The behaviour of the three-dimensional Hamiltonian −∆ + λ [δ(x + x0) + δ(x − x0)] as the distance between the two centres vanishes
https://doi.org/10.17586/2220-8054-2017-8-2-153-159
Abstract
In this note, we continue our analysis of the behavior of self-adjoint Hamiltonians with symmetric double wells given by twin point interactions perturbing various types of “free Hamiltonians” as the distance between the two centers shrinks to zero. In particular, by making the coupling constant to be renormalized and also dependent on 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 studied in detail by Albeverio and collaborators, 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 negative three-dimensional Laplacian 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 non-zero.
Supporting agencies: Fabio Rinaldi wishes to express his heartfelt gratitude to Prof. I. Popov for his 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 – Pierre Duclos Workshop”, held at ITMO University, St. Petersburg, Russian Federation (14–15 November 2016). S. Fassari gratefully acknowledges financial support from the “Grants for Visiting Researchers at the Campus of International Excellence Triangular-E3”, as part of the “Attraction of Excellent Researchers and Stays for Visiting Researchers Program”, carried out under the subvention of the Ministry of Education, Culture and Sports to the Campus of International Excellence Triangular-E3. Partial financial support is acknowledged to the Spanish Junta de Castilla y Leon (VA057U16) and MINECO (Project MTM2014-57129-C2-1-P). S. Fassari also wishes to thank the entire staff at Departamento de F´ısica Teorica, At´omica y ´Optica, Universidad de Valladolid, for their warm ´hospitality throughout his stay.
About the Authors
S. Albeverio
Institut fur Angewandte Mathematik, HCM, IZKS, BiBoS, Universitat Bonn; CERFIM
Germany
Germany
Endenicherallee 60, D-53115 Bonn; PO Box 1132, CH-6601 Locarno
S. Fassari
CERFIM; Universita’ degli Studi Guglielmo Marconi; Universidad de Valladolid
Switzerland
Switzerland
Departamento de F´ısica Teorica, At´omica y ´Optica
PO Box 1132, CH-6601 Locarno, Switzerland; Via Plinio 44, I-00193 Rome; E-47011 Valladolid
F. Rinaldi
CERFIM; Universita’ degli Studi Guglielmo Marconi
Switzerland
Switzerland
PO Box 1132, CH-6601 Locarno; Via Plinio 44, I-00193 Rome
References
1. Albeverio S., Fassari S., Rinaldi F. The discrete spectrum of the spinless Salpeter Hamiltonian perturbed by δ-interactions. Journal of Physics A: Mathematical and Theoretical, 2015, 48 (18), P. 185301.
2. 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. Nanosystems: Physics, Chemistry, Mathematics, 2016, 7 (2), P. 268–289.
3. 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), P. 803–815.
4. Fassari S., Rinaldi F. On the spectrum of the Schrodinger Hamiltonian with a particular configuration of three point interactions. Reports on Mathematical Physics, 2009, 64 (3), P. 367–393.
5. Fassari S., Rinaldi F. On the spectrum of the Schrodinger Hamiltonian of the one-dimensional harmonic oscillator perturbed by two identical attractive point interactions. Reports on Mathematical Physics, 2012, 69 (3), P. 353–370.
6. Albeverio S., Gesztesy F., Høegh-Krohn R., Holden H. Solvable models in Quantum Mechanics AMS (Chelsea Series) second ed., 2004.
7. Kato T. Perturbation Theory for Linear Operators. Springer Verlag, Heidelberg-NY, 1995.
8. Reed M., Simon B. Analysis of Operators – Methods of Modern Mathematical Physics IV. Academic Press NY, 1978.
9. Bhaduri R.K., Chatterjee A., Van Zyl B.P. An elementary exposition of the Efimov effect. American Journal of Physics, 2011, 79 (3), P. 274–281.
10. Glasser M.L., Nieto L.M. The energy level structure of a variety of one-dimensional confining potentials and the effects of a local singular perturbation. Canadian Journal of Physics, 2015, 93 (12), P. 1588.
11. Lorinczi J., Malecki J. Spectral properties of the massless relativistic harmonic oscillator. ¨ Journal of Differential Equations, 2012, 253 (10), P. 2846–2871.
Review
For citations:
Albeverio S., Fassari S., Rinaldi F. The behaviour of the three-dimensional Hamiltonian −∆ + λ [δ(x + x0) + δ(x − x0)] as the distance between the two centres vanishes. Nanosystems: Physics, Chemistry, Mathematics. 2017;8(2):153-159. https://doi.org/10.17586/2220-8054-2017-8-2-153-159
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