najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2016-7-2-268-289najo-743Research ArticleINVITED SPEAKERSINVITED SPEAKERSSpectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the originSpectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the originAlbeverioS.AlbeverioS.

Universitat Bonn, Endenicheralee 60, D-53115 Bonn

PO Box 1132, CH-6601 Locarno

EPFL, CH-1015 Lausanne

Dhahran, KSA

 

Endenicheralee 60, D-53115 Bonn

PO Box 1132, CH-6601 Locarno

CH-1015 Lausanne

Dhahran, KSA

silvestro.fassari@isr.chFassariS.

PO Box 1132, CH-6601 Locarno

Aeulistr.10, CH-9470 Buchs

Via Plinio 44, I-00193 Rome

RinaldiF.

PO Box 1132, CH-6601 Locarno

Via Plinio 44, I-00193 Rome

Punjab

Institut fur Angewandte Mathematik, HCM, IZKS, BiBoS, Universitat Bonn; CERFIM; Centre Interfacultaire Bernoulli; Chair Professorship, Department of Mathematics and Statistics, King Fahd University of Petroleum and MineralsGermanyCERFIM; ISR; Universita degli Studi Guglielmo MarconiSwitzerlandCERFIM; Universit`a degli Studi Guglielmo Marconi; BIS Group of Institutions, Gagra-Moga, Punjab under Punjab Technical UniversitySwitzerland20161308202572Special Issue268289Copyright © Albeverio S., Fassari S., Rinaldi F., 20252025Albeverio S., Fassari S., Rinaldi F.Albeverio S., Fassari S., Rinaldi F.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/743

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 δ-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 Bruning, 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). 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 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.

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 δ-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 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 [1] 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.

level crossingdegeneracypoint interactionsrenormalisationSchrodinger operatorsquantum dotsperturbed quantum oscillatorsSilvestro Fassari and Fabio Rinaldi wish to express their heartfelt gratitude to Prof. I. Popov and Prof. I. Lobanov for their kind invitation to present the main contents of this note during one of the sessions of the conference “Mathematical Challenge of Quantum Transport in Nanosystems”, held at ITMO University, St. Petersburg, Russian Federation (9 – 11 September 2015). Sergio Albeverio is very grateful to Prof. Nicolas Monod, Director of “Centre Interfac- ultaire Bernoulli – EPFL”, Lausanne, for the warm hospitality during his stay there as organiser jointly with Prof. A. B. Cruzeiro and Prof. D. Holm of the semester programme ”Geometric Mechanics, Variational and Stochastic Methods” (January – June 2015). S. Fassari and F. Rinaldi also wish to thank Prof. A. B. Cruzeiro and Prof. J. C. Zambrini for their warm hospitality on the occasion of their visit to CIB – EPFL.Silvestro Fassari and Fabio Rinaldi wish to express their heartfelt gratitude to Prof. I. Popov and Prof. I. Lobanov for their kind invitation to present the main contents of this note during one of the sessions of the conference “Mathematical Challenge of Quantum Transport in Nanosystems”, held at ITMO University, St. Petersburg, Russian Federation (9 – 11 September 2015). Sergio Albeverio is very grateful to Prof. Nicolas Monod, Director of “Centre Interfac- ultaire Bernoulli – EPFL”, Lausanne, for the warm hospitality during his stay there as organiser jointly with Prof. A. B. Cruzeiro and Prof. D. Holm of the semester programme ”Geometric Mechanics, Variational and Stochastic Methods” (January – June 2015). S. Fassari and F. Rinaldi also wish to thank Prof. A. B. Cruzeiro and Prof. J. C. 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The authors declare that there are no conflicts of interest present.