Hamiltonians arising from L-functions in the Selberg class

We establish a new equivalent condition for the Grand Riemann Hypothesis for L-functions in a wide subclass of the Selberg class in terms of canonical systems of differential equations. A canonical system is determined by a real symmetric matrix valued function called a Hamiltonian. To establish the equivalent condition, we solve and use an inverse spectral problem for canonical systems of special type.


Introduction
The Riemann Hypothesis (RH) asserts that all nontrivial zeros of the Riemann zetafunction ζ(s) lie on the critical line ℜ(s) = 1/2, and it had been generalized to wider classes of zeta-like functions. Especially, the analogue of RH for L-functions in the Selberg class is often called the Grand Riemann Hypothesis (GRH) 1 .
Briefly, we have two issues in this paper. The first is the resolution of an inverse spectral problem for canonical systems of differential equations. The second is the establishment of a new equivalent condition for GRH for L-functions in a wide subclass of the Selberg class in terms of canonical systems. These two issues are closely related with each other. We explain the relation by dealing with the case of the Riemann zeta function as the introduction. The present study was mainly stimulated by the works of J. C. Lagarias [25,26,27] and J.-F. Burnol [10,11,12,13].
The Riemann xi-function ξ(s) = 1 2 s(s − 1)π −s/2 Γ s 2 ζ(s) is an entire function taking real-values on the critical line such that the zeros coincide with nontrivial zeros of ζ(s). Therefore RH is equivalent that all zeros of the entire function ξ(s) lie on the critical line. Noting this, we start from the consideration on the additive decomposition by an entire function E, where i = √ −1, F ♯ (z) = F (z) for an entire function F and the bar stands for the complex conjugate. It is easily confirmed that (1.1) holds for by the functional equations ξ(s) = ξ(1 − s) and ξ(s) = ξ ♯ (s) and thus it also holds for infinitely many entire functions E = E ξ + F , where the prime stands for the derivative with respect to s and F is an entire function satisfying F ♯ = −F . The advantage of the decomposition (1.1) stands on the theory of the Hermite-Biehler class of entire functions. We denote by HB the set of all entire functions satisfying |E ♯ (z)| < |E(z)| for all z ∈ C + , (1.3) where C + = {z = u + iv | u, v ∈ R, v > 0} is the upper half-plane. The Hermite-Biehler class HB consists of all E ∈ HB having no real zeros 2 . An entire function F satisfying F = F ♯ is called a real entire function. In general, E ∈ HB implies that two real entire functions A(z) := 1 2 (E(z) + E ♯ (z)) and B(z) := i 2 (E(z) − E ♯ (z)) (1.4) have only real zeros, and these zeros interlace. If E ∈ HB, all (real) zeros of A and B are simple ( [4]). Therefore, the existence of E ∈ HB satisfying (1.1) implies that RH holds together with the Simplicity Conjecture (SC) which assert that all nontrivial zeros of ζ(s) are simple. Conversely, there exists E ∈ HB satisfying (1.1) if we assume that RH and SC hold. In fact, E ξ of (1.2) belongs to HB under RH and SC ( [26]).
The above discussion suggest the following strategy to the proof of RH with SC: first, find an entire function E satisfying (1.1); second, prove that E belongs to HB. Then these two conditions conclude RH and SC as the above. More simply, we may start from the second step by using E ξ of (1.2). The obvious difficulty of the above strategy is the second step, in other words, we do not know any reason why an entire function E satisfying (1.1) (or E ξ ) should belong to HB if RH holds. In this paper, we search for the reason of E ∈ HB in the theory of de Branges on canonical systems as with Lagarias [25,26,27] in which the applicability of the theory of de Branges to the study of the zeros of L-functions is suggested.
To start with, we explain that canonical systems generate functions of HB. Let Sym 2 (R) be the set of all 2 × 2 real symmetric matrices. A Sym 2 (R)-valued function H defined on I = [t 1 , t 0 ) (−∞ < t 1 < t 0 ≤ ∞) is called a Hamiltonian if cos 2 θ cos θ sin θ cos θ sin θ sin 2 θ holds on J for some positive function h on J and 0 ≤ θ < π. A point t ∈ I is called regular if it does not belong to any H-indivisible interval, otherwise t is called singular. The first-order system Woracek [46] and references there in for theoretical and historical details on canonical systems (of dimension two). If t (A(t, z), B(t, z)) is a solution of a canonical system on I = [t 1 , t 0 ) such that lim t→t 0 J(t; z, z) = 0 for all z ∈ C + , the function E(t, z) = A(t, z) − iB(t, z) is an entire function of HB for every regular t ∈ I (see Proposition 3.4 below for a special H). To derive more strong conclusion E(t, z) ∈ HB, we need information at t = t 0 in general. For instance, E(t, z) ∈ HB for every regular t ∈ I if (A(t, z), B(t, z)) tends to (c, 0) as t → t 0 for some constant c = 0. 2 The definitions of HB is equivalent to the definition of Levin [28, §1 and §2 of Chap. VII] if we replace the word "the upper half-plane" by "the lower half-plane", because (1.3) implies that E(z) has no zeros in C+. On the other hand, note that HB of this paper is different form HB of [28]. We adopt the above definition for the convenience to use the theory of canonical systems. Note that a member of HB is called a strict structure function in [26] and that a member of HB is called a de Brange structure function and a structure function in [25] and [26,27], respectively.
A fundamental and quite important result is the resolution of the inverse spectral problem that recovers H from E ∈ HB. This inverse spectral problem had been solved by L. de Branges as follows ( [9,Theorem 40], see also [7, Theorem II]): Theorem dB. For every E ∈ HB, there exists a Hamiltonian H on some (possibly unbounded) interval I = [t 1 , t 0 ) having the following properties: (1) For the unique solution t (A(t, z), B(t, z)) of the initial value problem (1.5) with (A(t 1 , z), B(t 1 , z)) = (A(z), B(z)), E(t, z) := A(t, z) − iB(t, z) belongs to HB for every regular t ∈ I. for (A(t, z), B(t, z)) of (1). Then lim t→t 0 J(t; z, z) = 0 for all z ∈ C.
We abbreviate to Inv(E) the triple (I, H(t), (A(t, z), B(t, z))) in Theorem dB. The triple Inv(E) is unique up to the scaling of t under the normalization E(0) = 1 and tr H(t) = 1, but we do not require such normalizations. If (I, H(t), (A(t, z), B(t, z))) satisfies the conditions in Theorem dB, the triple (J, H(t), (A(t, z), B(t, z))) for a subinterval J (⊂ I) satisfies the conditions in Theorem dB without (2). We abbreviate to Inv(E) ♭ such a triple. The only difference between Inv(E) and Inv(E) ♭ is the domain of H.
By Theorem dB, if we assume that RH and SC hold, there exists a Hamiltonian H ξ defined on some interval I such that E ξ of (1.2) is recovered from the solution of the canonical system associated with H ξ by applying de Branges' result to E ξ . Using the conjectural Hamiltonian H ξ , the above naive strategy to the proof of RH is now refined as follows: (1-1) Constructing the Hamiltonian H ξ on I = [t 1 , t 0 ) without RH; (1-2) Constructing the solution (A ξ (t, z), B ξ (t, z)) of the canonical system on I associated with H ξ satisfying E ξ (z) = A ξ (t 1 , z) − iB ξ (t 1 , z); (1-3) Showing that lim t→t 0 J ξ (t; z, z) = 0 for all z ∈ C + , where J ξ (t; z, w) is the function defined by (1.6) for (A ξ (t, z), B ξ (t, z)).
It is concluded that E ξ belongs to HB if the above three steps are completed (cf. Proposition 3.4). Therefore, all zeros of ξ( 1 2 − iz) = A ξ (z) = 1 2 (E ξ (z) + E ♯ ξ (z)) are real, this is nothing but the Riemann hypothesis. In this approach, we face a serious obstacle from the first step. That is, to carry out it, we need an explicit formula of H ξ obtained from E ξ under RH. However, the explicit construction of H for given E ∈ HB is difficult in general as well as the other inverse spectral problem, and it is usually not possible to obtain enough analytic information of H in known constructions. In fact, an explicit form of H is not known except for a few examples of E as in [9,Chapter 3], [16,Section 8] (and also [13], [27]) and some additional examples constructed from such known examples using transformation rules for Hamiltonians ( [44]). More seriously, known constructions of H is not applicable to E if it is not known whether E ∈ HB.
In order to avoid the above obstacles, we consider the family of entire functions E ω ξ (z) := ξ( 1 2 + ω − iz), ω > 0 instead of the single function E ξ . (Note that E ξ = E 0 ξ , but E ω ξ for small ω > 0 is similar to E ξ in the sense that E ω ξ (z) = ξ( 1 2 + ω − iz) + ω ξ ′ ( 1 2 + ω − iz) + O(ω 2 ) for small ω > 0 if z in a compact set.) Then we find that a necessary and sufficient condition for RH (not require SC) is that E ω ξ ∈ HB for every ω > 0 (Propositions 2.1 and 2.2). In particular, there exists a Hamiltonian H ω ξ for every ω > 0 under RH. Therefore, RH is proved by completing the following three steps for every ω > 0: (2-1) Constructing the Hamiltonian H ω ξ on I = [t 1 , t 0 ) without RH; (2-2) Constructing the solution (A ω ξ (t, z), B ω ξ (t, z)) of the canonical system on I associated with H ω ξ satisfying E ω ξ (z) = A ω ξ (t 1 , z) − iB ω ξ (t 1 , z); (2-3) Showing that lim t→t 0 J ω ξ (t; z, z) = 0 for all z ∈ C + . There are two advantages of the second strategy. The first advantage is that the Hamiltonian H ω ξ on I = [0, ∞) and the solution (A ω ξ (t, z), B ω ξ (t, z)) of the associate canonical system are explicitly constructed in [40] under the restriction ω > 1 by applying the method of Burnol [13] introduced for the study of the Hankel transform. The second advantage is the avoiding of SC and the central zero, that is, multiple zeros on the critical line and the zero at the central point s = 1/2 are allowed to ξ in the second strategy. This point is important to generalize the above strategy to the other L-functions, because they often have a multiple zero at the central point s = 1/2.
In [40], H ω ξ and (A ω ξ (t, z), B ω ξ (t, z)) are constructed by using solutions ϕ ± t of the integral equations where K ω ξ is the kernel defined by for large c > 0. The behavior of (A ω ξ (t, z), B ω ξ (t, z)) at t = ∞ and its role in the proof of E ω ξ ∈ HB were not studied in [40]. However, if the construction of H ω ξ and (A ω ξ (t, z), B ω ξ (t, z)) is extended to 0 < ω ≤ 1 together with an additional result on the behavior of (A ω ξ (t, z), B ω ξ (t, z)) at t = ∞, we obtain E ω ξ ∈ HB for every ω > 0, which implies RH. Unfortunately, there were several technical difficulties in [40] to extend the construction of H ω ξ and (A ω ξ (t, z), B ω ξ (t, z)) to 0 < ω ≤ 1. For instance, the above K ω L is far from continuous functions and L 2 -functions if ω > 0 is small, and this fact is a serious obstacle for the construction in [40].
In this paper, we resolve the above technical difficulties by introducing the additional discrete parameter ν: . The parameter ν does not affect to study whether E ω,ν L ∈ HB by definition of HB, but it plays an important role in the construction of Inv(E ω,ν ξ ) ♭ . The conjectural triple Inv(E ω,ν ξ ) will be obtained by applying a general construction of Inv(E) ♭ in Section 3 for entire functions E satisfying several conditions. We will find in Section 4 that E ω,ν L satisfies such conditions owing to the parameter ν. In this way, a large part of (2-1) and (2-2) are achieved successfully for each ω > 0.
Summarizing the above discussion, we obtain an equivalent condition for RH in terms of canonical systems associated with Hamiltonians. This framework to establish the equivalent condition of RH is applicable to more general zeta-and L-functions. We apply it to L-functions in the Selberg class which was introduced in Selberg [36] together with a sophisticated consideration about the question what is an L-function. Then we obtain an equivalent condition for the analogue of RH for L-functions in the Selberg class as in Section 2. This is the goal of this paper.
Before concluding the introduction, we comment on the Hilbert-Pólya conjecture, a conjectural possible approach to RH. It claims that the imaginary parts of the nontrivial zeros of ζ(s) are eigenvalues of some unbounded self-adjoint operator D acting on a Hilbert space H. The Montgomery-Odlyzko conjecture on the vertical distribution of the nontrivial zeros of ζ(s) and the resemblance between the Weil explicit formula and the Selberg trace formula are strong evidences to the Hilbert-Pólya conjecture. No pair (H, D) of space and operator had been found, although the conjectural pair were suggested by several authors. Among them, the idea of A. Connes [14] for the conjectural Hilbert-Pólya pair (H, D) is very attractive in the sense that it stands on the localglobal principle in number theory (adeles and ideles), it enables us to understand the Weil explicit formula as a trace formula, and it is stated not only for ζ(s) but also Dedekind zeta-functions and Hecke L-functions. In addition, his idea is compatible with the Berry-Keating model [2] which is an attempt to give an explanation for RH by using physical model.
If E ω,ν ξ ∈ HB, we can construct the family of pairs {(H ω,ν(ω) , D ω,ν(ω) )} ω>0 of Hilbert spaces and self-adjoint operators. This family may be regarded as a possible realization of Connes' Hilbert-Pólya pair (H, D) by allowing the perturbation parameter ω. This topic will be treated more precisely in Section 8.
The paper is organized as follows. In Section 2, we state the main results Theorem 2.1, Theorem 2.2, Theorem 2.3 and Theorem 2.4 after a small preparation of notation. The first two theorems are used to associate the theory of canonical systems to GRH via the inverse spectral problem Inv(E) ♭ . The third theorem is related with the necessity of GRH in terms of Inv(E). The fourth theorem is the goal of this paper which is an equivalent condition for GRH in terms of Inv(E). In Section 3, we solve the inverse spectral problem Inv(E) ♭ in Theorem 3.1 by assuming five conditions to an entire function E. The way of construction of Inv(E) ♭ is similar to [13] and [40]. In Section 4, we prove that E ω,ν ξ and its generalization E ω,ν L to L-functions in the Selberg class satisfy the first four conditions assumed in Theorem 3.1. Successively, we prove Theorem 2.1 which assert that E ω,ν L satisfies the fifth condition assumed in Theorem 3.1. Then, we obtain Theorem 2.2 by applying Theorem 3.1 to E ω,ν L . In Section 5, we review the theory of de Branges spaces as preparation for the proof of Theorem 2.3. In Section 6, we prove Theorem 2.3 by studying subspaces of certain de Branges spaces. In Section 7, we prove Theorem 2.4 by combining several results in former sections. In Section 8, we comment on the Hilbert-Pólya conjecture and Connes' approach about it from the viewpoint of the thoery of canonical systems. In Section 9, we state and prove several complementary or digressive results. In Section 10, we give miscellaneous remarks on results and contents of this paper.
Basically, we attempt as much as possible to prove the main results in Section 2 by applying general results to L-functions in the Selberg class for the convenience of applications to other class of L-functions.

