′
            r1 (z) = − Re ψ
            b                   +        + log |z| − log 2 = O(|z|−2 ) |z| → ∞.
                              4     2
It follows that
                              Z ∞                           
                          1                    1 iz
               QBa (v) =             Re ψ         +                v (z)|2 dz
                                                          − log π |b
                         2π     −∞             4     2
                                                                           
                                 Z ∞
                              1                    X      Λ(n)
                           −         |v̂(z)|2            √ 2 cos(z log n) dz
                             2π −∞                          n
                                                n≤exp(2a)
                                 Z ∞
                              1
                           +         r′′
                                     d (z)|b  v (z)|2 dz.
                             2π −∞ 0,a
12                                           M. SUZUKI


This formula precisely reflects the structure of the explicit formula. From the explicit
formula [1, Lemma 3 and its proof], we have
                        Z ∞                                   Z a
                      1              1 iz
             a
           QW (v) =           ℜ ψ      +       |b     2
                                                v (z)| dz − log π     |v(x)|2 dx
                     2π −∞           4    2                        −a
                       + O(a) · ∥v∥2L2 ,
where O(a) denotes a bounded quantity depending on a. Since
               Re [ψ(s/2)] = log |s| − log 2 + O(|s|−2 )      (ℜ(s) > 0)    |s| → ∞,
we obtain
                               Z ∞
                a          1
               QW (v) =                                   v (z)|2 dz + O(a) · ∥v∥2L2 .
                                     (log |z| − log(2π)) |b                               (2.8)
                          2π   −∞
This perfectly matches the result obtained from (2.7).

2.5. Distributional formulas. Since g is continuous, the distributional second deriv-
ative −g ′′ is a tempered distribution. By (1.8) and integration by parts,
                                 Z aZ a
                       QWa
                           (v) =         g(x − y)v ′ (y)v ′ (x) dx dy
                                 Z−a  −a
                                   a Z a                                        (2.9)
                                            ′′
                               =         (−g (x − y))v(y)v(x) dx dy
                                −a −a
         ∞
for v ∈ Cc (−a, a). This was already observed in [13, Section 3.5]. Combining this with
(1.1), we obtain                            Z a
                             (Aa v)(x) =          (−g ′′ (x − y))v(y) dy                 (2.10)
                                             −a
for v ∈ Cc∞ (−a, a). Since g ′′ is a tempered distribution, the right-hand side is well-
defined. Formula (2.10) provides an alternative representation of Bombieri’s Lagrangian
([1, Lemma 1], [2, Sections 5–6]). Let us compute the right-hand side of (2.10). Since
                            d2
                                                    
                                                     1
                                (|t| log |t|) = Pf      + 2δ(t)
                            dt2                     |t|
in the sense of distributions, it follows from (2.2) that
                                   
                    ′′      1        1
                −g (t) = − Pf            − (2A + 1)δ(t)
                            2       |t|
                            X Λ(n)
                                   √ δ(t − log n) + δ(t + log n) − r′′ (t).
                                                                
                          −
                                     n
                               n≥2

Hence
        Z a                              Z a
                ′′                  1           v(y)
            (−g (x − y))v(y) dy = − Pf                dy − (2A + 1)v(x)
         −a                         2     −a |x − y|
                                     X Λ(n)                               
                                  −         √ v(x − log n) + v(x + log n)                (2.11)
                                              n
                                    n≤e2a
                                    Z a
                                  −     r′′ (x − y)v(y) dy.
                                           −a
Here we note that
         Z a Z a                 
                         v(y)
               Pf               dy v(x) dx
              −a     −a |x − y|
                                                   Z a
                    1 a a |v(x) − v(y)|2
                      Z Z
                 =−                        dx dy +     log(a2 − x2 )|v(x)|2 dx
                    2 −a −a       |x − y|           −a
                  WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                          13


which follows by a direct symmetrization argument. Using this identity, one verifies that
(2.9) follows from (2.5).
   For a general element v ∈ D(Aa ) ⊂ D(QWa ), pointwise values need not be well defined,

so an expression such as (2.11) is not meaningful. However, by (2.7), identity (2.9)
remains valid for v ∈ D(Aa ) in the sense of limits of values on elements of Cc∞ (−a, a)
with respect to the form norm of QW a .


                              3. Proof of Theorem 1.1
3.1. The Relation between Aa and Ba . To compare the operators Aa and Ba , we
begin by recalling some basic facts concerning the quadratic form QW a . For general

background on quadratic forms, we refer to Schmüdgen [12, Chapter 10].
   Let Γ denote the set of all zeros of the function z 7→ ξ(1/2 − iz). For each γ ∈ Γ,
let mγ be the multiplicity of γ as a zero of ξ(1/2 − iz). By the functional equations
ξ(1 − s) = ξ(s) and ξ(s) = ξ(s̄), the set Γ is invariant under γ 7→ −γ and γ 7→ γ̄. It is
customary to write the zeros of ξ(s) as ρ = β + iγ (β, γ ∈ R), but our notation differs
from this convention. In particular, an element γ ∈ Γ may be non-real. RH is equivalent
to the inclusion Γ ⊂ R. In this notation, the Weil explicit formula yields
                                            X
                            QW (v1 , v2 ) =     mγ vb1 (γ)vb2 (γ̄)                 (3.1)
                                           γ∈Γ
                                                                                  a is
(cf. [14, (1.2), (3.3)]). The form domain of the closed symmetric quadratic form QW
defined by
                             a
                         D(QW  ) := { v ∈ L2 (−a, a) | |QW
                                                         a
                                                           (v)| < ∞ },
and becomes a Hilbert space with respect to the form norm
                           ∥v∥2Q a := QW
                                       a
                                         (v) + (1 − λa )∥v∥2L2 ,
                                W

where λa is the constant defined in (1.7). By (2.6) and (2.8), if we define
                                           Z ∞                             
            log                  2                      +            2
         H (−a, a) : = v ∈ L (−a, a)            (1 + log |z|)|bv (z)| dz < ∞
                                              −∞
                                    2
                        ⊃ { v ∈ L (−a, a) | La (v) < ∞ },
then
                                       a
                                  D(QW    ) ⊂ H log (−a, a).
  In [2, Theorem 5.1], Bombieri appears to state implicitly that D(QW   a ) coincides with
                    2
the set of all v ∈ L (−a, a) such that
                      Z aZ a                             Z a
                               |v(x) − v(y)|2
                                               dx dy +       |v(x)|2 dx
                       −a −a       |x −  y|               −a
is finite. However, taking into account the equivalence between (2.3) and (2.6), one finds
that this space is in fact slightly larger than D(QWa ).

   The fact that QW a (v) is lower bounded and lower semicontinuous on L2 (−a, a) was

proved in [3, Proposition 2.1]. It is also mentioned in the proof of [1, Lemma 3] and is
implicitly contained in [17, Section 2]. All of these arguments are based on the formula
(2.8). The lower semicontinuity follows immediately from an application of Fatou’s
lemma.
   We now recall the self-adjoint operator Aa associated with QW   a through (1.1). Such

a self-adjoint operator is uniquely determined. Moreover, v ∈ D(Aa ) ⊂ D(QW     a ) if and

only if the map w 7−→ QW (v, w) is continuous on D(QW ) with respect to the L2 -norm.
                          a                               a

By the Riesz representation theorem, this is equivalent to [12, Definition 10.4]. In this
                                           a (v, w) = ⟨A v, w⟩             a
case, Aa v is uniquely characterized by QW               a     L2 (w ∈ D(QW )).

Lemma 3.1. Ba ⊂ Aa . In particular, Aa is a self-adjoint extension of Ba .
14                                        M. SUZUKI


Proof. Let E ⊂ L2 (−a, a) be the linear subspace generated by
                            en (x) = exp(iπnx/a),            n ∈ Z.
The space E is a core for the quadratic form QW a [3, Proposition 2.3]. On the other

hand, if v ∈ D(Ba ) = H0 (−a, a), then vb(z) = O(|z|−1 ) as |z| → ∞. Hence, by (3.1),
                         1

D(Ba ) ⊂ D(QW a ). Moreover,


                                             2 sin(az + nπ)
                                 ebn (z) =                  ,
                                                z + nπ/a
        w(z) = O(|z|−1 ) for every w ∈ E. If u = Dv, then vb(z) = z −1 u
so that b                                                              b(z). Therefore,
                            a                                    1
by (3.1) and [13, (3.1)], QW (v, w) = ⟨Ba v, w⟩L2 for all v ∈ H0 (−a, a) = D(Ba ) and
w ∈ E. It follows from [12, Proposition 10.5 (v)] that Ba ⊂ Aa .                     □

