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

H. Falomir·P. A. G. Pisani·A. Wipf
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Abstract

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.

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H. Falomir, P. A. G. Pisani, A. Wipf. 2002-05-13. Pole structure of the Hamiltonian $ζ$-function for a singular potential. https://doi.org/10.1088/0305-4470%2F35%2F26%2F306

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