2.1.
Selberg class and GRH. Let s = σ + it be the complex variable. The Selberg class S consists of the Dirichlet series satisfying the following five axioms 3 : 3 As a matter of fact, (S1) is unnecessary to define S because it is derived from (S4). But we put it into the axiom of S according to other literature. A positive reason is that it is convenient to define the extended Selberg class which is the class of Dirichlet series satisfying (S1)∼(S3).
(S2) Analytic continuation -There exists an integer m ≥ 0 such that (s − 1) m L(s) extends to an entire function of finite order. (S3) Functional equation -L satisfies the functional equation Γ is the gamma function and r ≥ 0, where b L (n) = 0 unless n = p m with m ≥ 1, and b L (n) ≪ n θ for some θ < 1/2. The Riemann zeta-function and Dirichlet L-functions associated to primitive Dirichlet characters are typical members of the Selberg class S. As with these examples, it is conjectured that all major zeta-and L-functions appearing in number theory, such as automorphic L-functions and Artin L-functions, are members of S. Considering this conjecture, S is a proper class of L-functions in studying the analogue of RH for number theoretic L-functions. See the survey article [35] of A. Perelli for an overview of results, conjectures and problems relating to the Selberg class.
We define the degree d L of L ∈ S by d L = 2 r j=1 λ j , where λ j are numbers in (S3). From (S5), we have a L (1) = 1 and find that coefficients a L (n) define a multiplicative arithmetic function. From (S3) and (S5), L ∈ S has no zeros outside the critical strip 0 ≤ σ ≤ 1 except for zeros in the half-plane σ ≤ 0 located at poles of the involved gamma factors. The zeros lie in the critical strip are called the nontrivial zeros. The nontrivial zeros are infinitely many unless L ≡ 1 and coincide with the zeros of the entire function 4 ξ L (s) = s m L (s − 1) m L Λ L (s) (2.2) of order one, where m L is the minimal nonnegative integer m in (S2). It is conjectured that the analogue of RH holds for all L-functions in S: Grand Riemann Hypothesis (GRH). For L ∈ S, ξ L (s) = 0 unless ℜ(s) = 1/2.
For L ∈ S, we abbreviate to GRH(L) the assertion that all zeros of ξ L (s) lie on the critical line ℜ(s) = 1/2.
The main subject of this paper is the studying of GRH(L) for L-functions in the subclass S R of S defined by If L ∈ S R , the gamma factor γ L in (S3) is not an exponential function, ξ L are nonconstant entire functions satisfying functional equations where the root number ǫ L must be 1 or −1. The second equality of (2.3) means that ξ L is an real entire function. 4 The quantity 2 r j=1 (µj − 1 2 ) is usually referred to as ξ-invariant and is often written as ξL in the theory of the Selberg class, but we do not use the letter ξ for the ξ-invariant of L ∈ S to avoid confusion.

