The Birman-Schwinger Operator for a Parabolic Quantum Well in a Zero-Thickness Layer in the Presence of a Two-Dimensional Attractive Gaussian Impurity

Frontiers in Physics

Manuel Gadella²  Luis M. Nieto²  Silvestro Fassari³  Fabio Rinaldi³  Sergio Albeverio⁴ 

[1] CERFIM, Locarno, Switzerland;Departamento de Física Teórica, Atómica y Óptica and IMUVA, Valladolid, Spain;Dipartimento di Fisica Nucleare, Subnucleare e delle Radiazioni, Università degli Studi Guglielmo Marconi, Rome, Italy;Institut für Angewandte Mathematik, HCM, Universität Bonn, Bonn, Germany;

关键词: Gaussian potential;    Birman-Schwinger operator;    Hilbert-Schmidt operator;    contact interaction;    quantum well;   

DOI  :  10.3389/fphy.2019.00102

来源: DOAJ

【 摘 要 】

In this note we consider a quantum mechanical particle moving inside an infinitesimally thin layer constrained by a parabolic well in the x-direction and, moreover, in the presence of an impurity modeled by an attractive Gaussian potential. We investigate the Birman-Schwinger operator associated to 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 bound states created by the impurity. Furthermore, we consider the case where the Gaussian potential degenerates to a δ-potential in the x-direction and a Gaussian potential in the y-direction. We construct the corresponding self-adjoint Hamiltonian and prove that it is the limit in the norm resolvent sense of a sequence of corresponding Hamiltonians with suitably scaled Gaussian potentials. Satisfactory bounds on the ground state energies of all Hamiltonians involved are exhibited.

【 授权许可】

Unknown