Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin II
https://doi.org/10.17586/2220-8054-2016-7-5-803-815
Abstract
In this note, we continue our analysis (started in [1]) of the isotropic three-dimensional harmonic oscillator perturbed by a pair of identical attractive point interactions symmetrically situated with respect to the origin, that is to say, the mathematical model describing a symmetric quantum dot with a pair of point impurities. In particular, by making the coupling constant (to be renormalized) dependent also upon the separation distance between the two impurities, we prove that it is possible to rigorously define the unique self-adjoint Hamiltonian that, differently from the one introduced in [1], behaves smoothly as the separation distance between the impurities shrinks to zero. In fact, we rigorously prove that the Hamiltonian introduced in this note converges in the norm-resolvent sense to that of the isotropic three-dimensional harmonic oscillator perturbed by a single attractive point interaction situated at the origin having double strength, thus making this three dimensional model more similar to its one-dimensional analog (not requiring the renormalization procedure) as well as to the three-dimensional model involving impurities given by potentials whose range may even be physically very short but different from zero. Moreover, we show the manifestation of the Zeldovich effect, known also as level rearrangement, in the model investigated herewith. More precisely, we take advantage of our renormalization procedure to demonstrate the possibility of using the concept of ‘Zeldovich spiral’, introduced in the case of perturbations given by rapidly decaying potentials, also in the case of point perturbations.
Keywords
level crossing,
degeneracy,
point interactions,
renormalisation,
Schr¨ odinger operators,
quantum dots,
perturbed quantum oscillators,
Zeldovich effect,
level rearrangement
Supporting agencies: We wish to thank Prof. Igor Yu. Popov (Chair of Higher Mathematics, ITMO University, St. Petersburg, Russian Federation) for his kind invitation to contribute to this special issue dedicated to the memory of our dear friend Boris Pavlov. The first author had the great luck to meet him in the 80s, first during a meeting in Dubna, and was greatly impressed by his bright mind, his contagious enthusiasm for mathematics, that he transmitted to a large number of students. One of the topics we discussed was the subject of point interactions. He gave fundamental contributions to this area and, more generally, to spectral theory. The contacts with him and his research associates happily accompanied our further scientific life. The second author had the privilege of meeting him in February 1991 as we were both visiting Prof. Albeverio in Bochum. It was a great pleasure, shared also by the third author, to see him twenty-four years later in St. Petersburg on the occasion of the international conference ‘Mathematical Challenge of Quantum Transport in Nanosystems’, held at ITMO University, St. Petersburg, Russian Federation (9–11 September 2015). It is with great gratitude that we acknowledge his scientific legacy and dedicate this work to his memory.
About the Authors
S. Albeverio
Institut f¨ur Angewandte Mathematik, HCM, IZKS, BiBoS, Universit¨ at Bonn; CERFIM; Chair Professorship, Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals
Germany
Germany
Endenicheralee 60, D-53115 Bonn, Germany
PO Box 1132, CH-6601 Locarno, Switzerland
Dhahran, KSA
S. Fassari
CERFIM; Universit` a degli Studi Guglielmo Marconi
Italy
Italy
PO Box 1132, CH-6601 Locarno, Switzerland
Via Plinio 44, I-00193 Rome, Italy
F. Rinaldi
CERFIM; Universit` a degli Studi Guglielmo Marconi
Italy
Italy
PO Box 1132, CH-6601 Locarno, Switzerland
Via Plinio 44, I-00193 Rome, Italy
References
1. Albeverio S., Fassari S., Rinaldi F. Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive-impurities symmetrically situated around the origin. Nanosystems: Physics, Chemistry, Mathematics, 2016, 7 (2), P. 268–289.
2. Schmidtke H.-H. Quantenchemie. Weinheim: VCH, 1987 (in German).
3. Byers Brown W., Steiner E. On the electronic energy of a one-electron diatomic molecule near the united atom. Journal of Chemical Physics, 1966, 44, P. 3934.
4. Klaus M. On H+ 2 for small internuclear separation. Journal of Physics A: Mathematical and General, 1983, 16, P. 2709–2720.