2.2.
Auxiliary functions for GRH. Let z = u + iv be the complex variables relating with the variable s by s = 1/2 − iz. To work on GRH(L) for L ∈ S R , we introduce the two parameter family of entire functions parametrized by ω ∈ R >0 and ν ∈ Z >0 . Then, L ∈ HB for all ω > 0.
As easily found from (2.4) and definition of HB, for fixed ω > 0, if E ω,ν L ∈ HB for some ν ∈ Z >0 , then E ω,ν L ∈ HB for arbitrary ν ∈ Z >0 . Therefore, E ω,1 L in Proposition 2.1 can be replaced by E ω,νω L defined for positive integers ν ω indexed by ω > 0. On the other hand, HB in the statement can be replaced by HB without the changing of the meaning of the statement, although E ω,1 L ∈ HB is different from E ω,1 L ∈ HB for individual ω > 0. As the converse of Proposition 2.1, we have the following.
The value ω = 1/2 in Proposition 2.2 comes from the trivial zero-free region of L ∈ S. More precise relation between the zero-free region of L and the property E ω,ν L ∈ HB will be discussed in Proposition 4.3.
From Propositions 2.1 and 2.2, the validity of GRH(L) for L ∈ S R is equivalent to the existence of a section R >0 → R >0 × Z >0 ; ω → (ω, ν) such that E ω,ν L ∈ HB for every ω > 0. Hence GRH(L) will be established if the following three steps are completed for every point of a section: ). The main results stated below concern each step in (3-1)∼(3-3). . We define the function K ω,ν L on the real line by

Results on
and for L ∈ S R , (ω, ν) ∈ R >0 × Z >0 . The integral on the right-hand side of (2.6) converges absolutely if v > 0 is sufficiently large, and we have for x > 0 and K ω,ν L (x) = 0 for x < 0, where ǫ L is the root number in (2.3), q ω,ν L (n) is an arithmetic function determined by the Dirichlet coefficients a L (n) (non-archimedean information), and G ω,ν L is a certain explicit real-valued function having a support in [0, ∞) determined by the gamma factor γ L (archimedean information).
We find that K ω,ν L is a continuous function if (ω, ν) ∈ R >0 × Z >0 satisfies νωd L > 1, (2.8) where d L is the degree of L ∈ S R (Proposition 4.1). This condition for (ω, ν) is technical but essential to the following construction of H ω,ν L . If (ω, ν) ∈ R >0 × Z >0 satisfies (2.8), defines a bounded operator on L 2 (−∞, t) for every t ∈ R, where 1 A is the characteristic function of a set A. The study of K ω,ν L [t] yields a canonical system as follows.
Then, for every t > 0, (2.9) defines a Hilbert-Schmidt type self-adjoint operator on L 2 (−∞, t) having a continuous kernel, and K ω,ν L [t] = 0 for t ≤ 0. Moreover, there exists τ = τ (L; ω, ν) > 0 such that both ±1 are not the eigenvalues of K ω,ν L [t] for every t ∈ [0, τ ). In particular, the Fredholm determinant det(1 ± K ω,ν L [t]) does not vanish for every t ∈ [0, τ ). The first half will be proved by applying the argument in Section 3.1 to K ω,ν L under Proposition 4.1. The latter half is Proposition 4.2. Propositions 4.1 and 4.2 are proved in Section 4. By (2.9), to understand the operator K ω,ν L [t], we need only the values of K ω,ν L on [0, 2t) that are determined by the information on the gamma factor γ L and finitely many coefficient q ω,ν L (n)'s by (2.7). By Theorem 2.1, the Hamiltonian H ω,ν L on [0, τ ) is defined by and it defines the canonical system − d dt on [0, τ ). By definition, the Hamiltonian H ω,ν L has no H ω,ν L -indivisible intervals, that is, all points of [0, τ ) are regular. Next, we construct the unique solution of the canonical system recovering the entire function E ω,ν L .
2.4. Results on . Let τ be the number in Theorem 2.1. Then, two integral equations for unknown functions ϕ ± t ∈ L 2 (−∞, t) have unique solutions for every t ∈ [0, τ ), since 1 ± K ω,ν L [t] are invertible. We extend the solutions ϕ ± t to continuous functions on R by Then, we find that |φ ± (t, x)| ≪ e c|x| for some c > 0 which may depend on t (Lemma 3.7 and Proposition 4.1). Therefore, two functions defined by are analytic functions in the upper half-plane ℑ(z) > c ′ for t ∈ [0, τ ).
We define E ω,ν L by (2.4), and then define A ω,ν L and B ω,ν L by (1.4) for E = E ω,ν L . Let τ = τ (L; ω, ν) be the positive real number in Theorem 2.1. Let A ω,ν L (t, z) and B ω,ν L (t, z) be the families of functions defined by (2.12). Then, (1) A ω,ν L (t, z) and B ω,ν L (t, z) are extended to real entire functions as a function of z for every t ∈ [0, τ ), (2) A ω,ν L (t, z) is even and B ω,ν L (t, z) is odd as a function of z for every t ∈ [0, τ ), (3) A ω,ν L (t, z) and B ω,ν L (t, z) are continuous and piecewise continuously differentiable functions as functions of t for every z ∈ C, 2.5. Results on (3-3). By the above results, we obtain the Hamiltonian H ω,ν L defined on [0, τ ) and the solution t (A ω,ν L (t, z), B ω,ν L (t, z)) of the canonical system associated with H ω,ν L satisfying (2.13) without GRH(L) for L ∈ S R . However, it is not enough to conclude that E ω,ν L (z) ∈ HB for small ω > 0 because we have no information about the solution t (A ω,ν L (t, z), B ω,ν L (t, z)) at the right endpoint t = τ . However, we can not expect a nice result at t = τ , because it seems that the interval [0, τ ) in Theorem 2.1 is not the one in Inv(E ω,ν L ). In fact, the result [40,Theorem 2.3] suggests that the genuine interval in Inv(E ω,ν L ) is the half-line [0, ∞). If we suppose that E ω,ν L ∈ HB, the Fredholm determinant det(1 ± K ω,ν L [t]) does not vanish for every t ≥ 0 (Proposition 4.4. See also Proposition 2.2 and Section 9.3). Thus the Hamiltonian H ω,ν L of (2.10) extends to a Hamiltonian on I = [0, ∞) having no H ω,ν Lindivisible intervals and the solution t (A ω,ν L (t, z), B ω,ν L (t, z)) of (2.12) also extends to the solution of the canonical system (2.11) on I. Moreover we have the following result which will be proved in Section 6.
2.6. Equivalent condition for GRH. Considering Theorem 2.2, Theorem 2.3 and Proposition 3.4 below, E ω,ν L ∈ HB is equivalent to the condition that H ω,ν L is extended to a Hamiltonian on [0, ∞) and lim t→∞ J ω,ν L (t; z, w) = 0 if νωd L > 1. Hence we obtain the following equivalent condition for GRH(L) for L ∈ S R by noting that the range of ω in Proposition 2.1 can be relaxed to a decreasing sequence tending to 0.
Theorem 2.4. The validity of GRH(L) for L ∈ S R is equivalent to the condition that there exists a sequence (ω n , ν n ) ∈ R >0 × Z >0 , n ≥ 1, such that (1) ω m < ω n if m > n and ω n → 0 as n → ∞, The detailed proof of Theorem 2.4 will be given in Section 7. Also, a variant of Theorem 2.4 is stated in Section 9.1.