3.2. Completion of the proof. For simplicity, we write A = Aa , B = Ba , QB (v) =
⟨Bv, v⟩L2 , Q = QWa , and ∥v∥2 = Q(v)+(1−λ )∥v∥2 . Note that Q = Q on D(B ). The
                             Q               a    L2                   B         a
Friedrichs extension of the operator B is the uniquely determined self-adjoint extension
associated with the closure QB of the quadratic form QB . The domain D(QB ) of QB is
defined as the completion of the form domain D(QB ) := D(B) = H01 (−a, a) with respect
to the form norm ∥ · ∥Q . Since B ⊂ A (Lemma 3.1), we have D(QB ) ⊂ D(QA ) = D(Q).
Hence, D(QB ) is a closed subspace of D(Q) with respect to the form norm. On the
other hand, the space E (in the proof of Lemma 3.1) is a core for the quadratic form Q
       ∥·∥
(i.e. E Q = D(Q)). Since Cc∞ (−a, a) ⊂ H01 (−a, a), if
                                                      ∥·∥Q
                                   E ⊂ Cc∞ (−a, a)           ,                         (3.2)
then D(QB ) = D(Q). It follows that the Friedrichs extension of B coincides with A.
Indeed, by [12, Corollary 10.8], on a given Hilbert space, lower bounded self-adjoint op-
erators are in one-to-one correspondence with lower bounded closed symmetric quadratic
forms having dense domains.
    To prove the inclusion (3.2), it suffices to show that each function en ∈ E belongs
to the closure of Cc∞ (−a, a) with respect to the form norm ∥ · ∥Q . In other words, we
must construct a sequence vϵ ∈ Cc∞ (−a, a) such that ∥en − vϵ ∥Q → 0 as ϵ → 0. For a
sufficiently small parameter ϵ > 0, we define a cutoff function ηϵ (x) which decays to 0
near the boundary of the interval [−a, a]. More precisely, ηϵ (x) = 1 on [−a + ϵ, a − ϵ],
and ηϵ (±a) = 0. On the boundary intervals [−a, −a + ϵ] and [a − ϵ, a], ηϵ is chosen as a
smooth monotone cutoff interpolating between these values, with |ηϵ′ (x)| ≪ ϵ−1 . Define
vϵ (x) := en (x)ηϵ (x). Then vϵ belongs to Cc∞ (−a, a). It therefore remains to show that
the form norm of the error function wϵ (x) = en (x)(1 − ηϵ (x)) tends to zero as ϵ → 0.
    The function wϵ is supported only on the two intervals [−a, −a + ϵ] and [a − ϵ, a], each
of length ϵ. Moreover, |wϵ (x)| ≤ 1. Therefore,
                                    Z −a+ϵ         Z a
                               2
                          ∥wϵ ∥L2 ≤         1 dx +      1 dx = 2ϵ.
                                     −a               a−ϵ

Hence ∥wϵ ∥L2 → 0 as ϵ → 0.
  By (2.7) and Plancherel’s theorem, we have
                        Z ∞
                    2
               ∥wϵ ∥Q =      (1 + log+ |z|) w
                                           |cϵ (z)|2 dz + O(a) · ∥wϵ ∥2L2 .
                            −∞

Therefore, the principal term in ∥wϵ ∥2Q is
                                 Z ∞
                            Iϵ =                   wϵ (z)|2 dz.
                                      log(3 + |z|)|c
                                   −∞
                       WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                              15


To estimate it, we examine the behavior of w  cϵ . First, it follows immediately from the
definition of the Fourier transform that
                                          Z
                               |cϵ (z)| ≤
                                w           |wϵ (x)| dx ≤ 2ϵ.                        (3.3)
                                                R

                    |cϵ (z)| = O(|z|−1 ) as |z| → ∞, uniformly in ϵ. Indeed, integrating
On the other hand, Rw
                     a
            cϵ (z) = −a wϵ (x)eizx dx, we obtain
by parts in w
                                                a   Z a
                                           eizx                  eizx
                                   
                           w
                           cϵ (z) = wϵ (x)         −     wϵ′ (x)      dx.
                                            iz −a     −a          iz

Here, wϵ (a) = en (a) = (−1)n and wϵ (−a) = en (−a) = (−1)n . Moreover, wϵ′ (x) =
eR′n (x)(1 − ηϵ (x)) − en (x)ηϵ′ (x). Since |ηϵ′ (x)| ≪ ϵ−1 on intervals of total length 2ϵ, we have
    |ηϵ′ (x)| dx = O(1). Hence

                                   ∥wϵ′ ∥L1 ≤ C1 ϵ + O(1) ≤ C2 ,

where C2 is independent of ϵ. Consequently, there exists a constant M > 0 independent
of ϵ such that
                               |wϵ (a)| + |wϵ (−a)| + ∥wϵ′ ∥L1   M
                    |cϵ (z)| ≤
                     w                                         ≤     .           (3.4)
                                              |z|                |z|
   Finally, we show that Iϵ → 0. We split the domain of integration into two parts at
|z| = 1/ϵ. For |z| ≤ 1/ϵ, (3.3) gives
         Z                                       Z 1/ϵ
                                     2         2
                 log(3 + |z|)|c
                              wϵ (z)| dz ≤ (2ϵ)        log(3 + |z|) dz ≪ ϵ log(1/ϵ).
           |z|≤1/ϵ                                     −1/ϵ

For |z| > 1/ϵ, (3.4) gives
                                               Z ∞
                                                              M2
          Z
                                       2
                   log(3 + |z|)|c
                                wϵ (z)| dz ≤ 2      log(3 + z) 2 dz ≪ ϵ log(1/ϵ).
           |z|>1/ϵ                              1/ϵ           z

Combining these estimates, we obtain

                     ∥en − vϵ ∥2Q = Iϵ + ∥wϵ ∥2L2 ≪ ϵ log(1/ϵ) → 0      (ϵ → 0).

Therefore, vϵ ∈ Cc∞ (−a, a) converges to en with respect to the form norm ∥ · ∥Q . Hence
                                         ∥·∥Q
every en ∈ E belongs to Cc∞ (−a, a)             (⊂ D(QB )), and the inclusion (3.2) follows. □


                                  4. Proof of Theorem 1.3
4.1. Compact embedding result. The following proposition is the main analytic
ingredient in the proof of Theorem 1.3.

Proposition 4.1. For a bounded interval I, the embedding H log (I) ,→ L2 (I) is compact.
In other words, every bounded sequence in H log (I) admits a subsequence converging in
L2 (I).

Proof. The proof is essentially the same as that of [4, Theorem 3.6].                            □

   This compact embedding is also the key ingredient in the proof of the discreteness of
the spectrum of Aa in [4, Theorem 3.6]. By [12, Proposition 10.6], it suffices to prove
that the embedding (D(QW  a ), ∥ · ∥ a ) ,→ (L2 (−1, 1), ∥ · ∥ ) is compact. This follows
                                    QW                        L2
readily from Proposition 4.1.
16                                             M. SUZUKI


4.2. Scaling transformation of the Rayleigh quotient. Using (1.8), we transfer
the Rayleigh quotient in (1.7) to the fixed interval [−1, 1] by a scaling transformation.
From definition (1.6), the Rayleigh quotient associated with Ba for v ∈ H01 (−a, a) is
given by
                                            Z aZ a
                                                     g(x − y)v ′ (y)v ′ (x) dxdy
           ⟨Ba v, v⟩L2   ⟨Ga Dv, Dv⟩L2
                       =                 = −a −a Z a                             .
              ∥v∥2L2          ∥v∥2L2                              2
                                                          |v(x)| dx
                                                                    −a

Setting w(t) = v(at) on the fixed interval [−1, 1], we define
                              Z 1Z 1
                                      1
                    qa (w) :=           g(a(x − y))w′ (x)w′ (y) dx dy.                           (4.1)
                               −1 −1  a
Then a change of variables gives
                      ⟨Ba v, v⟩L2   qa (w)
                             2    =        ,         w(t) = v(at) ∈ H01 (−1, 1).                 (4.2)
                        ∥v∥L2       ∥w∥2L2
We write
                                                         qa (w)
                                           R(a, w) :=           .                                (4.3)
                                                         ∥w∥2L2
     For the second term on the right-hand side of (2.3), we have
           Z a                                        Z 1
                     2     2   2           2
               |v(x)| log(a − x ) dx = 2∥v∥L2 log a +     |v(ax)|2 log(1 − x2 ) dx.
            −a                                                      −1

Thus, defining
                      Z 1Z 1                                     Z 1
                  1            |w(x) − w(y)|2         1
        L(w) :=                               dx dy −                  |w(x)|2 log(1 − x2 ) dx   (4.4)
                  4    −1   −1     |x − y|            2           −1

for v ∈ H01 (−1, 1) (which should not be confused with the case a = 1 of (2.3)), we
deduce from (2.5) that
                                              L(w)
        R(a, w) = − log a − (2A + 1) +
                                              ∥w∥2L2
                                              Z 1− log n                  
                      1    X Λ(n)                    a          log n
                  −            √                           w t+                w(x) dx
                    ∥w∥2L2  2a
                                 n             −1                 a
                               n≤e
                                                    Z 1                         !              (4.5)
                                                                   log n
                                                  +           w x−         w(x) dx
                                                    −1+ log n        a
                                                             a
                               Z 1Z 1
                        a
                  −                       r′′ (a(x − y))w(y)w(x) dxdy
                      ∥w∥2L2    −1   −1

for w ∈ H01 (−1, 1).
   For the quadratic form L defined by (4.4) for w ∈ D(L) := H01 (−1, 1), one can show
in the same way as in the proof of (2.6) that
                                      Z ∞
                                    1
                                                          w(z)|2 dz.
                                                        
