Kodaira's Assumptions I and II: characterizing the potentials for which the Jost function of a half-line Schrödinger operator with Coulomb-type tail continues into the unphysical k-plane

Kodaira (1949) studied the half-line Schrödinger operator $H=-d^2/dx^2+\nu(\nu+1)/x^2+V(x)$ with an inverse-square term at the origin and a Coulomb tail $V(x)=\alpha/x+O(x^{-1-\epsilon})$ at infinity, and proved — conditionally on two analyticity hypotheses on the scattering solution $u(x,k)$ (Assumption I: continuation across the negative real $k$-axis into the lower half-plane; Assumption II: regularity in the cut plane $\mathbb{C}\setminus[0,\infty)$) — that Heisenberg's S-function $S(k)=A(-k)/A(k)$ is meromorphic in $\Im k>0$ with its only poles the simple poles at the bound states, so the negative eigenvalues are exactly the poles of $S$. Whether the assumptions hold throughout the class, and the necessary and sufficient condition on $V$ for them to hold, remain unestablished. The later literature maps only the sufficient side, and only for purely short-range potentials: compact support gives an entire Jost function; exponential decay $O(e^{-ax})$ gives continuation to a strip of width $a/2$; super-exponential decay gives the whole lower half-plane; power decay with sector analyticity gives only a sector. No necessity result, no explicit counterexample, and no treatment of the Coulomb-tail case is known. The problem asks for the characterization, or a counterexample, within Kodaira's class.

Background

On the half-line $x>0$ consider the radial Schrödinger equation

$$Hu=-\frac{d^{2}u}{dx^{2}}+\frac{\nu(\nu+1)}{x^{2}}u+V(x)u=k^{2}u,\qquad \nu\ge-\tfrac12,$$

which describes a quantum particle in one dimension with a centrifugal (inverse-square) barrier $\nu(\nu+1)/x^{2}$ indexed by the angular momentum $\nu$, together with a perturbation $V$ that is $O(x^{-2+\epsilon})$ at the origin and has a Coulomb tail $V(x)=\alpha/x+O(x^{-1-\epsilon})$ at infinity, $\alpha$ being the Coulomb charge. Writing the energy as $E=k^{2}$, the spectrum of the self-adjoint operator $H$ consists of negative eigenvalues (bound states, i.e. square-integrable solutions at $E=\lambda_m<0$) plus a continuous part on $E>0$.

Scattering theory studies solutions behaving like free waves at infinity; because of the tail, 'free' means Coulomb-modified. The scattering (or Jost) solution $u(x,k)$ is the unique solution of $Hu=k^{2}u$ with asymptotic behaviour $u(x,k)\sim\exp(ikx-\tfrac{i\alpha}{2k}\log x)$ as $x\to\infty$ for $\Im k\ge0$, $k\ne0$; the logarithmic phase is the distortion produced by the $1/x$ tail. Kodaira proved (his Theorem 5.1) that $u$ and its $x$-derivative $u'$ are analytic in $k$ throughout the upper half-plane $\Im k>0$. From $u$ one builds the Jost function $A(k)$: the Wronskian $A(k)=[u,s_{1}]$ (with $[f,g]=fg'-f'g$, constant in $x$) of $u$ with the solution $s_{1}$ that behaves like $x^{\nu+1}$ at the origin. $A$ is analytic in $\Im k>0$, and its zeros on the positive imaginary axis, $k=i\varkappa$, are exactly the bound states $E=-\varkappa^{2}$.

The motivation was Heisenberg's theory of S-matrices: the 1940s proposal that all bound-state data of a quantum system should be readable off the analytic structure of its scattering matrix (S-function), the object relating incoming to outgoing waves. Kodaira's Theorem 6.1 realizes this for the half-line Coulomb-tail problem: under two hypotheses on $u$, the S-function $S(k)=A(-k)/A(k)$ is meromorphic in $\Im k>0$, and its only poles there are simple poles on the imaginary axis located exactly at the bound-state momenta $k_{m}=i|k_{m}|$, with residues giving the norming constants, i.e. the normalization constants of the bound-state eigenfunctions ($c_{m}^{2}=\oint_{k_{m}}S(k)\,dk$). The hypotheses are: (Assumption I) $u(x,k)$ and $u'(x,k)$, a priori analytic only for $\Im k>0$, admit analytic continuation across the negative real $k$-axis into the lower half-plane $\Im k<0$; and (Assumption II) $u$ and $u'$ are regular (analytic) in the whole $k$-plane cut along the positive real axis $[0,\infty)$. Kodaira proved the theorem conditionally and stated that the validity of the assumptions throughout the class, and the necessary and sufficient condition on $V$ for them, were open. If Assumption I fails, $S$ cannot be defined off the real axis at all; if Assumption I holds but Assumption II fails, the numerator $A(-k)$ may contribute additional singularities in $\Im k>0$ (symmetric images of lower-half-plane singularities of $A$), and the negative eigenvalues could no longer be recovered from the singular points of $S(k)$ alone.

