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

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.

12 pages, 1 figure, RevTeX. References added. Version to appear in Jour. Phys. A: Math. Gen

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