Pole structure of the Hamiltonian ζ-function for a singular potential

Abstract

We study the pole structure of the ζ-function associated with the Hamiltonian H of a quantum mechanical particle living in the half-line R⁺, subject to the singular potential gx⁻² + x². We show that H admits nontrivial self-adjoint extensions (SAE) in a given range of values of the parameter g. The ζ-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.