najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2017-8-2-153-159najo-642Research ArticleMATHEMATICSМАТЕМАТИКАThe behaviour of the three-dimensional Hamiltonian −∆ + λ [δ(x + x0) + δ(x − x0)] as the distance between the two centres vanishesAlbeverioS.

Endenicherallee 60, D-53115 Bonn; PO Box 1132, CH-6601 Locarno

FassariS.

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

sifassari@gmail.comRinaldiF.

PO Box 1132, CH-6601 Locarno; Via Plinio 44, I-00193 Rome

Institut fur Angewandte Mathematik, HCM, IZKS, BiBoS, Universitat Bonn; CERFIMGermanyCERFIM; Universita’ degli Studi Guglielmo Marconi; Universidad de ValladolidSwitzerlandCERFIM; Universita’ degli Studi Guglielmo MarconiSwitzerland20171208202582153159Copyright © 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/642

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.

point interactionsrenormalisationSchrodinger operatorsquantum dotsFabio 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. 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The authors declare that there are no conflicts of interest present.