Inverse problem for some special canonical system
Let A and B be entire functions defined by (1.4) for an entire function E. The goal of this section is the construction of a triple Inv(E) ♭ consisting of some possibly infinite interval is the set of all positive definite matrices in Sym 2 (R). The way of construction is similar to [13] and [40].
The most important point is that we will impose several conditions for E but E ∈ HB is not necessary to the following construction of Inv(E) ♭ differ from de Branges' theory for the inverse spectral problem Inv(E).
3.1. Basic assumptions. Let L p (I) be the L p -space on an interval I with respect to the Lebesgue measure. If J ⊂ I, we regard L p (J) as a subspace of L p (I) by the extension by zero. We denote by the Fourier integral and inverse Fourier integral, respectively. We use the same notation for the Fourier transforms on L 1 (R) and L 2 (R) if no confusion arises. If we understand the right-hand sides in L 2 -sense, they provide isometries on L 2 (R) up to a constant multiple: . We impose several conditions for E throughout this section, because the existence of a Hamiltonian is not guaranteed for general entire function E.
The first is the following condition: (K1) There exists a real-valued continuously differentiable function ̺ on R such that |̺(x)| ≪ e −n|x| for any n > 0 and E(z) = (F̺)(z) for all z ∈ C. Here "≪" stands for the Vinogradov symbol which will be used often as well as the Landou symbols "O" and "o". We have E ♯ (z) = E(−z) from (K1). Thus, A(z) is even and B(z) is odd under (K1). We define Then, by definition and (K1). In addition we suppose the following conditions: (K2) There exists a real-valued continuous function K defined on the real line such that |K(x)| ≪ exp(c|x|) for some c ≥ 0 and Θ(z) = (FK)(z) holds for ℑ(z) > c. (K3) K vanishes on (−∞, 0), (K4) K is continuously differentiable outside a discrete subset Λ ⊂ R and |K ′ | is locally integrable on R. Under the above conditions, the map defines a Hilbert-Schmidt operator on L 2 (−∞, t) for every t > 0. In fact, the Hilbert- Finally, we suppose the following condition in addition to (K1)∼(K4): The set of entire functions satisfying (K1)∼(K5) is not empty. In fact, it will be shown in Section 4 that E ω,ν L of (2.4) satisfies these five conditions for every L ∈ S R = ∅ under (2.8). Here we should mention that (K5) plays the roll of the condition E ∈ HB in the following construction of Inv(E) ♭ , but (K5) does not imply E ∈ HB in general, because τ in (K5) may not be the right endpoint of the interval in Inv(E) and hence a nice information at the point τ can not be expected.
Under the above five conditions (K1)∼(K5), we will construct a triple Inv(E) ♭ by using solutions of some family of integral equations after the next subsection.

Meromorphic inner functions. Let
If an inner function Θ in C + is extended to a meromorphic function in C, it is called a meromorphic inner function in C + . Here E is not unique and we may choose E ∈ HB, because we can factorize E ∈ HB as E = E 0 E 1 such that E 0 ∈ HB, E 1 having only real zeros and E ♯ 1 = E 1 by using usual factorization theorem for functions in H ∞ .
for f in the space C ∞ c (R) of all compactly supported smooth function on R. Then Kf belongs to L 2 (R), and the linear map f → Kf is extended to the isometry K : Proof.
for ℑ(z) > c by (K2), where F = Ff . In the right-hand side, F (−z) is an entire function satisfying F (−z) = O(|z| −n ) as |z| → ∞ in any horizontal strip c 1 ≤ ℑ(z) ≤ c 2 for arbitrary fixed n > 0. Therefore, we find that Kf belongs to L 2 (R) by applying the Fourier inversion formula to Θ(z)F (−z) along a line ℑ(z) = c ′ > c and then moving the path of integration to the real line ℑ(z) = 0, since Θ is inner in C + . Moreover The extended operator is obviously isometric. Equality (3.3) holds for real z by the continuity, and it implies K 2 = id.
Suppose that f ∈ L 2 (R) has a support in (−∞, t] for some t ∈ R. Then Kf belongs to L 2 (R) and has a support in [−t, ∞) by (K3). Therefore the left-hand side of (3.3) is defined by the Fourier integral and analytic in C + . On the other hand, F (−z) in the right-hand side of (3.3) is also defined by the Fourier integral and analytic in C + . Hence both sides of (3.3) are analytic functions in C + , and they are equal on the real line. Thus equality (3.3) holds for ℑ(z) ≥ 0.

