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

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

2001-12-11

arxiv.org/abs/math-ph/0112019v2

J.Phys.A35:5427-5444,2002

Physics

Mathematical Physics

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

Scientific paper

10.1088/0305-4470/35/26/306

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.

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