NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2016, 7 (2), P. 268–289
Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin
S. Albeverio – Institut für Angewandte Mathematik, HCM, IZKS, BiBoS, Universität Bonn, Endenicheralee 60, D-53115 Bonn, Germany; CERFIM, PO Box 1132, CH-6601 Locarno; Centre Interfacultaire Bernoulli, EPFL, CH-1015 Lausanne, Switzerland; Chair Professorship, Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, KSA
S. Fassari – CERFIM, PO Box 1132, CH-6601 Locarno; ISR, Aeulistr.10, CH-9470 Buchs, Switzerland; Università degli Studi Guglielmo Marconi, Via Plinio 44, I-00193 Rome, Italy; firstname.lastname@example.org
F. Rinaldi – CERFIM, PO Box 1132, CH-6601 Locarno, Switzerland; Università degli Studi Guglielmo Marconi, Via Plinio 44, I-00193 Rome, Italy; BIS Group of Institutions, Gagra-Moga, Punjab under Punjab Technical University, Punjab, India
In this presentation, we wish to provide an overview of the spectral features for the self-adjoint Hamiltonian of the three-dimensional isotropic harmonic oscillator perturbed by either a single attractive δ-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 Brüning, Geyler and Lobanov  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). 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  in the case of a quantum dot with a single δ-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 3D-Hamiltonian with the singular double well (requiring renormalization to be defined) does not converge to the one with a single δ-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 the case of another model requiring the renormalization of the coupling constant, namely the one-dimensional Salpeter Hamiltonian perturbed by two twin attractive δ-interactions symmetrically situated at the same distance from the origin.
Keywords: level crossing, degeneracy, point interactions, renormalisation, Schrödinger operators, quantum dots, perturbed quantum oscillators.
PACS 02.30.Gp, 02.30.Hq, 02.30.Hq, 02.30.Lt, 02.30.Sa, 02.30.Tb, 03.65.Db, 03.65.Ge, 68.65.Hb