                          L(w) =            log |z| + C0 | b                          (4.6)
                                   2π −∞
                                                                R∞
Hence the form norm ∥w∥2L := L(w)+∥w∥2L2 is equivalent to −∞ (1+log+ |z|)| b     w(z)|2 dz.
Therefore, the completion of D(L) with respect to ∥·∥L can be identified with a subspace
of H log (−1, 1). In particular, formula (4.6) is valid not only for w ∈ D(L), but also for
w ∈ D(L).
                     WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                              17


  To prove Theorem 1.3, it remains to establish the upper semicontinuity
                                       lim sup λa ≤ λa0 .                                  (4.7)
                                         a→a0
and lower semicontinuity
                                       lim inf λan ≥ λa0 .                                 (4.8)
                                        n→∞
From (4.2) and Theorem 1.1, we obtain for the Rayleigh quotient associated with the
closure q̄a of qa that
                       QaW (v)   q̄a (w)
                               =         , v(at) = w(t) ∈ D(q̄a ).            (4.9)
                       ∥v∥2L2    ∥w∥2L2
By (4.5), (4.6), and the definition of H log (−1, 1), we obtain D(q̄a ) ⊂ H log (−1, 1) for
every a > 0. Hence, by (1.7) and (4.9), it suffices to study the infimum of
                                                    q̄a (w)
                                      R̄(a, w) :=           .
                                                    ∥w∥2L2
4.3. Proof of upper semicontinuity. Let qa be the quadratic form defined in (4.1),
and let R(a, w) be the Rayleigh quotient defined in (4.3). We choose w0 ∈ D(q̄a0 ) ⊂
H log (−1, 1) such that λa0 = R̄(a0 , w0 ), and normalize it by ∥w0 ∥L2 = 1. To avoid
referring to pointwise values of w0 and w   c0 , we choose a sequence wn ∈ Cc∞ (−1, 1)
such that wn → w0 with respect to the form norm ∥ · ∥0 (the form norm associated
with qa0 ), and ∥wn ∥L2 = 1. Such a sequence exists by the proof of Theorem 1.1. Since
Cc∞ (−1, 1) ⊂ D(qa ), each wn belongs to D(qa ) for every a, and R(a, wn ) can be expressed
in terms of wn and w cn .
   Fix n. By the definition of the infimum λa ≤ R(a, wn ) for every a. Moreover, (4.5)
implies that R(a, wn ) → R(a0 , wn ) as a → a0 . Since ∥wn ∥L2 = 1, we obtain
                             lim sup λa ≤ R(a0 , wn ) = qa0 (wn ).
                               a→a0
Passing to the limit as n → ∞, we obtain
                 lim sup λa ≤ lim qa0 (wn ) = q̄a0 (w0 ) = R̄(a0 , w0 ) = λa0 .
                   a→a0         n→∞

Thus, (4.7) holds.                                                                            □

4.4. Proof of lower semicontinuity. By (4.5), we decompose q̄a = q̄ 0 + q̄a1 , where
q̄ 0 (w) = L(w) − (2A + 1)∥w∥2L2 and q̄a1 denotes the remaining terms in (4.5). Let (an ) be
a sequence with an → a0 , and let wn ∈ D(q̄an ) ⊂ H log (−1, 1) be a sequence such that
λan = R̄(an , wn ) and ∥wn ∥L2 = 1. We first show that there exists a subsequence of (wn )
which converges in L2 to some limit w∗ . Fix an arbitrary w ∈ H01 (−1, 1) ⊂ H log (−1, 1).
Then λan ≤ R(an , w), and since R(a, w) is continuous in a by the formula (4.5), the
sequence (λan ) is bounded above.
     Since λan = q̄an (wn ) by λan = R̄(an , wn ) and ∥wn ∥L2 = 1, the sequence q̄an (wn )
is bounded above. As (an ) remains in a compact neighborhood of a0 , the estimate
q̄a1 (v) = O(∥v∥2L2 ) holds uniformly in a. It follows that the terms other than L(wn )
in (4.5) are uniformly bounded. Consequently, L(wn ) is bounded above. Since L is
bounded below by definition, the sequence L(wn ) is bounded. Together with (4.6), this
shows that (wn ) is bounded in H log (−1, 1). Proposition 4.1 therefore implies that (wn )
is relatively compact in L2 , and we may extract a subsequence converging in L2 . We
relabel it again as (wn ) and denote its limit by w∗ .
     The explicit expression for q̄a1 shows that it is a finite sum of quadratic forms associated
with translation operators and integral operators with continuous kernels. Thus q̄a1 (w)
depends continuously on (a, w) near (a0 , w∗ ), and therefore
                                    lim q̄ 1 (wn ) = q̄a10 (w∗ ).
                                   n→∞ an
18                                                M. SUZUKI


On the other hand, q̄ 0 is an a-independent lower bounded closed quadratic form on
L2 (−1, 1). Hence, by [12, Proposition 10.1], it is lower semicontinuous with respect to
L2 -convergence:
                                      lim inf q̄ 0 (wn ) ≥ q̄ 0 (w∗ ).
                                       n→∞
Combining these estimates, we obtain
         lim inf λan = lim inf q̄ 0 (wn ) + q̄a1n (wn ) ≥ q̄ 0 (w∗ ) + q̄a10 (w∗ ) = q̄a0 (w∗ ).
                                                       
          n→∞             n→∞

Since wn → w∗     in L2 (−1, 1) and ∥w   n ∥L2 = 1 for all n, we have ∥w∗ ∥L2 = 1. Therefore,
R̄(a0 , w∗ ) = q̄a0 (w∗ ), and by the definition of λa0 as the infimum of R̄(a0 , ·), q̄a0 (w∗ ) =
R̄(a0 , w∗ ) ≥ λa0 . Consequently,
                                    lim inf λan ≥ q̄a0 (w∗ ) ≥ λa0 ,
                                     n→∞

which proves (4.8).                                                                                   □


4.5. A remark on the continuity of λa via parity decomposition. Since the kernel
g(x − y) is even, Ga commutes with the parity operator (Ju)(x) = u(−x), i.e. Ga J =
JGa . It follows that both Ba = D∗ Ga D and its Friedrichs extension Aa commute with
J. In particular, each eigenspace E(λ) of Aa admits a decomposition into even and odd
subspaces: E(λ) = E + (λ) ⊕ E − (λ). For the lowest eigenvalue space E(λa ) = E + (λa ) ⊕
E − (λa ), the assumption in [4, Theorem 5.10] amounts to requiring that dim E + (λa ) = 1
and E − (λa ) = {0}.
   In view of the parity decomposition and Corollary 1.2, we define
                                                  ⟨Ba v, v⟩L2
                       λ•a :=        inf                      ,      • ∈ {+, −}.
                                0̸=v∈H•1 (−a,a)     ∥v∥2L2
Then the global infimum λa in (1.7) can be expressed as
                                                          −
                                           λa = min(λ+
                                                     a , λa ).                                     (4.10)
Thus the continuity of λa follows from that of λ±a , since the minimum of two continuous
functions is continuous. The continuity of λ±  a is asserted in [1, Theorem 5], although
the details of the proof are not fully provided there.
   We now verify that (4.10) holds. Every v ∈ H01 (−a, a) admits a unique decomposition
into even and odd components: v = v + + v − . Since differentiation reverses parity and
Ga preserves it, Dv + and Ga (Dv + ) are odd, whereas Dv − and Ga (Dv − ) are even.
Expanding the quadratic form in the numerator,
                    ⟨Ga Dv, Dv⟩ = ⟨Ga (Dv + ), Dv + ⟩ + ⟨Ga (Dv − ), Dv − ⟩,
since the cross terms vanish due to the orthogonality. Similarly, ∥v∥2L2 = ∥v + ∥2L2 +
∥v − ∥2L2 for the denominator. Hence the Rayleigh quotient on H01 (−a, a) can be written
as
                    ⟨Ga Dv, Dv⟩   ⟨Ga (Dv + ), Dv + ⟩ + ⟨Ga (Dv − ), Dv − ⟩
                                =                                           .
                      ∥v∥2L2                  ∥v + ∥2 + ∥v − ∥2
For each component we have ⟨Ga (Dv • ), Dv • ⟩ ≥ λ•a ∥v • ∥2 for • ∈ {+, −} by definition.
Therefore,
               ⟨Ga Dv, Dv⟩   λ+ ∥v + ∥2 + λ−      − 2
                                              a ∥v ∥                   −
                           ≥ a + 2                      ≥ min(λ+  a , λa ),
                     2
                  ∥v∥L2          ∥v ∥ + ∥v − ∥2
which yields λa ≥ min(λ+    −                                                      1
                       a , λa ). Conversely, restricting the Rayleigh quotient to H+ (−a, a)
and H−1 (−a, a) immediately yields λ ≤ min(λ+ , λ− ). Thus, (4.10) is established.
                                     a          a    a
                   WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                              19