Existence and uniqueness of solutions of integral equations.
We suppose that E satisfies (K1)∼(K5) throughout this and the later subsections. In particular, we understand that τ is the number in (K5). Lemma 3.3. Let ε ∈ {±1} and t ∈ (0, τ ). Then the integral equation Proof. By the assumption for E, K[t] is a compact operator on L 2 (−∞, t) such that both ±1 belong to its resolvent set. Therefore, integral equation (3.4) has the unique solution ϕ ε t in L 2 (−∞, t) by the Fredholm alternative. We easily obtain the other properties of the solution ϕ ε t from (3.4), (K2) and (K3).
Lemma 3.4. Let ε ∈ {±1} and t ∈ (0, τ ). Then, for arbitrary t < s < τ , the equation has the unique solution X =φ ε The right-hand side belongs to L 2 (−∞, s) by the Cauchy-Schwartz inequality, since K belongs to L 2 (−∞, s ′ ) for every s ′ ∈ R and the integral on the right-hand side vanishes for almost every x < −t by (K2) and (K3). Clearly,φ ε t = ϕ ε t on (−∞, t]. Conversely, equality (3.6) shows that every solution of (3.4) on L 2 (−∞, s) is determined by its restriction on (−∞, t). Hence the uniqueness of solutions follows from Lemma 3.3. By the way of the extension,φ ε t is real-valued.
In what follows, we denote by φ ε (t, x) the extension of ϕ ε t (x) to R for t ∈ (0, τ ) if no confusion arises. That is, φ ε (t, x) is given by The extended solution φ ε (t, x) are real-valued, since K(x) and ϕ ε t (x) are real-valued. We take the convention that This convention is compatible with Lemma 3.3 and 3.4, since integral equation (3.4) for (3.8) We note that (3.8) and the integral equation for ̺ in (K1) have the same solution φ ε (t, x), since the Fourier transforms of (3.8) and (3.9) give the same equation x)e izx dx is defined for every z ∈ C. 3.4. Differentiability of solutions. We handle the differentiability of the extended solution φ ε (t, x) for both variables under (K1)∼(K5).
Proof. The continuity of φ ε (t, x) in x is obvious from the definition (3.7) because of the continuity of K and ϕ ε t . On the other hand, . The integral on the right hand side defines a continuous function of x, since We investigate the differentiability of φ ε (t, x) with respect to t by using the kernel R(x, y; µ; t) of the resolvent (1 − µK[t]) −1 , because the solution ϕ ε t of (3.4) is related to R(x, y; µ; t) as follows.
Let Ω t = (−∞, t] × (−∞, t]. We introduce the notation K x 1 , x 2 , · · · , x n y 1 , y 2 , · · · , y n = det as usual for the kernel K(x, y) = K(x + y). The Fredholm determinant d(µ; t) and the first Fredholm minor D(x, y; µ; t) of the continuous kernel K(x, y) on Ω t are defined by where d 0 (t) = 1, D 0 (x, y; t) = K(x, y) and Integrals on the right hand sides are converges absolutely by (K2) and (K3). In particular, the kernel D n (x, y; t) is continuous in (x, y). Note that d n (t) = D n (x, y; t) = 0 for every n ≥ 1 if t ≤ 0 by (K3). It is well-known that the series (3.11) and (3.12) converge uniformly and absolutely in µ and (x, y, µ) respectively, when µ is confined in a compact subset of C (see [37,Theorem 5.3.1], for example). A standard way of the proof of such facts also provides the continuity for t as follows. Put This is defined well by (K2) and (K3), and defines a continuous function of t. We have for (x, y) ∈ Ω t and t ∈ [0, τ ) by (K3), definitions (3.13), (3.14) and Hadamard's inequality ([37, Theorem 5. . Therefore the series of (3.11) (resp. (3.12)) converges absolutely and uniformly on a compact subset of (µ, t) In particular, D(x, y; µ; t) and d(µ; t) are continuous in all variables. If d(µ; t) = 0, that is 1/µ is not an eigenvalue of K[t], the resolvent kernel R(x, y; µ; t) is defined by Note that d(±1; t) = 0 for every t ∈ [0, τ ) by the assumption (K5). By the general theory of integral equations, R(x, y; µ; t) satisfies integral equations [31,Chap. 24], for example). By taking y = t and µ = −ε in the first equation of (3.16), we have Therefore, we obtain for x ∈ (−∞, t] by the uniqueness of solutions of (3.4). In particular, we obtain the continuity of ϕ ε t (x) for x again and the continuity of ϕ ε t (x) for t. We have lim Proof. The first half of the lemma follows from (3.7) and the above argument. If d(µ; t) = 0 for every t ∈ [0, τ ) as well as µ = ±1, we have by (3.15). Therefore, in order to prove the latter half of the lemma, it is sufficient to prove (i) the existence and the continuity of ∂ ∂t d(λ; t) and (ii) the existence, the continuity and the integrability of ∂ ∂t D(x, t; λ; t) in x by (3.7), (3.17) and We prove (i). By definition (3.11) and (3.13), we have Clearly, each term in the series is continuous in t by (K2) and (K3). By using Hadamard's inequality, the right-hand side is bounded by The series on the right-hand side converges uniformly on a compact subset of (µ, t) ∈ C × (0, τ ). Hence d(µ; t) is continuously differentiable in t (with no exceptional points). Successively, we prove (ii). We have where D y (resp. D t ) means the partial derivative with respect to the second (resp. the fourth) variable. We have by differentiating (3.16). The integrals on the right hand sides are continuous function of (x, y) on Ω t by (K2), (K3) and (K4), since R(x, y; µ; t) is continuous in (x, y). Therefore, D(x, y; µ; t) is continuously differentiable with respect to x and y unless x + y ∈ Λ by (K4) and (3.15). Thus D y (x, t; µ; t) is continuous in t except for points in {λ − x | λ ∈ Λ} for fixed x. In addition, (3.19) shows that |D y (x, t; µ; t)| is integrable on (−∞, t] with respect to x by (K4). On the other hand, by definition (3.12) and (3.14), Clearly, each term in the series is continuous in (x, y, t) by (K2) and (K3). By the row expansion of the determinant and Hadamard's inequality, the right hand side is bounded by when y ≤ t. The series on the right-hand side converges uniformly on a compact subset of (µ, t) ∈ C × (0, τ ). Thus D t (x, t; µ; t) is continuous in t. In addition, the right-hand side shows that |D t (x, t; µ; t)| is integrable on (−∞, t] with respect to x. 3.5. The first order differential system. As in the previous section, we assume that t ∈ [0, τ ) under (K1)∼(K5). Then φ ε (t, x) is continuously differentiable with respect to t and x outside a discrete subset of [0, τ ) × R. Under this situation, we derive a first order differential system arising from φ ε (t, x), ε ∈ {±1}, start from (3.8). We often denote the extended solutions φ ε (t, x) by φ ε t (x) for the simplicity of writing. First, we operate ∂/∂t on both sides of (3.8). Then, Second, we operate ∂/∂x on both sides of (3.8): Using the identity ∂ ∂x K(x + y) = ∂ ∂y K(x + y) and then applying integration by parts to the integral of the left-hand side, we obtain Then µ(t) is a real-valued function on [0, τ ), since φ ± (t, x) are real-valued. By (3.7), (3.18) and Lemmas 3.5 and 3.6, µ(t) is continuous on [0, τ ), satisfies lim t→0 + µ(t) = 0, and is continuously differentiable is a multiple of the solution of (3.8). Hence, by comparing (3.8) with (3.22), we obtain by the uniqueness of solutions (Lemmas 3.3 and 3.4). We often use (3.24) in the form 3.6. Construction of the canonical system associated with E. We suppose (K1)∼(K5) as well as previous subsections.
Now we introduce two special functions defined for (t, x) ∈ [0, τ ) × R and define A(t, z) and B(t, z) by Fourier integrals (3.27) The right-hand sides of (3.27) analytic functions on the upper half plane ℑ(z) > c for any fixed t ∈ [0, τ ) by (K1) and Lemma 3.7.
Proposition 3.1. We have (3.28) In particular, F(t, x) is even and G(t, x) is odd in x. Moreover, (F(t, x), G(t, x)) are real-valued and |F(t, x)|, |G(t, x)| ≪ e −n|x| for any n > 0, where the implied constant depending on t.
if |x| ≫ t by (K1). Hence the first equalities of (3.28) implies the assertion. Proof. This is an immediate consequence of Proposition 3.1 and (3.27).
We denote by Φ ε (t, z) the Fourier transforms They are defined for ℑ(z) > c by Lemma 3.7 and extends to meromorphic functions on C by Corollary 3.1. Moreover, we have formulas (3.27). By the second equations in (3.27) and parities of A(t, z) and B(t, z) as functions of z, we have Substituting the left-hand sides of (3.10) to E(−z)e −izt of the right-hand sides with E ♯ (z) = E(−z), and then noting that φ ± (t, x) = 0 for x < −t, we obtain where we understand Φ ± (t, z) as extended meromorphic functions on C.
Proposition 3.2. F(t, x) and G(t, x) are continuous and continuously differentiable functions in both variables. In addition, they satisfy partial differential equations Proof. F(t, x) and G(t, x) are continuous and continuously differentiable functions in both variables by (3.24) and Lemmas 3.5 and 3.6. By definition (3.26), where we used (3.25) in the second equation. Hence we obtain the first line of (3.30).
The second line of (3.30) is proved in a similar way.
We define This formula is proved in a way similar to the proof of Theorem 12 of Chapter 24 in [31]. In fact, this is a well-known formula for an integral operator defined on a finite interval with a continuous kernel. The formula (3.34) implies for t ∈ (0, τ ) by definition (3.23) and (3.32). This also holds for t ≤ 0 if we define m(t) = 1 for t ≤ 0, since K[t] = 0 for t ≤ 0 and m(0) = 1.
Using m(t), we define Then they satisfy the pair of partial differential equations by (3.30). In addition, we define and Then we have for t ∈ [0, τ ) and z ∈ C by (3.27) and (3.36). A(t, z) and B(t, z) are entire functions, since A(t, z) and B(t, z) are entire functions by Proposition 3.1. Moreover, we can verify that (3.31) and (3.37) imply that t (A(t, z), B(t, z)) satisfies the canonical system associated with H(t) on [0, τ ) with by elementary ways. To summarize the above discussion, we obtain the following.
Proof for any 0 ≤ t < s < ∞ in a way similar to the proof of Lemma 2.1 of [16]. Taking w = z ∈ C + and then tending s to ∞, we have by lim t→∞ J(t; z, z) = 0. On the other hand, we have by substituting (1.4) into (1.6). Hence we obtain This shows E(t, z) satisfy (1.3) and hence belongs to HB.

Proof of Theorem 2.2
In this section, we prove that E ω,ν L of (2.4) satisfies (K1)∼(K5) in Section 3 (Propositions 4.1 and 4.2). Then we obtain Theorem 2.2 as the consequence of Theorem 3.1.

Analytic properties of
by taking some c ∈ R, then it is a real-valued function in C ∞ (R) satisfying the estimate |̺ ω,ν L (x)| ≪ n e −n|x| for every n ∈ N such that holds for all z ∈ C.
In the right-hand side, where the implied constant depends on ω, λ, µ and δ > 0. This implies the first estimate of (4.5) by (4.4) and the definition of d L . On the other hand, the right-hand side of (4.6) takes the maximum [λ( holds uniformly for u ∈ R and v ≥ 1/2 + ω + δ, where the implied constant depends on ω, λ, µ and δ > 0. This implies the second estimate of (4.5) by (4.4) and the definition of d L .
We prove (4) and the last line of Proposition 4.1. By (4.2), Θ ω,ν L (u + iv) belongs to L 1 (R) as a function of u if v is sufficiently large. Thus, K ω,ν L is uniformly continuous on R by (2.6). Moreover, by (2.6) and (4.2), the formula holds together with the absolute convergence of the integral on the right-hand side. Therefore, this shows that K ω,ν L is C k if νωd L > k + 1. Finally, we prove (6). The derivative d dx K ω,ν L is locally integrable by (4). On the other hand, the set of possible singularities of d dx K ω,ν L is discrete in R by (2), and d dx K ω,ν L does not change its sign infinitely often around any possible singularity except for x = +∞ by definition of g ω,ν L . Therefore, the local integrability of d dx K ω,ν L implies the local integrability of | d dx K ω,ν L |.

