2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/185733We 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. GenMathematical PhysicsHigh Energy Physics - TheoryFunctional AnalysisSpectral TheoryQuantum Physics81Q10, 34L05, 34L40text