A remarkable spectral feature of the Schrödinger Hamiltonian of the harmonic oscillator perturbed by an attractive δ′-interaction centred at the origin: double degeneracy and level crossing**Dedicated to Professor Gianfausto Dell'Antonio on the occasion of his 80th birthday.

Abstract

We rigorously define the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ′-interaction, of strength β, centred at 0 (the bottom of the confining parabolic potential), by explicitly providing its resolvent. Our approach is based on a ‘coupling constant renormalization’, related to a technique originated in quantum field theory and implemented in the rigorous mathematical construction of the self-adjoint operator representing the negative Laplacian perturbed by the δ-interaction in two and three dimensions. The way the δ′-interaction enters in our Hamiltonian corresponds to the one originally discussed for the free Hamiltonian (instead of the harmonic oscillator one) by P Sěba. It should not be confused with the δ′-potential perturbation of the harmonic oscillator discussed, e.g., in a recent paper by Gadella, Glasser and Nieto (also introduced by P Sěba as a perturbation of the one-dimensional free Laplacian and recently investigated in that context by Golovaty, Hryniv and Zolotaryuk). We investigate in detail the spectrum of our perturbed harmonic oscillator. The spectral structure differs from that of the one-dimensional harmonic oscillator perturbed by an attractive δ-interaction centred at the origin: the even eigenvalues are not modified at all by the δ′-interaction. Moreover, all the odd eigenvalues, regarded as functions of β, exhibit the rather remarkable phenomenon called ‘level crossing’ after first producing the double degeneracy of all the even eigenvalues for the value (B( ·, ·) being the beta function).