The Hamiltonian of the harmonic oscillator with an attractive $\delta ^{\prime} $-interaction centred at the origin as approximated by the one with a triple of attractive $\delta $-interactions

Abstract

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.

Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin

“…"by using instead the alternative renormalization (3.6), the location of all the level crossings would be exactlyα (3) =0, leading to the graph shown in Fig. 4(a) of the aforementioned paper by Brüning, Geyler and Lobanov".…”

“…In this work, we wish to further extend our previous research on various types of point perturbations of Schrödinger Hamiltonians with or without harmonic confinement (see, e.g., [2][3][4][5][6][7][8][9][10][11]) to the physical models utilized to describe three-dimensional quantum dots.…”

“…In Section 2, by analyzing the behavior of the new energy levels created by a point perturbation as functions of the parameter labeling the self-adjoint extensions, physically characterized by being proportional to the inverse scattering length (see [1,4]), we are going to stress the analogy of the presence of the so-called "level crossings" between the newly created eigenenergies of bound states with a given symmetry and those of the unperturbed harmonic oscillator unaffected by the singular perturbation pertaining to bound states having the opposite symmetry. We will also underline the difference between this three-dimensional model and the one-dimensional one, studied in [2,3], in which the harmonic oscillator is perturbed by a so-called δ -interaction, whose quadratic form is given by the expression |δ δ |.…”

“…Given that such Hamiltonians represent the mathematical model for quantum dots with sharply localized impurities, we cannot help having the renowned article by Brüning, Geyler and Lobanov [1] as our key reference. We shall also compare the spectral features of the aforementioned three-dimensional models with those of the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ -interaction in one dimension, fully investigated in [2,3], given the existence in both models of the remarkable spectral phenomenon called "level crossing". The rigorous definition of the self-adjoint Hamiltonian for the singular double well model will be provided through the explicit formula for its resolvent (Green's function).…”

Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin II

“…The new self-adjoint Hamiltonian H {β, x0} , clearly dependent on x 0 and obtained as the norm resolvent limit after removing the energy cut-off (Theorem 2.1), is shown to approach smoothly H 2β in the norm resolvent limit as x 0 → 0 + (Theorem 2.2). We would like to stress that, although this is exactly the strategy employed also in papers such as [12][13][14] to obtain the self-adjoint operator with the δ -interaction perturbing either the negative Laplacian or the Hamiltonian of the harmonic oscillator in one dimension as the norm resolvent limit of Hamiltonians with the perturbation consisting of a triple of δ-interactions, the dependence on x 0 is completely different.…”