                                 5. Proof of Theorem 1.4
5.1. Positivity of the lower bound. By (4.9), the behavior of λa for small a > 0 is
determined by the asymptotics of the Rayleigh quotient R(a, w) in (4.3) as a → 0+.
From (4.5), we have
                                         1              L(v)
                        R(a, v) = log      − (2A + 1) +      + O(a)                      (5.1)
                                         a              ∥v∥2
for 0 < a < (1/2) log 2 and v ∈ H01 (−1, 1), so it suffices to study L(v)/∥v∥2 . Note that
this quantity is independent of a. In general, the infimum of L(v)/∥v∥2 need not coincide
with that of R(a, v), and hence their minimizers need not agree, but this distinction will
not be relevant in what follows. By (4.6), the domain D(L̄) of the closure L̄ is contained
in H log (−1, 1). Furthermore, the self-adjoint operator T associated with L̄ has discrete
spectrum. This can be proved by the same argument as in the proof of [4, Theorem
3.6], using Proposition 4.1.
   The term log(1/a) on the right-hand side of (5.1) diverges to +∞ as a → 0+. Since
the closure L̄ is a lower bounded closed quadratic form, it is lower semicontinuous [12,
Proposition 10.1]. We claim that inf ∥v∥=1 L̄(v) > 0. Indeed, it follows directly from the
definition that L(v) ≥ 0. If there existed a sequence (vn ) such that L(vn ) → 0, then
the first term in (4.4) would also tend to 0, hence a subsequence of (vn ) converges in
L2 (−1, 1) to a constant function by (4.6) and Proposition 4.1. However, the second term
in (4.4) is strictly positive for constant functions. Therefore, for sufficiently small a > 0,
we obtain λa = inf w R(a, w) > 0.

5.2. Positivity improving property. The positivity improving property of the semi-
group associated with L̄ established in this subsection will play a key role in proving
the simplicity of the lowest eigenvalue. From (4.4), it is easy to see that L(|v|) ≤ L(v)
holds for all v ∈ D(L). Hence a minimizer v0 of the Rayleigh quotient L̄(v)/∥v∥2 may
be chosen to be nonnegative.
    To show that v0 is positive, rather than merely nonnegative, we apply the theory of
Dirichlet forms [5]. A symmetric closed quadratic form E with domain F is called a
Dirichlet form if it satisfies the Markov property. That is, for every v ∈ F , the function
v ♯ := ϕ◦v = (0∨v)∧1, ϕ(t) := min(1, max(0, t)), belongs to F and satisfies E(v ♯ ) ≤ E(v).
Note that ϕ is Lipschitz continuous.
    We claim that v ♯ ∈ D(L̄) and L̄(v ♯ ) ≤ L̄(v) for every v ∈ D(L̄). As the domain
D(L) is dense in D(L̄), it suffices to verify these properties on D(L) using the Beurling–
Deny representation (4.4) of L̄ [5, Theorem 3.2.1]. For the jumping part, the Lipschitz
continuity of ϕ yields
                 |v ♯ (x) − v ♯ (y)|2 = |ϕ(v(x)) − ϕ(v(y))|2 ≤ |v(x) − v(y)|2 .
Multiplying by the kernel |x − y|−1 and integrating, we obtain
           1 1 1 |v ♯ (x) − v ♯ (y)|2         1 1 1 |v(x) − v(y)|2
             Z Z                               Z Z
                                      dx dy ≤                      dx dy.
           2 −1 −1       |x − y|              2 −1 −1      |x − y|
For the killing part, |v ♯ (x)|2 ≤ |v(x)|2 by |ϕ(a)| ≤ |a|, and κ(x) := − log(1 − x2 ) ≥ 0.
Hence                     Z 1                    Z    1
                                κ(x)|v ♯ (x)|2 dx ≤       κ(x)|v(x)|2 dx.
                          −1                −1
Combining these estimates, we obtain v ∈ D(L̄) and L̄(v ♯ ) ≤ L̄(v). Therefore (L̄, D(L̄))
                                      ♯

satisfies the Markov property and hence defines a Dirichlet form.
   We next show that (L̄, D(L̄)) is irreducible. Recall that a Dirichlet form is called
irreducible if every invariant set is trivial, that is, either it or its complement has measure
zero. Let A ⊂ (−1, 1) be an L̄-invariant set. By the characterization of invariant sets in
20                                         M. SUZUKI


[9, p. 173], L̄(1A u, 1Ac u) = 0 for all u ∈ D(L̄), where Ac = (−1, 1) \ A. Since D(L) is a
core of D(L̄), it suffices to test the condition on D(L). We have
                                             Z Z
                                           1        u(x)u(y)
                        L(1A u, 1Ac u) = −                   dx dy = 0
                                           2 A Ac |x − y|
for all u ∈ D(L), since
           1 1 1 (u(x) − u(y))(v(x) − v(y))         1 1
             Z Z                                     Z
L(u, v) =                                   dx dy −      u(x)v(x) log(1 − x2 ) dx.
           4 −1 −1         |x − y|                  2 −1
Because the jumping kernel |x − y|−1 is positive on (−1, 1)2 , the above identity cannot
hold when both A and Ac have positive measure. Thus either A or Ac has measure zero,
and L̄ is irreducible.
   By [9, Theorem 1.4], the irreducibility implies that the associated semigroup {e−tT |
t > 0} is positivity improving. In other words, for every nonnegative function f ∈
L2 (−1, 1) with f ̸≡ 0, one has (e−tT f )(x) > 0 for a.e. x ∈ (−1, 1). In particular, the
nonnegative eigenfunction v0 of T is positive almost everywhere.
5.3. Simplicity of the lowest eigenvalue and parity of eigenfunctions. Suppose
that the lowest eigenvalue of T admits two orthogonal eigenfunctions. Then, by the result
of the previous subsection, eigenfunctions corresponding to the lowest eigenvalue may
be chosen to be positive almost everywhere. However, two functions that are positive
almost everywhere cannot be orthogonal to each other, which leads to a contradiction.
Hence the eigenspace corresponding to the lowest eigenvalue of T is one-dimensional,
and the lowest eigenvalue is simple.
   Next, decompose the quadratic form q̄a (v) according to (5.1). The term (log(1/a) −
(2A + 1))∥v∥2L2 merely shifts the spectrum, so the asymptotic behavior of λa is governed
by the operator associated with L̄(v) + O(a)∥v∥2L2 . By [7, Chapter VIII, Section 3,
Theorems 3.6 and 3.15], as a → 0+, this operator has the same multiplicity for its
lowest eigenvalue as T . Therefore, the lowest eigenvalue of the perturbed operator, and
hence λa , is simple for sufficiently small a > 0.
   As discussed in Section 4.5, the eigenspaces of the self-adjoint operator associated
with q̄a decompose into even and odd parts. In particular, since the lowest eigenvalue is
simple, its eigenfunction must be either even or odd. However, since the eigenfunction
corresponding to the lowest eigenvalue has a fixed sign, it must be even.

                                6. Proof of Theorem 1.5
6.1. Symmetry of the minimal operator. Choose λ < λa , and let H(Ta ) denote the
Hilbert space obtained as the completion of Cc∞ (−a, a) with respect to the norm ∥ · ∥Ta
defined by (1.9). Define
                                 ⟨v1 , v2 ⟩Ta := ⟨Ta v1 , v2 ⟩L2
so that ∥v∥Ta = ⟨v, v⟩Ta . Let Da be the operator on H(Ta ) with domain (1.10) acting as
           2

Da = i d/dx. Since the canonical map Cc∞ (−a, a) → H(Ta ) is injective and the image is
dense, the operator Da is densely defined.
Lemma 6.1. The operator Da is symmetric on H(Ta ).
Proof. We show that ⟨Da u, v⟩Ta = ⟨u, Da v⟩Ta for all u, v ∈ D(Da ). By the definition of
the inner product on H(Ta ) and the self-adjointness of Aa , we have
          ⟨Da u, v⟩Ta = ⟨Aa (iu′ ), v⟩L2 − λ⟨iu′ , v⟩L2 = ⟨iu′ , Aa v⟩L2 − λ⟨iu′ , v⟩L2 .
According to the distribution kernel representation (2.10), the function Aa v is the con-
volution of v with the tempered distribution k := −g ′′ . Hence, by the general theory
of tempered distributions, Aa v ∈ C ∞ (R) whenever v ∈ D(Da ). Using this fact to-
gether with the boundary conditions u(a) = u(−a) = 0, integration by parts gives
                     WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                                      21