In modern language, continuing $A$ (equivalently the resolvent of $H$) from the physical sheet $\Im k>0$ across the real $k$-axis into the lower half-plane $\Im k<0$ — called the unphysical sheet — exposes resonances: poles of the continued resolvent off the real axis, which describe decaying states and dominate the long-time behaviour of waves. The modern theory of scattering resonances (systematized by Dyatlov and Zworski for compactly supported potentials) is complete in dimension one when $V$ has compact support: the Jost function is entire, and Korotyaev's inverse resonance theorem on the half line shows compactly supported potentials are in bijection with their Jost functions. For merely decaying potentials the picture is fragmentary. Dolph, McLeod and Thoe (1966) proved that exponential decay $V(x)=O(e^{-ax})$ of the short-range part yields continuation into a strip $\Im z>-a/2$, so the attainable depth of continuation is governed by the exponential decay rate; McLeod (1967) extended this resolvent-kernel continuation to the unphysical sheet. Froese (1997) and Hitrik (1999) showed that super-exponential decay yields continuation into the whole lower half-plane. Brown, Eastham and McCormack (2000) obtained strip continuation and resonance-free regions for special exponentially decaying classes. Brown and Eastham (2002) proved that power decay $V(x)=O(x^{-\gamma})$ combined with analyticity of $V$ in a sector yields continuation only into a sector $2\pi-\theta_{0}<\arg z<2\pi$ of the unphysical sheet (where $\theta_{0}$ is the angle of that sector of analyticity in their result), stating explicitly that all existing results require exponential decay. Christiansen and Cunningham (2024) tied the density of resonances to the convex hull of the singular support of the potential. All of these results concern short-range tails; none covers the Coulomb-tail class, and none gives a necessary and sufficient condition — that is precisely Kodaira's still-open question.

Problem Statement

Fix $\nu\ge-\tfrac12$ and a real-valued potential $V$ on $(0,\infty)$ with $V(x)=O(x^{-2+\epsilon})$ at the origin and $V(x)=\alpha/x+O(x^{-1-\epsilon})$ at infinity for some $\epsilon>0$ and $\alpha\in\mathbb{R}$ (Kodaira's class), and let $H=-\frac{d^{2}}{dx^{2}}+\frac{\nu(\nu+1)}{x^{2}}+V(x)$ be the Schrödinger operator on $(0,\infty)$. Let $u(x,k)$ be the scattering solution of $Hu=k^{2}u$ with $u(x,k)\sim\exp(ikx-\tfrac{i\alpha}{2k}\log x)$ as $x\to\infty$ ($\Im k\ge0$, $k\ne0$), let $s_{1}(\cdot,k^{2})$ be the solution of the same equation with $s_{1}\sim x^{\nu+1}$ at the origin, let $A(k)=[u,s_{1}]$ be the Jost function (Wronskian $[f,g]=fg'-f'g$, which is constant in $x$), and let $S(k)=A(-k)/A(k)$ be the Heisenberg S-function. Kodaira's Assumption I asserts that $u(x,\cdot)$ and $u'(x,\cdot)$, a priori analytic only for $\Im k>0$, admit analytic continuation across the negative real $k$-axis into the lower half-plane $\Im k<0$; Assumption II asserts that $u$ and $u'$ are regular (analytic) in the whole cut plane $\mathbb{C}\setminus[0,\infty)$. Under I and II, the negative eigenvalues of $H$ are exactly the poles of $S$ in $\Im k>0$.

Determine the following, within Kodaira's class. The three parts are distinct and each is required: (1) concerns Assumption I alone, (2) is the stronger condition for Assumptions I and II jointly, and (3) is the corollary of (1) and (2).

