Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive delta-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 three-dimensional isotropic harmonic oscillator perturbed by either a single attractive delta- interaction centered at theorigin or by a pair of identical attractive delta- interactions symmetrically situated with respect to the origin. Given thatsuch Hamiltonians represent the mathematical model for quantum dots with sharply localized impurities, we cannothelp having the renowned article by Bruening, Geyler and Lobanov [1] as our key reference. We shall also comparethe spectral features of the aforementioned threedimensionalmodels with those of the selfadjointHamiltonian of the harmonic oscillator perturbed by an attractive delta-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 selfadjointHamiltonian for the singular double well model will be provided through the explicitformula for its resolvent (Green’s function). Furthermore, by studying in detail the equation determining the groundstate 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 impuritiesin 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 3D-Hamiltonianwith 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 one-dimensional analog for which no renormalization is required. It is worth stressing that this phenomenon has also been recently observed in thecase of another model requiring the renormalization of the coupling constant, namely the one-dimensional SalpeterHamiltonian perturbed by two twin attractive delta- interactionssymmetrically situated at the same distance from the origin.