⟨iu′ , Aa v⟩L2 = ⟨u, i(Aa v)′ ⟩L2 , and ⟨iu′ , v⟩L2 = ⟨u, iv ′ ⟩L2 . Furthermore, using (2.10) and
the boundary conditions v(a) = v(−a) = 0, we obtain by integration by parts that
                  Z a                            Z a                         Z a
        ′       d                                     d
(Aa v) (x) =           k(x − y)v(y) dy = −              k(x − y)v(y) dy =        k(x − y)v ′ (y) dy.
               dx −a                              −a dy                       −a
Therefore, ⟨u, i(Aa v)′ ⟩L2 = ⟨u, Aa (iv ′ )⟩L2 , and hence
   ⟨Da u, v⟩Ta = ⟨u, Aa (iv ′ )⟩L2 − λ⟨u, iv ′ ⟩L2 = ⟨Aa u, iv ′ ⟩L2 − λ⟨u, iv ′ ⟩L2 = ⟨u, Da v⟩Ta .
This proves the desired identity.                                                                      □
6.2. Adjoint of the minimal operator. We now derive the adjoint operator Da∗ of
Da on H(Ta ). For a vector v ∈ H(Ta ) to belong to the domain D(Da∗ ) of Da∗ , the linear
functional u 7→ ⟨Da u, v⟩Ta must be continuous on D(Da ) with respect to the norm of
H(Ta ). Since Da is symmetric, D(Da ) ⊂ D(Da∗ ). Moreover, by the calculation in the
previous section, we have ⟨Da u, v⟩Ta = ⟨u, i(Aa v)′ ⟩L2 − λ⟨u, iv ′ ⟩L2 for u ∈ D(Da ). If
this functional is continuous with respect to ∥ · ∥Ta , then it is in particular continuous
with respect to the L2 -norm. Hence it is necessary that v ′ ∈ L2 (−a, a) and (Aa v)′ ∈
L2 (−a, a). Therefore,
               D(Da∗ ) ⊂ { v ∈ H(Ta ) | v ∈ H 1 (−a, a), Aa v ∈ H 1 (−a, a) }.
The condition that v ∈ D(Da∗ ) and Da∗ v = g ∈ H(Ta ) means that
                      ⟨Da u, v⟩Ta = ⟨u, g⟩Ta (= ⟨Ta u, g⟩L2 = ⟨u, Ta g⟩L2 )
holds for every u ∈ D(Da ). By the calculation in the previous section, we have
                                   ⟨Da u, v⟩Ta = ⟨u, i(Ta v)′ ⟩L2
for u ∈ D(Da ). Since D(Da ) is dense in H(Ta ), it follows that
                                        Ta (Da∗ v) = i(Ta v)′ .
If v ∈ D(Da ), then this identity becomes
                           Ta (Da∗ v) = i(Ta v)′ = Ta (iv ′ ) = Ta (Da v),
and hence Da∗ agrees with Da . The eigenvalue equations Da∗ v = ±i v are equivalent to
i(Ta v)′ = ±i(Ta v). Hence
                          (Ta v)(x) = C+ ex ,        (Ta v)(x) = C− e−x ,
and these functions span the eigenspaces corresponding to the eigenvalues ±i. Therefore
we obtain the following result.
Lemma 6.2. Da has the deficiency indices (1, 1).
6.3. Self-adjoint extensions of the minimal operator. We compute the self-adjoint
extensions of Da using von Neumann’s theory of deficiency indices ([11, Section X.1], [12,
Section 3.2]). By the results of the previous subsection, the deficiency spaces Ker (Da∗ ∓i)
are one-dimensional and are spanned by vectors v± satisfying (Ta v± )(x) = exp(±x). It is
immediate that ∥v+ ∥Ta = ∥v− ∥Ta . Thus, every self-adjoint extension D a,θ parametrized
by θ ∈ [0, 2π) has domain
                D(D a,θ ) = {v0 + c(v+ + eiθ v− ) | v0 ∈ D(D a ), c ∈ C}
                       ⊂ D(Da∗ ) = {v0 + av+ + bv− | v0 ∈ D(D a ), a, b ∈ C}.
These domains are characterized by the boundary form
                              W (u, v) = ⟨Da∗ u, v⟩Ta − ⟨u, Da∗ v⟩Ta ,
namely, v ∈ D(Da∗ ) belongs to D(D a,θ ) if and only if W (v, wθ ) = 0 for wθ = v+ + eiθ v− .
The action of D a,θ is given by
                                           
                  D a,θ v0 + c(v+ + eiθ v− ) := D a v0 + ic(v+ − eiθ v− ).
22                                         M. SUZUKI


6.4. Proof of Theorem 1.5. Let vz be an eigenfunction of Da∗ corresponding to an
eigenvalue z ∈ R. The equation Da∗ vz = zvz implies that (Ta vz )(x) = exp(−izx). The
basis vectors of the deficiency spaces satisfy Da∗ v± = ±iv± and (Ta v± )(x) = exp(±x).
On the other hand, applying Da∗ to wθ = v+ + eiθ v− yields
                                    Da∗ wθ = iv+ − ieiθ v− .
Substituting these equalities into the boundary form W , we obtain
                 W (vz , wθ ) = ⟨zvz , v+ + eiθ v− ⟩Ta − ⟨vz , iv+ − ieiθ v− ⟩Ta .
Expanding the right-hand side by linearity of the inner product and collecting the terms
involving the inner products with v+ and v− , we obtain
                W (vz , wθ ) = (z + i)⟨vz , v+ ⟩Ta + (z − i)e−iθ ⟨vz , v− ⟩Ta = 0.     (6.1)
Using the definition of the inner product on the space H(Ta ) and the identity (Ta vz )(x) =
exp(−izx), we obtain
                                                       Z a
                      ⟨vz , v± ⟩Ta = ⟨Ta vz , v± ⟩L2 =     e−izx v± (x) dx.
                                                        −a

Because v± ∈ H(Ta ) ⊂ L2 (−a, a) ⊂ L1 (−a, a), the above Fourier integrals define entire
functions of z. Consequently, upon rewriting the orthogonality condition (6.1), we find
that the eigenvalue z must satisfy the equation W (a, θ; z) = 0, where W (a, θ; z) is defined
in (1.11). This proves the first assertion of the theorem.
  Next, we show that all zeros of W (a, θ; z) are real. Even when λ ∈ C is not real, the
function (Ta vλ )(x) = exp(−iλx) is infinitely differentiable on the interval (−a, a) and
belongs to H 1 (−a, a). Since Ta is injective, the relation (Ta vλ )(x) = exp(−iλx) uniquely
determines an element vλ ∈ H(Ta ), which belongs to D(Da∗ ). Applying Da∗ to vλ , and
then applying Ta to the result, we obtain
          Ta (Da∗ vλ ) = i(Ta vλ )′ = i(−iλ exp(−iλx)) = λ exp(−iλx) = Ta (λvλ ).
It follows that Da∗ vλ = λvλ . Thus, vλ serves as an eigenfunction of Da∗ even for λ ∈ C.
   Suppose that a complex number λ0 ∈ C satisfies W (a, θ; λ0 ) = 0. By the character-
ization established above, this means that the corresponding vλ0 is orthogonal to the
deficiency vector wθ with respect to the boundary form, that is, W (vλ0 , wθ ) = 0. By
definition, every element satisfying this condition belongs to D(D a,θ ). Consequently,
vλ0 is a nonzero element of the domain of the self-adjoint extension D a,θ and satis-
fies D a,θ vλ0 = λ0 vλ0 . Hence vλ0 is a genuine eigenvector of the self-adjoint operator
D a,θ with eigenvalue λ0 . However, λ0 must be real, since every eigenvalue of a self-
adjoint operator is real. This contradiction shows that any complex number λ satisfying
W (a, θ; λ) = 0 must lie on the real axis.                                              □

         7. Heuristic justification of the formula in Corollary 1.6
   In this section, we assume RH. By Weil’s positivity criterion, we have QW ≥ 0 and
hence Aa > 0 for all a > 0. Under this assumption, for each a > 0 we may take λ = 0
in the definition (1.9) of Ta = Ta,λ . Hence we work with Aa in place of Ta , and consider
the Hilbert spaces H(Aa ) := H(Ta,0 ).

