A remarkable spectral feature of the Schrodinger Hamiltonian of the harmonic oscillator perturbed by an attractive delta %26apos;-interaction centred at the origin: double degeneracy and level crossing

作者：Albeverio Sergio^*; Fassari Silvestro; Rinaldi Fabio

来源：Journal of Physics A-Mathematical and Theoretical, 2013, 46(38): 385305.

DOI：10.1088/1751-8113/46/38/385305

We rigorously define the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive delta%26apos;-interaction, of strength beta, centred at 0 (the bottom of the confining parabolic potential), by explicitly providing its resolvent. Our approach is based on a %26apos;coupling constant renormalization%26apos;, 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 delta-interaction in two and three dimensions. The way the delta%26apos;-interaction enters in our Hamiltonian corresponds to the one originally discussed for the free Hamiltonian (instead of the harmonic oscillator one) by P Seba. It should not be confused with the delta%26apos;-potential perturbation of the harmonic oscillator discussed, e.g., in a recent paper by Gadella, Glasser and Nieto (also introduced by P Seba 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 delta%26apos;-interaction centred at the origin: the even eigenvalues are not modified at all by the delta%26apos;-interaction. Moreover, all the odd eigenvalues, regarded as functions of beta, exhibit the rather remarkable phenomenon called %26apos;level crossing%26apos; after first producing the double degeneracy of all the even eigenvalues for the value beta = beta(0) = 2 root pi/B(3/4, 1/2) congruent to 1.47934(B(., .) being the beta function).

• 出版日期2013-9-27
• 单位中国石油大学（北京）