5. Albeverio S., Fassari S., Rinaldi F. The discrete spectrum of the spinless Salpeter Hamiltonian perturbed by-interactions. Journal of Physics A: Mathematical and Theoretical, 2015, 48 (18), P. 185301.
6. Albeverio S., Gesztesy F., Hegh-Krohn R., Holden H. Solvable models in Quantum Mechanics. AMS (Chelsea Series) second edition, 2004.
7. Albeverio S., Kurasov P. Singular Perturbations of Differential Operators: Solvable Type Operators. Cambridge University Press, 2000.
8. Albeverio S., Fassari S., Rinaldi F. A remarkable spectral feature of the Schr¨ odinger Hamiltonian of the harmonic oscillator perturbed by an attractive-interaction centred at the origin: double degeneracy and level crossing. Journal of Physics A: Mathematical and Theoretical, 2013, 46 (38), P. 385305.
9. Fassari S., Inglese G. Spectroscopy of a three-dimensional isotropic harmonic oscillator with a-type perturbation. Helvetica Physica Acta, 1996, 69, P. 130–140.
10. Fassari S., Rinaldi F. On the spectrum of the Schr¨ odinger Hamiltonian of the one-dimensional harmonic oscillator perturbed by two identical attractive point interactions. Reports on Mathematical Physics, 2012, 69 (3), P. 353–370.
11. Reed M., Simon B. Fourier Analysis, Self-adjointness– Methods of Modern Mathematical Physics II, Academic Press NY, 1975.
12. Exner P., Neidhardt H., Zagrebnov V.A. Potential approximations to ’: an inverse Klauder phenomenon with norm resolvent convergence. Communications in Mathematical Physics, 2001, 22, P. 4593–4612.
13. Fassari S., Rinaldi F. On the spectrum of the Schr¨ odinger Hamiltonian with a particular configuration of three point interactions. Reports on Mathematical Physics, 2009, 64 (3), P. 367–393.
14. Albeverio S., Fassari S., Rinaldi F. The Hamiltonian of the harmonic oscillator with an attractive ’-interaction centred at the origin as approximated by the one with a triple of attractive-interactions. Journal of Physics A: Mathematical and Theoretical, 2016, 49 (2), P. 025302.
15. Fassari S., Inglese G. On the spectrum of the harmonic oscillator with a-type perturbation. Helvetica Physica Acta, 1994, 67, P. 650–659.
16. Fassari S., Inglese G. On the spectrum of the harmonic oscillator with a-type perturbation II. Helvetica Physica Acta, 1997, 70, P. 858–865.
17. Combescure M., Khare A., et al. Level rearrangement in exotic atoms and quantum dots. International Journal of Modern Physics B, 2006, 21 (22), P. 3765.
18. Zeldovich Ya.B. Energy levels in a distorted Coulomb field. Soviet Journal Solid State, 1960, 1, P. 1497.
19. Farrell A., MacDonald Z., van Zyl B. The Zeldovich effect in harmonically trapped, ultra-cold quantum gases. Journal of Physics A: Mathematical and Theoretical, 2012, 45, P. 045303.
20. Br¨uning J., Geyler V., Lobanov I. Spectral properties of a short-range impurity in a quantum dot. Journal of Mathematical Physics, 2004, 45, P. 1267–1290.
21. Plancherel M., Rotach W. Sur les valeurs asymptotiques des polynomes d’Hermite. Commentarii Mathematici Helvetici, 1929, 1 (1), P. 227–254.
22. Reed M., Simon B. Functional Analysis– Methods of Modern Mathematical Physics I. Academic Press, NY, 1972.
23. Mityagin B., Siegl P. Root system of singular perturbations of the harmonic oscillator type operators. Letters in Mathematical Physics, 2016, 106 (2), P. 147–167
Review
For citations:
Albeverio S., Fassari S., Rinaldi F. Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin II. Nanosystems: Physics, Chemistry, Mathematics. 2016;7(5):803-815. https://doi.org/10.17586/2220-8054-2016-7-5-803-815
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