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We study the pole structure of the $\zeta$-function associated to theHamiltonian $H$ 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 valuesof the parameter $g$. The $\zeta$-functions of these operators present poleswhich depend on $g$ and, in general, do not coincide with half an integer (theycan even be irrational). The corresponding residues depend on the SAEconsidered.Comment: 12 pages, 1 figure, RevTeX. References added. Version to appear in  Jour. Phys. A: Math. Ge

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