7.1. The Hilbert Space H(A∞ ). In order to study the behavior of the differential
operator Da on the Hilbert space H(Aa ) as a → ∞, we first introduce the Hilbert space
H(A∞ ) and the corresponding differential operator D∞ . Here A∞ is only a formal
symbol and does not denote an actual operator.
   Let D(QW ) denote the completion of Cc∞ (R) with respect to the form norm
                                 ∥v∥2QW := QW (v) + ∥v∥2L2 .
                    WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                               23


This space is a Hilbert space with respect to the form norm. N := {v ∈ D(QW ) |
QW (v) = 0} is a closed subspace of D(QW ) with respect to the form norm. We then
consider the quotient space
                                  H(A∞ ) := D(QW )/N .
For v ∈ D(QW ), we denote its equivalence class by [v] ∈ H(A∞ ). We define the norm
of [v] by
                                     ∥[v]∥2A∞ := QW (v).
The Hilbert space H(A∞ ) can be identified with the completion HW of Cc∞ (R) with
respect to QW (v) studied in [14].
   Since QW (v) ̸= 0 for any nonzero v ∈ Cc∞ (R), the map v 7→ [v] embeds Cc∞ (R)
injectively into H(A∞ ), and its image is dense. Hence, by fixing representatives on
Cc∞ (R), we define a densely defined operator D∞ on H(A∞ ) by D∞ [v] := [D∞ v] = [iv ′ ].
This is well-defined since [u] = [v] for u, v ∈ Cc∞ (R) implies u = v.
   We now verify that D∞ is symmetric on H(A∞ ) and hence closable. If QW (w) = 0, it
follows that QW (w, v) = 0 for all v by the assumption QW ≥ 0 and the Cauchy-Schwarz
inequality |QW (u, v)|2 ≤ QW (u)QW (v). This implies QW (u + w, v) = QW (u, v) and
QW (u, v + w) = QW (u, v). Hence the sesquilinear form
                                          Z ∞Z ∞
             QW ([u], [v]) = QW (u, v) =            (−g ′′ (x − y)) u(y) v(x) dxdy (7.1)
                                                 −∞    −∞
is well-defined (cf. (2.9)). Moreover, the translation operator τt (t ∈ R) defined by
                             τt [u] := [τt u],    (τt u)(x) = u(x − t)
is well-defined, since τt [u + w] = [τt (u + w)] = [τt u] + [τt w] and QW ([τt w]) = QW (τt w) =
QW (w) = 0 if QW (w) = 0. Furthermore, by (7.1), the norm in H(A∞ ) is invariant
under translations:
                                     ∥[v]∥A∞ = ∥τt [v]∥A∞ .                                 (7.2)
Since D∞ commutes with convolution, it follows from (7.1) that
                             QW (D∞ [u], [v]) = QW ([u], D∞ [v])
for u, v ∈ Cc∞ (R). Hence D∞ is symmetric with respect to ⟨·, ·⟩A∞ . Since every symmet-
ric operator is closable, its closure will be denoted by D̄∞ . Thus, for [v] ∈ D(D̄∞ ), one
can choose a sequence vn ∈ Cc∞ (R) such that [vn ] → [v] and D∞ [vn ] = [ivn′ ] → D̄∞ [v] in
H(A∞ ).

7.2. Isomorphism with a de Branges space. We now recall the isometric isomor-
phism between H(A∞ ) (denoted by HW in [14]) and a certain de Branges space B
established in [14]. Here it is sufficient to understand B as a reproducing kernel Hilbert
space consisting of entire functions.
   Let M be the operator of multiplication by an independent variable on B defined by
D(M ) = {F ∈ B | zF (z) ∈ B} and (M F )(z) = zF (z). The operator M is a densely
defined closed symmetric operator with deficiency indices (1, 1), and it admits a one-
parameter family of self-adjoint extensions parametrized by ψ ∈ [0, π). In particular, the
self-adjoint extension Mπ/2 has purely discrete spectrum Γ, the set of zeros of ξ(1/2−iz)
in Section 3, and one can choose an orthonormal basis of B consisting of the correspond-
ing eigenfunctions Fγ for γ ∈ Γ [14, Section 6]. In this setting, defining U : H(A∞ ) → B
by                                          X√
                                 U ([v]) :=      mγ vb(γ)Fγ                           (7.3)
                                                 γ∈Γ
yields an isometric isomorphism of Hilbert spaces [14, Theorems 1.1 and 1.4]:
                              (QW (v) =) ∥[v]∥2A∞ = ∥U ([v])∥2B .                          (7.4)
This U is precisely the operator denoted by π −1/2 Pc
                                                    D =π
                                                         −1/2 PD
                                                              b in [14].
24                                           M. SUZUKI


7.3. Correspondence between D∞ and M . Via the isomorphism U : H(A∞ ) → B,
the operator M induces a self-adjoint operator on H(A∞ ). However, its relation to
the differential operator D̄∞ is not immediately clear. On the other hand, the relation
between the differential operator D̄∞ and the self-adjoint extension Mπ/2 of M can be
described in a simple way as follows.
   If v ∈ Cc∞ (R), then vb is rapidly decreasing. Hence [v] belongs to the domain of Mπ/2 .
For each γ ∈ Γ, Fγ is an eigenfunction of Mπ/2 with eigenvalue γ, and therefore
                          X               X
          Mπ/2 U ([v]) =       γb
                                v (γ)Fγ =      d′ )(γ)Fγ = U ([iv ′ ]) = U (D∞ [v]).
                                              (iv
                           γ                  γ

Let Dπ/2 := U −1 Mπ/2 U . Then the above computation shows that Dπ/2 [v] = [iv ′ ] =
D∞ [v] for v ∈ Cc∞ (R). Since Dπ/2 is closed and extends D∞ , it follows that
                                         Dπ/2 [v] = D̄∞ [v]
for all [v] ∈ D(D̄∞ ). Thus, Dπ/2 is a self-adjoint extension of D̄∞ .
Remark. More generally, for any self-adjoint extension Mψ of M , one may define Dψ :=
U −1 Mψ U . Since Mψ is unitarily equivalent to Dψ , the operator Dψ is self-adjoint. It is
natural to ask whether Dψ can be characterized directly as a self-adjoint extension of
D̄∞ . We do not pursue this question here.

7.4. An operator unitarily equivalent to M . We now consider the operator on
H(A∞ ) induced by M through U . More precisely, we define
                                          D := U −1 M U.
By construction, D and M are unitarily equivalent via U , and have the same deficiency
indices. In particular, the deficiency indices of D are (1, 1).
   Writing the Fourier transform as Fv = vb, there exists a subspace V (0) ⊂ L2 (0, ∞)
such that the maps B ∋ F 7→ vF := F −1 F ∈ V (0) and V (0) ∋ v 7→ [v] ∈ H(A∞ )
are isomorphisms [14, Theorem 5.5]. In this setting, the operator on V (0) unitarily
equivalent to M via F −1 : B → V (0) is denoted by D = F −1 M F . Then, via the
isomorphism ι : V (0) → H(A∞ ), the operator corresponding to D is given by D = ιDι−1 .
In particular, we have U = F ι−1 . If F ∈ D(M ) and F = F v, then
                               Dv = F −1 (M F ) = F −1 (zF ) = iv ′ .
Moreover, for v ∈ V (0),
                                  D[v] = ι(Dv) = ι(iv ′ ) = [iv ′ ].
We claim that for a given function u, the decomposition u = v + w with v ∈ V (0) and
[w] = 0 is uniquely determined. Once this is established, we may define D[u] := D[v] by
writing u = v + w with v ∈ V (0) for [u] ∈ D(D) (defined via the unitary correspondence
with D(M )).
   The uniqueness of v in the above decomposition is as follows. If u = v + w and
u = ṽ + w̃ are two such decompositions, then v − ṽ = w̃ − w. Hence, if v0 := v − ṽ ̸= 0,
then 0 ̸= v0 ∈ V (0) and [v0 ] = 0. It follows from (7.3) and (7.4) that vb0 (γ) = 0 for all
γ. Since F : V (0) → B is an isomorphism, we obtain v0 = 0, a contradiction.
     With this definition, the relation between
                                      Cc∞ (R)     and    D(D)
is not clear. Indeed, for u ∈ Cc∞ (R) decomposed as u = v + w with v ∈ V (0) and
[w] = 0, it is not clear whether F v ∈ D(M ) holds.
                   WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                           25


7.5. Infinitesimal generator of translations (a = ∞). The previously defined fam-
ily (τt )t∈R forms a strongly continuous one-parameter unitary group. Indeed, τt τs = τt+s
is clear from the definition, and each τt is a unitary operator by (7.2). It remains to ver-
ify strong continuity, namely limt→0 ∥τt [v] − [v]∥A∞ = 0. To this end, using the isometric
property of U , we estimate
                                               X
                        ∥U (τt [v] − [v])∥2B =   mγ |e−itγ − 1|2 |b
                                                                  v (γ)|2 .
                                           γ∈Γ

Since A∞ > 0 implies RH, all elements of Γ are real. Since |e−itγ −1|2 ≤ 4, the dominated
convergence theorem implies that the right-hand side tends to 0 as t → 0. Hence τt is
strongly continuous with respect to the norm of H(A∞ ).
   Therefore, there exists a self-adjoint operator D on H(A∞ ) such that
                                     τt = eitD   (t ∈ R)
by Stone’s theorem, where
                                                
