Pole structure of the Hamiltonian $\zeta$-function for a singular potential

TL;DR

This paper investigates the pole structure of the $ta$-function for a quantum Hamiltonian with a singular potential, revealing how self-adjoint extensions influence pole positions and residues, which can be irrational or non-half-integers.

Contribution

It characterizes the pole structure of the $ta$-function for Hamiltonians with singular potentials, highlighting the dependence on self-adjoint extensions and the resulting complex pole behavior.

Findings

$ta$-functions have poles depending on the extension parameter $g$

Poles can be irrational and not half-integers

Residues vary with the chosen self-adjoint extension

Abstract

We study the pole structure of the $\zeta$-function associated to the Hamiltonian $H$ of a quantum mechanical particle living in the half-line $\mathbf{R}^+$, subject to the singular potential $g x^{-2}+x^2$. We show that $H$ admits nontrivial self-adjoint extensions (SAE) in a given range of values of the parameter $g$. The $\zeta$-functions of these operators present poles which depend on $g$ and, in general, do not coincide with half an integer (they can even be irrational). The corresponding residues depend on the SAE considered.