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

We rigorously define the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ '− interaction centred at 0 (the bottom of the confining parabolic potential), formally written as.... by explicitly providing its resolvent. Our approach is based on a “coupling constant renormalisation”, 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. Furthermore, we investigate in detail the spectrum of such a perturbed harmonic oscillator. The spectral structure is the opposite of the one 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 (the parameter representing the “strength” of the δ '−interaction), exhibit the rather remarkable phenomenon called “level crossing” after first producing the double degeneracy of all the even eigenvalues for the value βo approx.=1.47934. 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. Available from: https://www.researchgate.net/publication/235662695_A_remarkable_spectral_feature_of_the_Schrodinger_Hamiltonian_of_the_harmonic_oscillator_perturbed_by_an_attractive_d%27interaction_centred_at_the_origin_double_degeneracy_and_level_crossing [accessed May 14, 2017].