                              τϵ [v] − [v]                              τϵ [v] − [v]
       D(D) = [v] ∈ H∞ lim                 exists ,        D[v] = −i lim             .
                          ϵ→0       ϵ                               ϵ→0       ϵ
For [v] ∈ D(D), we have
                                  D[v] = [iv ′ ] = D[v]
by considering the action on the B side. Thus, the self-adjoint operator D extends the
closed symmetric operator D, whose deficiency indices are (1, 1). Moreover,
                                  U (D[v]) = Mπ/2 U ([v])
holds, that is, D = Dπ/2 . Indeed,
                                          (U τϵ [v])(z) − (U [v])(z)
                    (U D[v])(z) = −i lim
                                      ϵ→0              ϵ
                                          X√         eiγϵ − 1
                                 = −i lim        mγ           vb(γ)Fγ
                                      ϵ→0                ϵ
                                          γ∈Γ
                                   X√
                                 =       mγ γbv (γ)Fγ = (Mψ U [v])(z).
                                     γ∈Γ

Viewed on H(A∞ ), D is characterized as the unique self-adjoint extension of D that
commutes with the (usual) translation. Moreover, the equality D = Dπ/2 asserts that a
Hilbert–Pólya operator is realized as the infinitesimal generator of the translation group
under the assumption A∞ > 0 (RH).
  Since D acts as v 7→ iv ′ on Cc∞ (−a, a), to approximate the behavior as a → ∞, it is
natural to consider the minimal operator Da for each a > 0. Considering the self-adjoint
extensions Da,θ of Da , it is expected that by choosing θ = θ(a) appropriately,
                                  D a,θ → D      (a → ∞)
in the sense of strong resolvent convergence.

7.6. Operator picture via embedding. We denote by H(Aa ) the completion of
Cc∞ (−a, a) with respect to (QW |L2 (−a,a) )(v) = QW
                                                   a (v) = ⟨A v, v⟩ . Since U (v) ̸= 0
                                                              a    L2
                          ∞                     ∞
for every nonzero v ∈ Cc (−a, a), the space Cc (−a, a) embeds injectively into H(A∞ )
via v 7→ [v] as in the case of Cc∞ (R). This embedding extends to an injective map
                                     H(Aa ) ,→ H(A∞ ).
Through this embedding, we may regard Da and D a,θ as operators on H(A∞ ), although
they are no longer densely defined in H(A∞ ). However, we still have
                                      Da = D∞ |H(Aa )
26                                       M. SUZUKI


                                    D̄∞ ⊂ Dπ/2 = D
and thus, as a → ∞, it is natural to expect that D a,θ approximates the self-adjoint
extension Dπ/2 = D of D̄∞ . Since the spectrum of Dπ/2 is given by Γ, one further
expects that the eigenvalues of D a,θ approximate Γ as a → ∞.
7.7. Factorization of A∞ . By (7.4),
                   QW (v) = ⟨A∞ v, v⟩L2 = ⟨U v, U v⟩L2 = ⟨U ∗ U v, v⟩L2 .
Using the notation of [14] and the factorization U = π −1/2 PD,
                                                            b we obtain
                                            b∗ b
                       A∞ = U ∗ U = π −1 D∗ P PD,               b∗ P
                                                       G = π −1 P  b
under RH. Conversely, if one could show unconditionally that the kernel of U ∗ U coincides
with −g ′′ (x − y), or equivalently that the kernel of π −1 P  b∗ P
                                                                  b coincides with the kernel
g(x−y)−g(x)−g(−y)+g(0) of G, then it would follow that A∞ > 0, and hence RH would
follow immediately. More precisely, using the notation of [14], one has unconditionally
                                             Z ∞
                                        −1/2
                           (U v)(z) = π          Sx (z̄) v ′ (x) dx,
                                              −∞
and therefore RH would follow if one could establish unconditionally that
               Z ∞
                   Sx (z)Sy (z) dz = g(x − y) − g(x) − g(−y) + g(0).
                  −∞
We note that, as in [14, (1.5), (1.6)], the function Sx (z) can also be represented without
reference to Γ, just as g can.

             8. From distribution kernels to continuous kernels
8.1. Passing from Aa to Ga . So far, we have developed the theory of QW       a and A using
                                                                                      a
the screw function associated with ζ(s). However, in the theory of screw functions, as
presented for example in [8, Section 5], it is more natural to consider the integral operator
Ge defined by
                             Z ∞
                (e
                 Gu)(x) :=       (g(x − y) − g(x) − g(−y) + g(0))u(y) dy                (8.1)
                              −∞
rather than the integral operator G in (1.4). As shown in [13, Theorem 1.3], if QGe (v) :=
⟨Gv,
  e v⟩L2 denotes the quadratic form associated with G,   e then the condition Q e (u) ≥ 0
                                                                                 G
               ∞         2
for all u ∈ Cc (R) ∩ L0 (−a, a) is equivalent to RH. This follows from the relation
QW (v) = QGe (Dv), v ∈ Cc∞ (R) [13, Proposition 3.1]. Note that if u = Dv with v ∈
Cc∞ (R), then u ∈ L20 (−a, a). If QG (v) := ⟨Gv, v⟩L2 denotes the quadratic form associ-
ated with the integral operator G in (1.4), then QGe (u) = QG (u) for every u ∈ L20 (−a, a),
and hence QW (v) = QGe (Dv) = QG (Dv). Motivated by this relation, we chose to study
QW through the quadratic form QG , whose kernel is considerably simpler.
   To represent QW directly on the finite interval (−a, a), one must use the operator
Aa as in (1.1). However, if one is interested only in the positivity of QW , there is no
essential difference in working with Ga , since QW (v) = QG (Dv) and the restriction of
QG to (−a, a) is represented by the operator Ga through (QG |L2 (−a,a) )(u) = ⟨Ga u, u⟩L2 .
Moreover, Ga is an integral operator with a continuous kernel and is therefore much
easier to handle than the unbounded operator Aa . As we shall see below, the Hilbert
space obtained by completing with respect to ⟨Aa u, u⟩L2 and the Hilbert space obtained
by completing with respect to ⟨Ga u, u⟩L2 are related through the identity QW (v) =
QG (Dv), and Theorem 1.5 can be interpreted within the framework of Ga .
   Although Aa and Ga are very different as operators, the former being an unbounded
operator with discrete spectrum accumulating at +∞ and the latter being a compact
operator with discrete spectrum accumulating at 0, the situation is analogous from the
                    WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                                     27


viewpoint of positivity of quadratic forms, since in both cases the relevant quantity is the
bottom of the spectrum. (There is, of course, the distinction that the infimum may or
may not be attained.) Furthermore, as we shall see below, this difference does not affect
the spectrum of the corresponding self-adjoint extensions of the differential operator.
  To discuss these matters, we first review the relation between the norms arising
through the differential operator D.

8.2. The Inverse Neumann Laplacian. The operator D : H01 (−a, a) → L20 (−a, a) is
bijective. Therefore, if u = Dv with v ∈ H01 (−a, a), then v = D−1 u, and hence
                        ∥v∥2L2 = ⟨D−1 u, D−1 u⟩L2 = ⟨(DD∗ )−1 u, u⟩L2 .
Note that −∆N = DD∗ coincides with the Neumann Laplacian, which is the operator
acting as −∆N = −d2 /dx2 with domain D(−∆N ) = {u ∈ H 2 (−a, a) | u′ (a) = u′ (−a) =
0}, and its range is L20 (−a, a). Its inverse (−∆N )−1 : L20 (−a, a) → L2 (−a, a) is a
compact, self-adjoint, and positive integral operator with kernel
                                           x2 + y 2 |x − y| a
                              N (x, y) =           −       + .
                                             4a        2    6
                                                               Rx
On the other hand, if u = Dv with v ∈ H01 (−a, a), then v(x) = −a u(t)dt, and therefore
          Z a              Z a Z a             Z a              
     2              2
 ∥v∥L2 =      |v(x)| dx =           u(t)1t≤x dt         u(s)1s≤x ds dt
           −a               −a   −a                  −a
                               !
          Z Z  a   a Z    a                       Z Z      a     a
        =                           dx u(t)u(s)dtds =                 (a − max(t, s))u(t)u(s)dtds.
            −a     −a    max(t,s)                         −a     −a

Accordingly, defining
                                           Z a
                           (K
                            e a u)(x) :=         (a − max(x, y))u(y)dy,
                                            −a
we obtain
                                       ∥v∥2L2 = ⟨K
                                                 e a u, u⟩L2 .

The operator K
             e a is compact, self-adjoint, and positive. Moreover, the identity
                                     e a u, u⟩L2 = ⟨(−∆N )−1 u, u⟩L2
                           ∥v∥2L2 = ⟨K                                                         (8.2)
holds as an equality of quadratic forms on L20 (−a, a). Since the operator-theoretic prop-
erties of (−∆N )−1 are more transparent, we shall henceforth write
                                                                      x2 + y 2 |x − y| a
       Ka := (−∆N )−1 ,       with integral kernel N (x, y) =                 −       + ,
                                                                        4a        2    6
and work with Ka rather than K
                             ea.

