Spectral zeta functions of a 1D Schr\"odinger problem

TL;DR

This paper analyzes spectral zeta functions of a 1D Schrödinger problem with specific potentials, deriving closed-form evaluations and identities, and extending methods to PT-symmetric problems with applications in quantum field theory.

Contribution

It introduces a novel approach using the quantum Wronskian to evaluate spectral zeta functions and applies it to PT-symmetric problems and integrable quantum field theories.

Findings

Closed-form evaluations for second zeta functions.

Derived identities involving hypergeometric series.

Extended methods to PT-symmetric eigenvalue problems.

Abstract

We study the spectral zeta functions associated to the radial Schr\"odinger problem with potential V(x)=x^{2M}+alpha x^{M-1}+(lambda^2-1/4)/x^2. Using the quantum Wronskian equation, we provide results such as closed-form evaluations for some of the second zeta functions i.e. the sum over the inverse eigenvalues squared. Also we discuss how our results can be used to derive relationships and identities involving special functions, using a particular 5F_4 hypergeometric series as an example. Our work is then extended to a class of related PT-symmetric eigenvalue problems. Using the fused quantum Wronskian we give a simple method for calculating the related spectral zeta functions. This method has a number of applications including the use of the ODE/IM correspondence to compute the (vacuum) nonlocal integrals of motion G_n which appear in an associated integrable quantum field theory.