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 β = β 0 = being the beta function).

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In this note we provide an alternative way of defining the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive - interaction, of strength , centred at 0 (the bottom of the confining parabolic potential), that was rigorously defined in a previous paper by means of a ‘coupling constant renormalisation’. Here we get the Hamiltonian as a norm resolvent limit of the harmonic oscillator Hamiltonian perturbed by a triple of attractive -interactions, thus extending the Cheon–Shigehara approximation to the case in which a confining harmonic potential is present.

We rigorously define the self-adjoint one-dimensional Salpeter Hamiltonian perturbed by an attractive of strength centred at the origin, by explicitly providing its resolvent. Our approach is based on a ‘coupling constant renormalization’, a technique used first heuristically in quantum field theory and implemented in the rigorous mathematical construction of the self-adjoint operator representing the negative Laplacian perturbed by the in two and three dimensions. We show that the spectrum of the self-adjoint operator consists of the absolutely continuous spectrum of the free Salpeter Hamiltonian and an eigenvalue given by a smooth function of the parameter The method is extended to the model with two twin attractive deltas symmetrically situated with respect to the origin in order to show that the discrete spectrum of the related self-adjoint Hamiltonian consists of two eigenvalues, namely the ground state energy and that of the excited antisymmetric state. We investigate in detail the dependence of these two eigenvalues on the two parameters of the model, that is to say both the aforementioned strength and the separation distance. With regard to the latter, a remarkable phenomenon is observed: differently from the well-behaved Schrödinger case, the 1D-Salpeter Hamiltonian with two identical Dirac distributions symmetrically situated with respect to the origin does not converge, as the separation distance shrinks to zero, to the one with a single centred at the origin having twice the strength. However, the expected behaviour in the limit (in the norm resolvent sense) can be achieved by making the coupling of the twin deltas suitably dependent on the separation distance itself.