8.3. The Hilbert Space H(Sa ). Let λ < λa . Motivated by (8.2) and by the operator
Ta = Aa − λI considered earlier, we define
                                      Sa := Ga − λ(−∆N )−1 .
Since both Ga and (−∆N )−1 are self-adjoint with respect to the L2 inner product, so is
Sa , and
            ∥u∥2Sa = ⟨Sa u, u⟩L2 = ⟨Ta v, v⟩L2 = ∥v∥2Ta ,        u = Dv ∈ L20 (−a, a).
Furthermore,
               h                                                                  i
      H(Sa ) := the completion of Cc∞ (−a, a) ∩ L20 (−a, a) with respect to ∥u∥Sa
28                                           M. SUZUKI


is isometrically isomorphic to H(Ta ), since D : Cc∞ (−a, a) → Cc∞ (−a, a) ∩ L20 (−a, a) is
bijective. The operator D therefore extends to an isometric isomorphism
                                                   ∼
                                  D̄ : H(Ta ) → H(Sa ).
The operator Ga extends naturally to a bounded operator on H(Sa ). Using the same
notation for this extension, we have
                                        Aa = (D̄)∗ Ga D̄.
Note that H(Ta ) ,→ L2 (−a, a), whereas H(Sa ) ̸⊂ L2 (−a, a). It should also be noted
that 1(−a,a) ∈ H(Aa ) \ D(Da ), while D̄(1(−a,a) ) ̸= 0.
  Through the isomorphism D̄, operators on H(Ta ) can be transferred to operators on
H(Sa ). Accordingly, we define an operator D ea on H(Sa ) by
                                        ea := DDa D−1
                                        D
with domain Cc∞ (−a, a) ∩ L20 (−a, a). For u ∈ Cc∞ (−a, a) ∩ L20 (−a, a),
                                 Z x
                  −1
               (D u)(x) = −i          u(t) dt, DDa (D−1 u)(x) = iu′ (x),
                                      −a

and hence Dea may be regarded as the operator with domain C ∞ (−a, a) ∩ L2 (−a, a)
                                                                    c             0
acting by Da = id/dx. By construction, D
          e                               ea on H(Sa ) is unitarily equivalent to Da on
H(Ta ). Therefore, D
                   ea has deficiency indices (1, 1), and its self-adjoint extensions have
the same spectra as those of Da .
     The eigenvalue equation for the adjoint D
                                             e ∗ is determined by
                                              a
                                 e ∗ − z)v⟩S = ⟨u, Sa (D
                        0 = ⟨u, (D                     e ∗ − z)v⟩L2
                                  a         a           a

for every u ∈ Cc∞ (−a, a) ∩ L20 (−a, a). Here u ranges not over all of Cc∞ (−a, a), but only
over those functions satisfying u b(0) = 0. Since the orthogonal complement in L2 (−a, a)
of the subspace of functions satisfying u b(0) = 0 is spanned by the constant function 1,
it follows that
                          1 = Sa (D e ∗ − z)v = i(Sa v)′ − z(Sa v).
                                      a

using Sa (D v) = i(Sa v) (which is derived exactly as in the case of Ta ). Hence
           e ∗
             a
                        ′

                                                1
                            (Sa v)(x) = Ce−izx − ,          (z ̸= 0),
                                                z
and
                              (Sa v)(x) = −ix + C,        (z = 0).
On the other hand,
                                      Z a
                        (Sa v)(x) =         (g(x − y) − λr(x, y))v(y) dy.
                                       −a

Here r(x, y) may be taken to be either of the kernels considered in Section 8.2, since in
either case                      Z a
                              d2
                           − 2        r(x, y)u(y)dy = u(x).
                             dx −a
Therefore,
         Z a                                Z a
      d2
   − 2       (g(x − y) − λr(x, y))v(y) dy =     (−g ′′ (x − y))v(y) dy − λv(x) = Ta v.
     dx −a                                   −a

It follows from the injectivity of Sa (equivalently, of Ta ) that the eigenfunctions cor-
responding to z = ±i coincide with those for Da∗ up to multiplication by the constant
                   WEIL’S QUADRATIC FORM VIA THE SCREW FUNCTION                             29


−λ2 . Consequently, the boundary form determining the eigenvalues should also be the
same for D
         e ∗ and D ∗ . Therefore, it may be advantageous to work with the equation
          a         a
                             Z a
               (Sa v± (x) =)     (g(x − y) − λr(x, y))v± (y) dy = e±x ± i,
                                −a
whose kernel is an ordinary function rather than a distribution.
   Assume RH. Then Ga > 0, so one may take λ = 0 and hence Sa = Ga . Extending
va,± by zero outside [−a, a], we obtain Ga va,± = g ∗va,± , and the right-hand side extends
naturally to a function on R. Thus the equation Sa va,± = e±x ± i may be written as
g ∗ va,± = ha,± , where ha,± (x) = e±x ± i for x ∈ [−a, a]. Taking Fourier transforms
yields
                                                  hda,± (z)
                                       vd
                                        a,± (z) =           .
                                                    gb(z)
Since gb(z) = z −2 ξ ′ (1/2 − iz)/ξ(1/2 − iz) by [13, (1.2), (1.8)], we obtain
                                ξ(1/2 − iz)                      iθ
                                                                                  
             W (a, θ; z) = z 2 ′              (z − i)vda,+ (z) + e (z + i)vd
                                                                           a,− (z) .
                                ξ (1/2 − iz)
This suggests the limiting relation (1.12).

8.4. Relation to Bombieri’s Problem 1. Using G
                                             e in (8.1) and Pa in (2.1), define

                          G        GPa : L20 (−a, a) → L20 (−a, a).
                          e a := Pa e
Recall that Ga = Pa GPa in (1.5), and Ba = D∗ Ga D in (1.6). Then
      e a Dv, Dv⟩L2 = ⟨Ga Dv, Dv⟩L2 = ⟨Ba v, v⟩L2 = QW (v),
     ⟨G                                                            v ∈ Cc∞ (−a, a),     (8.3)
see [13, Proposition 3.1]. For finite a, the norm on H01 (−a, a) defined by ∥v∥L2 +∥v ′ ∥L2 is
equivalent to that defined by ∥v ′ ∥L2 . Adopting the latter, the linear map H01 (−a, a) →
L20 (−a, a) given by v 7→ Dv becomes an isometric isomorphism. Hence the Rayleigh
quotient QW (v)/∥v∥2H 1 on H01 (−a, a) appearing in [1, Problem 1] (and [2, Problem B])
can be rewritten as the Rayleigh quotient on L20 (−a, a)
                              QW (v)   QG (u)
                                 2   =        ,       u = Dv,
                              ∥v∥H 1   ∥u∥2L2
by [13, Proposition 3.1]. In other words, [1, Problem 1] (and [2, Problem B]) is nothing
but the eigenvalue problem for Ga = Pa GPa = Pa GP      e a on L2 (−a, a). [1, Theorem
                                                                  0
4] asserts that if this infimum is negative, then it is attained in L2 (−a, a). From the
viewpoint of the eigenvalue problem for Ga , this is immediate.
   Since QW is bounded with respect to the H 1 -norm, the Riesz representation theorem
implies that it is represented by a bounded operator. That operator is precisely G
                                                                                 e a = Ga
                      2
(they coincide on L0 (−a, a)). Problem 1 in [1] corresponds to this formulation.

8.5. Reformulation as a generalized eigenvalue problem. By (8.2),
                                     ∥v∥2L2 = ⟨Ka u, u⟩L2
with u = Dv and Ka = (−∆N )−1 . Combining this with (8.3), the Rayleigh quotient in
(1.7) can be rewritten as
                                a (v)
                             QW         ⟨Ga u, u⟩L2
                                  2   =                                       (8.4)
                              ∥v∥L2     ⟨Ka u, u⟩L2
for v ∈ H01 (−a, a) and u = Dv ∈ L20 (−a, a). The right-hand side of (8.4) leads to the
generalized eigenvalue problem
                               Ga u = λKa u,       u ∈ H(Sa ).
30                                             M. SUZUKI


The spectrum of this generalized eigenvalue problem coincides with that of Aa . Hence
it is discrete, bounded from below, and accumulates only at +∞. (The case λ = 0 is
exceptional: then the contribution of Ka disappears, and one is simply looking at the
kernel (the 0-eigenspace) of Ga .)
   Although Aa is unbounded, both Ga and Ka are compact operators with continuous
integral kernels. In the generalized eigenvalue problem, one must analyze not only the
spectra of Ga and Ka individually but also the interaction between them. Thus it
cannot be said a priori that the spectral analysis of Aa becomes simpler. Nevertheless,
the availability of compact-operator techniques may provide certain advantages.

Acknowledgment
This work was supported by JSPS KAKENHI Grant Number JP23K03050. This work
was also supported by the Research Institute for Mathematical Sciences, an International
Joint Usage/Research Center located in Kyoto University.

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Department of Mathematics,
School of Science,
Institute of Science Tokyo
2-12-1 Ookayama, Meguro-ku,
Tokyo 152-8551, Japan
Email: msuzuki@math.sci.isct.ac.jp
