Spectral Zeta Functions for Spherical Aharonov-Bohm Quantum Bags

TL;DR

This paper investigates spectral zeta functions for quantum systems in spherical domains with magnetic flux, deriving exact sum rules for Bessel function zeros to improve numerical ground state energy calculations.

Contribution

It introduces a novel approach to find exact sum rules for Bessel zeros, aiding spectral analysis in quantum systems with magnetic flux, especially for non-integer and non-half-integer orders.

Findings

Derived exact sum rules for zeros of Bessel functions.

Provided methods for numerical approximation of ground state energies.

Enhanced understanding of spectral properties in Aharonov-Bohm quantum bags.

Abstract

We study the sum $\ds\zeta_H(s)=\sum_j E_j^{-s}$ over the eigenvalues $E_j$ of the Schrdinger equation in a spherical domain with Dirichlet walls, threaded by a line of magnetic flux. Rather than using Green's function techniques, we tackle the mathematically nontrivial problem of finding exact sum rules for the zeros of Bessel functions $J_{\nu}$, which are extremely helpful when seeking numerical approximations to ground state energies. These results are particularly valuable if $\nu$ is neither an integer nor half an odd one.