4.2.
Non-vanishing of Fredholm determinants: Unconditional cases. In this section, we understand K ω,ν L f by the integral on the right-hand side of (3.2) for K = K ω,ν L if it converges absolutely and locally uniform for a function f , because we do not assume that E ω,ν L ∈ HB (which implies that K ω,ν L f belongs to L 2 (R) by Lemmas 3.1 and 3.2). Proposition 4.2. Let L ∈ S R . Suppose that (ω, ν) satisfies (2.8) and define the operator K ω,ν L [t] on L 2 (−∞, t) by (2.9). Then, there exists τ = τ (L; ω, ν) > 0 such that both ±1 are not eigenvalues of K ω,ν L [t] for every 0 ≤ t < τ , that is, both 1 ± K ω,ν L [t] are invertible operator on L 2 (−∞, t) for every 0 ≤ t < τ . Thus E ω,ν L satisfies (K5) for [0, τ ). Proof. The spectrum of K ω,ν L [t] is discrete and consists of eigenvalues, since K ω,ν L [t] is a Hilbert-Schmidt operator on L 2 (−∞, t) by (K2) and (K3). The statement of the proposition is equivalent that K ω,ν Suppose that P t K ω,ν L f = ±f for some 0 = f ∈ L 2 (−∞, t). We have for −∞ < x < −t and f ∈ L 2 (−∞, t). Therefore, P t K ω,ν L f is a function on R having a support in [−t, ∞), and hence the assumption implies that f has a compact support contained in [−t, t].
The value ω = 1/2 corresponds to the abscissa σ = 1 of the absolute convergence of the Dirichlet series (2.1). The non-vanishing of L ∈ S on the line σ = 1 is an important problem because it relates with the analogue of the prime number theorem of L ∈ S for example. Conrey-Ghosh [15] proved the non-vanishing of L ∈ S on the line σ = 1 subject to the truth of the Selberg orthogonality conjecture. Kaczorowski-Perelli [21] obtained the non-vanishing of L ∈ S on the line σ = 1 under a weak form of the Selberg orthogonality conjecture. As mentioned before, it is conjectured that S consists only of automorphic L-functions. The non-vanishing for automorphic L-functions on the line σ = 1 had been proved unconditionally in Jacquet-Shalika [20].

4.4.
Non-vanishing of Fredholm determinants: Conditional cases. We suppose that E ω,ν L ∈ HB throughout this subsection, otherwise it will mentioned. Then, Θ ω,ν L is inner in C + by Lemmas 3.1. This assumption is satisfied unconditionally for ω > 1/2, and also for all ω > 0 under GRH(L) by Proposition 2.2 and Lemma 3.1 (see also Proposition 4.3). We denote by K ω,ν L the isometry on L 2 (R) defined by (3.2) for K = K ω,ν L (cf. Lemma 3.2). It is not obvious whether the integral (3.2) for K = K ω,ν L defines an operator on L 2 (R) if we do not assume that E ω,ν L ∈ HB. If (ω, ν) satisfies (2.8) in addition, we have (2.9) and the orthogonal projection P t from L 2 (R) to L 2 (−∞, t). Lemma 4.3. Let L ∈ S R and ω > 0. Then there exists entire functions f ω 1 (s) and f ω 2 (s) such that they have no common zeros, satisfy , and the number of zeros of f ω 2 (s) in |ℑ(s)| ≤ T is approximated by c T log T for large T > 0, where c > 0 is some constant.
Proof. We denote by Z L the set of all zeros of ξ L (s) and by m(ρ) the multiplicity of a zero ρ ∈ Z L . Then any zero of ξ L (s − ω) has the form s = ρ + ω for some ρ ∈ Z L and has the multiplicity m(ρ). On the other hand, if s = ρ + ω for some ρ ∈ Z L and ξ L (s + ω) = 0, we have ρ + 2ω ∈ Z L . Considering this, we set Z ω L = {ρ ∈ Z L | ρ + 2ω ∈ Z L }, and define an entire function by the Weierstrass product: where the right-hand side converges uniformly on compact subsets in C, since ξ L (s) is an entire function of order one. In addition, we put Then, by definition, f ω 1 (s) and f ω 2 (s) are entire functions such that they have no common zeros and satisfy . Therefore, the remaining task is to show that f ω 2 (s) has approximately c T log T many zeros in |ℑ(s)| ≤ T for some c > 0.
We denote by N L (T ) (resp. N ω L (T )) the number of zeros in Z L (resp. Z ω L ) with |ℑ(s)| ≤ T counting with multiplicity: Then, M ω L (T ) ∼ N L (T ), because N L (T ) ∼ N ω L (T ) and functional equations (2.3) imply that Z ω L is closed under ρ → 1−ρ and ρ →ρ except for a relatively small subset counting with multiplicity. If we take a zero ρ ∈ Σ ω L , then 1 − ρ ∈ Z ω L by the definition of Σ ω L , and thus 1−ρ+2ω ∈ Z L by the definition of Z ω L . Therefore, ρ−2ω = 1−(1−ρ+2ω) ∈ Z L by the first functional equation of (2.3). As a consequence, ρ ∈ Σ ω L implies ρ−2ω ∈ Z ω L . On the other hand, M ω L (T ) ∼ N ω L (T ) ∼ N L (T ) shows that ρ − 2ω ∈ Z ω L implies ρ − 2ω ∈ Σ ω L almost surely. Taken together, ρ ∈ Σ ω L implies ρ − 2ω ∈ Σ ω L almost surely and this process is continued repeatedly. However, it is impossible, because all zeros of ξ L (s) must lie in the critical strip. Hence, N L (T ) ∼ n ω L (T ). Lemma 4.4. Let t ≥ 0. Suppose that (ω, ν) satisfies (2.8). Then the support of K ω,ν L P t f is not compact for every f ∈ L 2 (R) unless K ω,ν L P t f = 0. Proof. We prove this by contradiction. Suppose that K ω,ν L P t f = 0 and has a compact support. Then FK ω,ν L P t f is an entire function of exponential type by the Paley-Wiener theorem. On the other hand, we have The entire function on the right-hand side has at least c T log T many zeros in the disk of radius T around the origin as T → ∞ for some c > 0 by Lemma 4.3. However all entire functions of exponential type have at most O(T ) zeros in the disk of radius T around the origin, as T → ∞, because of the Jensen formula ([29, §2.5 (15)]). This is a contradiction.
As the above, it is not necessary to assume that Θ is inner in C + for Lemma 4.4. , t), and iii) H ω,a < 1. In particular, 1 ± K ω,ν L [t] are invertible operator on L 2 (−∞, t) for every t ≥ 0. Proof. First, we note that K ω,ν L f is defined for every f ∈ L 2 (−∞, t) by Lemma 3.2 for K = K ω,ν L . Because t −∞ K ω,ν L (x + y)f (y) dy = 0 for x < −t by (K3), we obtain i). To prove ii), it is sufficient to show K ω,ν Thus K ω,ν L f (x) = 0 for almost every x > t. On the other hand, we have Hence K ω,ν L f has a compact support contained in [−t, t]. However, it is impossible for any f = 0 by Lemma 4.4. As the consequence Finally, we prove iii). As found in Section 3.1, K ω,ν L [t] is a self-adjoint compact operator (because the Hilbert-Schmidt operator is compact). Therefore, K ω,ν L [t] has purely discrete spectrum which has no accumulation points except for 0, and one of ± K ω,ν L [t] is an eigenvalue of K ω,ν L [t]. However, by ii), every eigenvalue of K ω,ν L [t] has an absolute value less than 1. Hence K ω,ν L [t] < 1.

