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

In this presentation, we wish to provide an overview of the spectral features for the selfadjoint Hamiltonian of the threedimensional isotropic harmonic oscillator perturbed by either a single attractive interaction centered at the origin or by a pair of identical attractive interactions symmetrically situated with respect to the origin. Given that such Hamiltonians represent the mathematical model for quantum dots with sharply localized impurities, we cannot help having the renowned article by Br¨uning, Geyler and Lobanov [1] as our key reference. We shall also compare the spectral features of the aforementioned threedimensional models with those of the selfadjoint Hamiltonian of the harmonic oscillator perturbed by an attractive 0interaction 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 selfadjoint Hamiltonian for the singular double well model will be provided through the explicit formula for its resolvent (Green’s function). Furthermore, by studying in detail the equation determining the ground state energy for the double well model, it will be shown that the concept of “positional disorder”, introduced in [1] in the case of a quantum dot with a single impurity, can also be extended to the model with the twin impurities in the sense that the greater the distance between the two impurities is, the less localized the ground state will be. Another noteworthy spectral phenomenon will also be determined; for each value of the distance between the two centers below a certain threshold value, there exists a range of values of the strength of the twin point interactions for which the first excited symmetric bound state is more tightly bound than the lowest antisymmetric bound state. Furthermore, it will be shown that, as the distance between the two impurities shrinks to zero, the 3DHamiltonian with the singular double well (requiring renormalization to be defined) does not converge to the one with a single interaction centered at the origin having twice the strength, in contrast to its onedimensional analog for which no renormalization is required. It is worth stressing that this phenomenon has also been recently observed in the case of another model requiring the renormalization of the coupling constant, namely the onedimensional Salpeter Hamiltonian perturbed by two twin attractive interactions symmetrically situated at the same distance from the origin.