Spectral Zeta Functions of a 1D Schrödinger Problem

DOI

10.1142/S140292511250026XHow to use a DOI?

Keywords

Spectral zeta; hypergeometric series; 𝒫𝒯-symmetric; integrals of motion

Abstract

We study the spectral zeta functions associated to the radial Schrödinger problem with potential V(x) = x^2M + αx^M−1 + (λ² − 1/4)/x². After directly computing some of the zeta functions, we use the quantum Wronskian equation to give sum rules between them, allowing for instances where the explicit form of the zeta functions can be simplified. An immediate application of this work is to derive functional relations and identities involving hypergeometric series, allowing for known identities to be found as instances of more general results. 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 indirectly calculating the associated spectral zeta functions. This method is then applied to calculate the nonlocal integrals of motion G_n which appear in an associated integrable quantum field theory.

Copyright

© 2012 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Joe Watkins
PY  - 2012
DA  - 2012/09/20
TI  - Spectral Zeta Functions of a 1D Schrödinger Problem
JO  - Journal of Nonlinear Mathematical Physics
SP  - 428
EP  - 444
VL  - 19
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1142/S140292511250026X
DO  - 10.1142/S140292511250026X
ID  - Watkins2012
ER  -