(1) Validity of Assumption I. Either prove that for every real $V$ in the class the Jost function $A(k)$ (equivalently $u(x,k)$, $u'(x,k)$ for fixed $x$) admits analytic continuation across the negative real $k$-axis into $\Im k<0$, and identify the maximal domain of continuation; or exhibit a specific $V$ in the class for which continuation across some point of the negative real axis provably fails (for example, an arc of the negative real axis that is a natural boundary for $A$, meaning no analytic continuation across it exists).

(2) Characterization. Give a necessary and sufficient condition on $V$ for Assumptions I and II to hold — that is, for $A(k)$ to be analytic in the whole cut plane $\mathbb{C}\setminus[0,\infty)$ — so that $S(k)=A(-k)/A(k)$ is meromorphic in $\Im k>0$ with no singularities other than the simple bound-state poles at $k=i\varkappa_{m}$ (the bound states being the negative eigenvalues $E_{m}=-\varkappa_{m}^{2}$ of $H$).

(3) Consequences for the eigenvalue-pole correspondence. Deduce from (1)-(2) for which potentials of the class the negative eigenvalues can be recovered from the singular points of $S(k)$ alone.

References

1. Kunihiko Kodaira, The Eigenvalue Problem for Ordinary Differential Equations of the Second Order and Heisenberg's Theory of S-Matrices, American Journal of Mathematics 71 (1949), 921-945. DOI: 10.2307/2372377. URL: https://doi.org/10.2307/2372377
2. C. L. Dolph, J. B. McLeod, and D. Thoe, The analytic continuation of the resolvent kernel and scattering operator associated with the Schroedinger operator, Journal of Mathematical Analysis and Applications 16 (1966), 311-332. DOI: 10.1016/0022-247x(66)90174-0. URL: https://doi.org/10.1016/0022-247x(66)90174-0
3. J. B. McLeod, The analytic continuation to the unphysical sheet of the resolvent kernel associated with the Schroedinger operator, The Quarterly Journal of Mathematics 18 (1967), 219-231. DOI: 10.1093/qmath/18.1.219. URL: https://doi.org/10.1093/qmath/18.1.219
4. Richard Froese, Asymptotic Distribution of Resonances in One Dimension, Journal of Differential Equations 137 (1997), 251-272. DOI: 10.1006/jdeq.1996.3248. URL: https://doi.org/10.1006/jdeq.1996.3248
5. Michael Hitrik, Bounds on Scattering Poles in One Dimension, Communications in Mathematical Physics 208 (1999), 381-411. DOI: 10.1007/s002200050763. URL: https://doi.org/10.1007/s002200050763
6. B. M. Brown, M. S. P. Eastham, and D. K. R. McCormack, Resonances and analytic continuation for exponentially decaying Sturm–Liouville potentials, Journal of Computational and Applied Mathematics 116 (2000), 181-193. DOI: 10.1016/s0377-0427(99)00324-6. URL: https://doi.org/10.1016/s0377-0427(99)00324-6
7. B. M. Brown and M. S. P. Eastham, Analytic continuation and resonance-free regions for Sturm–Liouville potentials with power decay, Journal of Computational and Applied Mathematics 148 (2002), 49-63. DOI: 10.1016/s0377-0427(02)00572-1. URL: https://doi.org/10.1016/s0377-0427(02)00572-1. Preprint: arXiv:math/0006064, https://arxiv.org/abs/math/0006064
8. Evgeni Korotyaev, Inverse resonance scattering on the half line, Asymptotic Analysis 37 (2004), 215-226. DOI: 10.3233/asy-2004-605. URL: https://doi.org/10.3233/asy-2004-605
9. Semyon Dyatlov and Maciej Zworski, Mathematical Theory of Scattering Resonances, Graduate Studies in Mathematics 200, American Mathematical Society, Providence, RI, 2019. DOI: 10.1090/gsm/200. URL: https://doi.org/10.1090/gsm/200 (author's draft: https://math.mit.edu/~dyatlov/res/)
10. T. J. Christiansen and T. Cunningham, Singularities and Asymptotic Distribution of Resonances for Schrödinger Operators in One Dimension, Asymptotic Analysis 143 (2024), 619-640. DOI: 10.3233/asy-241928. URL: https://doi.org/10.3233/asy-241928. Preprint: arXiv:2311.00804, https://arxiv.org/abs/2311.00804
11. Denis A. W. White, Meromorphic Continuation of Scattering Matrices: Long Range, Stark Case, Communications in Partial Differential Equations 33 (2008), 131-156. DOI: 10.1080/03605300701588755. URL: https://doi.org/10.1080/03605300701588755