Theory of de Branges spaces
In this section, we review several basic notions and properties of de Branges spaces as a preparation to the next section. General theory of de Branges spaces is given in the book [9], but the proofs of results are more accessible in de Branges' earlier papers [4,5,6,7,8]. See also [45], [46] and references there in. 5.1. Hardy spaces. The Hardy space H 2 = H 2 (C + ) in the upper half-plane C + is defined to be the space of all analytic functions f in C + endowed with norm f 2 H 2 := sup v>0 R |f (u + iv)| 2 du < ∞. The Hardy spaceH 2 = H 2 (C − ) in the lower half-plane C − is defined in a similar way. As usual we identify H 2 andH 2 with subspaces of L 2 (R) via nontangential boundary values on the real line such that L 2 (R) = H 2 ⊕H 2 . The Fourier transform provide an isometry of L 2 (R) up to a constant such that H 2 = FL 2 (0, ∞) andH 2 = FL 2 (−∞, 0) by the Paley-Wiener theorem.
The reproducing formula f (z) = f, J(z, ·) for f ∈ B(E) and z ∈ C + remains true for z ∈ R if Θ = E ♯ /E is analytic in a neighborhood of z, where f, g = R f (u)g(u)du.

Axiomatic characterization of de Branges spaces.
The Hilbert space H consisting of entire functions forms a de Branges space if and only if it satisfies the following three axioms: (dB1) For each z ∈ C the point evaluation Φ → Φ(z) is a continuous linear functional on H.
If H satisfies the above axioms, there exists E ∈ HB such that H = B(E) and f H = f B(E) for all f ∈ H. The possibility of E is not unique. In fact, B(E) = B(E θ ) for any θ ∈ [0, π) and E θ (z) = e iθ E(z).
) solve the canonical system on [0, c) associated with some Hamiltonian H(t) and E 0 (z) = E(z) (see [9,Theorem 40], but note that the result is formulated in terms of integral equations and that we need a changing of variables to apply the result.) For 0 ≤ t < s < c, there exists a transfer matrix M (t, s; z) which is a matrix of entire functions having the property that and det M (t, s; z) = 1 ([9, Theorem 33]).

Model subspaces.
We review a few results on model spaces according to 18] (and also Baranov [1]). For an inner function Θ, a model subspace (or coinvariant subspace) K(Θ) is defined by the orthogonal complement It has the alternative representation It is known that every meromorphic inner function is expressed as Θ = E ♯ /E by using some E ∈ HB. If Θ is a meromorphic inner function such that Θ = E ♯ /E, the model subspace K(Θ) is isomorphic and isometric to the de Branges space B(E) as a Hilbert The changing of consideration from B(E) to K(Θ) has the advantage that spaces ΘH 2 , ΘH 2 , H 2 ⊖ (H 2 ∩ ΘH 2 ) and H 2 ∩ ΘH 2 are defined even if Θ is not necessarily a meromorphic inner function in C + , and it allows us to study these spaces without assuming E ∈ HB. Note that ΘH 2 ⊂ H 2 in general if Θ is not necessary a inner function in C + . As developed in [10,11,12,13], the Hankel type operator K with the kernel K(x + y) is quite useful to study K(Θ) and the above spaces via Fourier analysis. This fact was already used implicitly in Sections 3 and 4 and will be used in Sections 6 (and also Section 9.2).

Proof of Theorem 2.3
We prove Theorem 2.3 by studying de Branges subspaces of B(E ω,ν L ). 6.1. Formulas for reproducing kernels of de Branges subspaces. We start from general theory for entire functions E ∈ HB satisfying (K1)∼(K4). Then Θ = E ♯ /E is inner in C + , f → Kf defines an isometry on L 2 (R) satisfying K 2 = id by Lemmas 3.1 and 3.2. In addition, we assume (K5) with τ = ∞. As proved in Section 4, E ω,ν L satisfies all these conditions for ω > 1/2 unconditionally and for all ω > 0 under GRH(L). Under the above setting, we study de Branges subspaces of B(E).
Proof. The case of t = 0 is Lemma 6.1. We suppose t > 0. By the general theory of Hilbert spaces, the orthogonal complement V ⊥ t of V t in L 2 (R) is equal to the closure of L 2 (−∞, t) + K(L 2 (−∞, t)). We show that L 2 (−∞, t) + K(L 2 (−∞, t)) is closed.
We suppose that w = u + Kv for u, v ∈ L 2 (−∞, t). Then we have . Therefore, if w n = u n + Kv n is L 2 -convergent, it implies that both u n and v n are also L 2 -convergent, and hence the space L 2 (−∞, t) + K(L 2 (−∞, t)) is closed. Hence we obtain To prove V t = {0}, it is sufficient to show that L 2 (−∞, t) + K(L 2 (−∞, t)) is a proper closed subspace of L 2 (R). Suppose that f ∈ K(L 2 (−∞, t)). Then the restriction of f to (t, ∞) is a continuous on (t, ∞). In fact, if we put g(x) := (Kf )(x) = t −x K(x+y)f (y) dy for f ∈ L 2 (−∞, t), the continuity of K(x) implies the continuity of g by where 0 < δ < x and we used the mean value theorem and the Schwartz inequality.
8. Spectral realization of zeros of A ω,ν L and B ω,ν L In this part, we mention that the zeros of A ω,ν L and B ω,ν L can be regarded as eigenvalues of self-adjoint extensions of a differential operator for ω > 1/2 unconditionally and for 0 < ω ≤ 1/2 under GRH(L). In general, dom(M) has codimension at most one. Hereafter, we suppose that dom(M) has codimension zero, that is, dom(M) is dense in B(E). Then, all self-adjoint extensions M θ of M are parametrized by θ ∈ [0, π) and their spectrum consists of eigenvalues only. The self-adjoint extension M θ is described as follows. We introduce and the operation is defined by where w 0 is a fixed complex number with S θ (w 0 ) = 0 and dom(M θ ) does not depend on the choice of w 0 . The set forms an orthogonal basis of B(E), and each F θ,γ is an eigenfunction of M θ with the eigenvalue γ: M θ F θ,γ = γF θ,γ ([23, Proposition 6.1, Theorem 7.3]). We have S π/2 (z) = 2i A(z) and S 0 (z) = −2i B(z) by definition. Therefore, {A(z)/(z − γ) | A(γ) = 0} and {B(z)/(z − γ) | B(γ) = 0} are orthogonal basis of B(E) consisting of eigenfunctions of M π/2 and M 0 , respectively. 8.2. Transform to differential operators. Considering the isometric isomorphism of the Hilbert spaces we define the differential operator D on V 0 by is the operator of multiplication by 1/E(z). Then we have (Df )(x) = i d dx f (x) for f ∈ C 1 (R) ∩ L 2 (0, ∞). All self-adjoint extensions of D are given by forms an orthogonal basis of V 0 consisting of eigenfunctions f θ,γ of D θ for eigenvalues γ, where ie −iθ is the constant for the simplicity of f θ,γ . We have On the other hand, V 0 is isomorphic to the quotient space L 2 (0, ∞)/K(L 2 (−∞, 0)) by Lemma 6.1 and (5.1). This structure of V 0 is similar to Connes' suggestion for the Pólya-Hilbert space as explained below. 8.3. Comparison with Connes' Pólya-Hilbert space. Connes [14] suggests a candidate of the Pólya-Hilbert space by interpreting the critical zeros of the Riemann zeta function as an absorption spectrum as follows. Let S(R) 0 be the subspace of the Schwartz space S(R) consisting of all even functions φ ∈ S(R) satisfying φ(0) = (Fφ)(0) = 0. For a function φ ∈ S(R) 0 , the function Zφ on R + = (0, ∞) is defined by (Zφ)(y) = y 1/2 ∞ n=1 φ(ny). Then Zφ is of rapid decay as y → +0 and y → +∞. In particular, Z(S(R) 0 ) ⊂ L 2 (R + , dy/y). Then the "orthogonal complement" L 2 (R + , dy/y)⊖Z(S(R) 0 ) is spanned by generalized eigenfunctions y −iγ (log y) k , 0 ≤ k < m γ , of the differential operator iyd/dy attached to the critical zeros 1/2 + iγ of the Riemann zeta function with multiplicity m γ . That is, iyd/dy, L 2 (R + , dy/y) ⊖ Z(S(R) 0 ) forms a "Pólya-Hilbert space". The differential operator iyd/dy may be regarded as the shift of the Hamiltonian (1/2)(y[−i d/dy]+[−i d/dy]y) = −i (yd/dy+1/2) of the Berry-Keating model [2].
Rigorously, the above argument does not make sense, since y −iγ (log y) k are not members of L 2 (R + , dy/y) and L 2 (R + , dy/y) = Z(S(R) 0 ). However, the above naive idea is justified by several manners ( [14] and R. Meyer [32,33]), but some nice property such as the self-adjointness of the operator, the spectral realization of zeros, the Hilbert space structure is lost by known justification.
Contrast with justifications so far, the family π/2 )} ω>0 justifies Connes' idea preserving the self-adjointness of the operator, the spectral realization of zeros and the Hilbert space structure by considering the perturbation family A ω,ν ξ of ξ.
Major objects of the above naive model of Connes' idea correspond to objects attached to (V 0 , D θ ) as follows under the changing of variables y = e x : where "Mellin" means the usual Mellin transform and ≈ means "is equal up to domain". 9.2. Inner property and isometry. We prove that the converse of Lemma 3.2 holds in the following sense. For Θ = E ♯ /E, the multiplication F → ΘF defines a map from L 2 (R) into L 2 (R) by (3.1). We denote it also by Θ if no confusion arises, and define where (Jf )(x) = f (−x). If Θ is an inner function in C + , images Θ(H 2 ) and K Θ (L 2 (R)) are subspaces of H 2 and L 2 (R), respectively. Obviously the map K Θ is related to the function K by (K2). Lemma 9.1. Let E be an entire function satisfying (K1)∼(K3). Suppose that Θ = E ♯ /E is uniformly bounded on ℑ(z) ≥ c ′ for some c ′ > 0. Then ΘH 2 ⊂ H 2 implies that Θ is inner in C + .
Proof. We know (3.1), and the assumption implies that Θ has no poles in C + . Hence, by applying the Phragmén-Lindelöf convexity principle to Θ in the strip 0 ≤ ℑ(z) ≤ c ′ , we find that Θ is bounded on 0 ≤ ℑ(z) ≤ c ′ . Therefore Θ is a bounded analytic function in C + satisfying (3.1). This is the definition of an inner function in C + .
Theorem 9.2. Let E be as in Lemma 9.1. Then Θ = E ♯ /E is a meromorphic inner function in C + if and only if one of the following conditions holds: (1) Kf is defined as a function on R and K Θ f = Kf for every f ∈ L 2 (−∞, 0); Kf is defined as a function of L 2 (R) for every f ∈ L 2 (−∞, 0), where we understand Kf by the integral of (3.2). In particular, Θ = E ♯ /E is a meromorphic inner function in C + if and only if f → Kf defines an isometry on L 2 (R).
The above equivalence can be applied to E = E ω,ν L for L ∈ S R and (ω, ν) ∈ R >0 × Z >0 . Proof. The last line of Theorem 9.2 will be established if the other assertions are proved, since E ω,ν L satisfies (K1)∼(K3) by Proposition 4.1 and Θ ω,ν L is uniformly bounded on ℑ(z) ≥ 1/2 + ω + δ by Lemma 4.2. Therefore, it is sufficient to prove the following three assertions: i) condition (1) is equivalent that Θ is inner in C + , ii) condition (2) implies that Θ is inner in C + , and iii) condition (3) implies that Θ is inner in C + , since (1) implies (2) and (3) by (K3) and definition of K Θ , respectively. i) Suppose that Θ is inner in C + . Then Θ(z)F (−z) ∈ H 2 for every F ∈H 2 . Thus the inverse Fourier transform along the line ℑ(z) = a is independent of a > 0, and belongs to L 2 (0, ∞), where f = F −1 F ∈ L 2 (−∞, 0) and the integral converges in the sense of L 2 . On the other hand, we have for a > c by (3.3) and [43,Theorem 65], where the integral converges also in the sense of L 2 . Comparing these two formula for large a, we obtain (1). Conversely, suppose that (1) holds. Write g = K Θ f = Kf for arbitrary fixed f ∈ L 2 (−∞, 0). Then g belongs to L 2 (R), since K Θ maps L 2 (R) to L 2 (R) by definition. In addition, g has a support in [0, ∞), since both K(x) and f (−x) have support in [0, ∞). Therefore g belongs to L 2 (0, ∞). Because f was arbitrary, we have ΘH 2 ⊂ H 2 . Hence Θ is inner in C + by Lemma 9.1.

