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The exotic structure of the spectral $ζ$-function for the Schrödinger operator with Pöschl--Teller potential

Guglielmo Fucci, Jonathan Stanfill

TL;DR

The paper analyzes the spectral $\zeta$-function for a Schrödinger operator with the Pöschl--Teller potential, revealing that analytic continuation can yield an exotic meromorphic structure with countably many logarithmic branch points and parameter-dependent poles. By constructing a contour representation with the characteristic function $F_{\mu,\nu}^{(A,B)}$, it shows how different self-adjoint extensions yield markedly different $\zeta$-function structures, ranging from the usual simple-pole behavior to intricate combinations of poles and logarithmic branches. The authors derive explicit results for several extensions (notably Friedrichs and various separated/coupled types), relate some to Hurwitz or Riemann zeta-functions, and develop a regularized determinant framework when branch points occur at the origin. A thorough analytic continuation is carried out using asymptotic expansions and Mellin-type techniques, with special attention to cases where $\nu=p/q$ is rational. The findings illustrate how smoothing or perturbing the potential, as well as endpoint boundary conditions, can dramatically alter the spectral meromorphic structure, offering new examples of unusual zeta-function behavior in one dimension and guiding future work on related Sturm–Liouville systems.

Abstract

This work focuses on the analysis of the spectral $ζ$-function associated with a Schrödinger operator endowed with a Pöschl--Teller potential. We construct the spectral $ζ$-function using a contour integral representation and, for particular self-adjoint extensions, we perform its analytic continuation to a larger region of the complex plane. We show that the spectral $ζ$-function in these cases can possess a very unusual and remarkable structure consisting of a series of logarithmic branch points located at every nonpositive integer value of $s$ along with infinitely many additional branch points (and finitely many simple poles) whose locations depend on the parameters of the problem. By comparing the Pöschl--Teller potential to the classic Bessel potential, we further illustrate that perturbing a given potential by a smooth potential on a finite interval can greatly affect the meromorphic structure and branch points of the spectral $ζ$-function in surprising ways.

The exotic structure of the spectral $ζ$-function for the Schrödinger operator with Pöschl--Teller potential

TL;DR

The paper analyzes the spectral -function for a Schrödinger operator with the Pöschl--Teller potential, revealing that analytic continuation can yield an exotic meromorphic structure with countably many logarithmic branch points and parameter-dependent poles. By constructing a contour representation with the characteristic function , it shows how different self-adjoint extensions yield markedly different -function structures, ranging from the usual simple-pole behavior to intricate combinations of poles and logarithmic branches. The authors derive explicit results for several extensions (notably Friedrichs and various separated/coupled types), relate some to Hurwitz or Riemann zeta-functions, and develop a regularized determinant framework when branch points occur at the origin. A thorough analytic continuation is carried out using asymptotic expansions and Mellin-type techniques, with special attention to cases where is rational. The findings illustrate how smoothing or perturbing the potential, as well as endpoint boundary conditions, can dramatically alter the spectral meromorphic structure, offering new examples of unusual zeta-function behavior in one dimension and guiding future work on related Sturm–Liouville systems.

Abstract

This work focuses on the analysis of the spectral -function associated with a Schrödinger operator endowed with a Pöschl--Teller potential. We construct the spectral -function using a contour integral representation and, for particular self-adjoint extensions, we perform its analytic continuation to a larger region of the complex plane. We show that the spectral -function in these cases can possess a very unusual and remarkable structure consisting of a series of logarithmic branch points located at every nonpositive integer value of along with infinitely many additional branch points (and finitely many simple poles) whose locations depend on the parameters of the problem. By comparing the Pöschl--Teller potential to the classic Bessel potential, we further illustrate that perturbing a given potential by a smooth potential on a finite interval can greatly affect the meromorphic structure and branch points of the spectral -function in surprising ways.

Paper Structure

This paper contains 23 sections, 2 theorems, 151 equations.

Table of Contents

  1. Introduction and main results
  2. Pöschl--Teller potential and spectral zeta-function
  3. The unusual structure of the spectral zeta-function associated with certain extensions
  4. The zeta-regularized functional determinant
  5. Structure of paper
  6. The characteristic function and some extensions with explicit spectrum
  7. The analytic continuation of the spectral zeta-function
  8. Case notzero
  9. zero
  10. zero
  11. zero
  12. zero
  13. zero
  14. The derivative of the spectral zeta-function at zero
  15. Concluding remarks
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $T_{\mu,\nu}^{(\alpha,\beta)}$ denote the self-adjoint extension of the separated boundary condition for the minimal operator associated with 2.1. Then the following $(i)$--$(iv)$ hold$:$ Let $\nu=p/q$ with $p,q\in{\mathbb N}$, $0< p\leqslant q-1$, and $p,q$ relatively prime. Then, generically,See Sections sub3.1.1, sub3.1.2, and Remark RemB3$)$ for the exact notion of generic in this setting

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Remark B.1
  • Remark B.2
  • Remark B.3