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

Falomir, H.; Pisani, P. A. G.; Wipf, A.

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

Authors

Falomir, H.

Pisani, P. A. G.

Wipf, A.

Description

We study the pole structure of the $ζ$-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 $ζ$-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.
12 pages, 1 figure, RevTeX. References added. Version to appear in Jour. Phys. A: Math. Gen