A sufficient condition for the invertibility of 1± K[t].
In general, for an entire function E satisfying (K1)∼(K3), the existence of τ > 0 in (K5) is not obvious even if we assume that Θ = E ♯ /E is inner in C + . We may need an additional condition for Θ to conclude the existence of τ > 0 in (K5) as well as the role of Lemma 4.3 in the proof of Proposition 4.4. A simple sufficient condition for (K5) with τ = ∞ is as follows.
Proposition 9.1. Let E be an entire function satisfying (K1)∼(K3). In addition, we suppose that Θ is inner in C + and there is no entire functions F and G of exponential type such that Θ = G/F . Then both ±1 are not eigenvalues of K[t] for every t > 0.

Miscellaneous Remarks
(1) Concerning the size of Dirichlet coefficients a L (n), the polynomial bound |a L (n)| ≪ n A for some A ≥ 0 is enough to prove Theorems 2.1, 2.2 and 2.3. In other words, the Ramanujan conjecture (S4) is not necessary to prove these theorems. Therefore, the method of Sections 3 and 4 about the construction of Inv(E ω,ν L ) ♭ can be applicable to more general L-functions, in particular, to L-functions associated to self-dual irreducible cuspidal automorphic representations of GL n (A Q ) with unitary central characters.
(2) In contrast with the Ramanujan conjecture (S4), the Euler product (S5) is essential to the construction of Inv(E ω,ν L ) ♭ . In fact, the explicit formula of the kernel K ω,ν L coming from (S5) was critical to proved that E ω,ν L satisfies (K2) and (K3). It seems that it is not easy even to prove that K ω,ν L is a function if we do not have (S5). It is an interesting problem to extend the construction of Inv(E ω,ν L ) ♭ to the class of L-data which is an axiomatic framework for L-functions introduced by A. Booker [3]. Superficially, Booker's L-datum does not require the Euler product, but it is based on the Weil explicit formula of L-functions in the Selberg class. Roughly, the Weil explicit formula is a result of (S3) and (S5), but the theory of L-data suggests that the Weil explicit formula is more essential than (S3) and (S5).
(3) By the Euler product (S5), L ∈ S is expressed as a product of local p-factors L p , and often, there exists polynomial F p of degree at most d L for each prime p such that L p (s) = 1/F p (p −s ). The Ramanujan conjecture (S4) is understood as the analogue of the Riemann hypothesis for F p (p −s ). The Hamiltonian H L,p attached to F p (p −s ) was constructed in [41] by using a way analogous to the method in Section 3 if F p is a real self-reciprocal polynomial (for details, see [41, Section 1, Section 7.6]). It is an interesting problem to find a relation among the perturbation family of global Hamiltonians H ω,ν L , the family of local Hamiltonians H L,p and the conjectural Hamiltonian H L corresponding to E(z) = ξ L (s) + ξ ′ L (s). (4) The method of [41] for the construction of Inv(E) (not Inv(E) ♭ ) for exponential polynomials E is useful to observe H ω,ν L by numerical calculation of computer for concrete given L ∈ S R , because an entire function satisfies (K1) is approximated by exponential polynomials by approximating the Fourier integral by Riemann sums (cf. the final part of the introduction of [41]).
(5) A sharp estimate of K ω,ν L (x) for large x > 0 is not necessary to prove the main results of this paper. In fact, we do not know the role of the behavior of K ω,ν L (x) at x = +∞ in the equivalent condition of Theorem 2.4. we obtain a sufficient condition for GRH(L), since (4)' implies (4). It is ideal if this is also a necessary condition, but we have no plausible evidence to support the necessity of (4)'. On the contrary, it is not clear whether lim t→+∞ A ω,ν L (t, z) defines a functions of z contrast with the fact lim t→+∞ B ω,ν L (t, z) = 0 under ω > 1/2 or GRH(L) (see Section 9.7). The limit behavior may be related with the arithmetic properties of L(s) in a deep level, because we need information of all q ω,ν L (n)'s to understand it differ from the situation that we need only finitely many q ω,ν L (n)'s to understand K ω,ν L [t] for a finite range of t ∈ R. We do not touch this problem further in this paper.