Original Research ARTICLE
The Birman-Schwinger operator for a zero-thickness layer in the presence of an attractive Gaussian impurity
- 1Università degli Studi Guglielmo Marconi, Italy
- 2University of Bonn, Germany
- 3Institute for Applied Mathematics, University of Bonn, Germany
- 4Departamento de Fisica Teorica, Atomica y Optica, Universidad de Valladolid, Spain
In this note we are concerned with the limiting case of a zero-thickness layer with harmonic conﬁnement along one of the two available dimensions. We investigate the Birman-Schwinger operator for such a model assuming the presence of a Gaussian impurity inside the layer and prove that such an integral operator is Hilbert-Schmidt, which allows the use of the modiﬁed Fredholm determinant in order to compute the impurity bound states. Furthermore, we consider the Hamiltonian H0 −λ√πδ(x)e−y2, that is to say the energy operator with the interaction term having a point interaction in place of the Gaussian along the x-direction, and prove that such an operator is self-adjoint as well as that it is the limit in the norm resolvent sense of the sequence H0 −λne−(n2x2+y2) as n →∞.
Keywords: Quantum wire, compact operator, Birman-Schwinger method, 2D Hamiltonian, Contact interaction
Received: 30 Mar 2019;
Accepted: 02 Jul 2019.
Edited by:Manuel Asorey, University of Zaragoza, Spain
Copyright: © 2019 Fassari, Albeverio, Rinaldi, Gadella and Nieto. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
* Correspondence: Dr. Silvestro Fassari, Università degli Studi Guglielmo Marconi, Rome, Italy, firstname.lastname@example.org