The Energy of the Ground State of the Two-Dimensional Hamiltonian of a Parabolic Quantum Well in the Presence of an Attractive Gaussian Impurity

Abstract

In this article, we provide an expansion (up to the fourth order of the coupling constant) of the energy of the ground state of the Hamiltonian of a quantum mechanical particle moving inside a parabolic well in the x-direction and constrained by the presence of a two-dimensional impurity, modelled by an attractive two-dimensional isotropic Gaussian potential. By investigating the associated Birman–Schwinger operator and exploiting the fact that such an integral operator is Hilbert–Schmidt, we use the modified Fredholm determinant in order to compute the energy of the ground state created by the impurity.

On Hermite Functions, Integral Kernels, and Quantum Wires

“…We have already shown in [11] that, although each summand K 0 ,n (x, x , y, y ) is the integral kernel of a trace class operator, its sum N 0 (x, x , y, y ) is not. Instead, N 0 (x, x , y, y ) is the integral kernel of a Hilbert-Schmidt operator.…”

“…was achieved in [11] by using the explicit Expression (11). Here, we wish to generalise that argument by estimating the following scalar product instead:…”

“…As was done in [11], for a given value of the coupling constant λ, the energy of the ground state is given by E 0 (λ) = 1 2 − 0 (λ), with 0 > 0. Then, γ 0 ( 0 ) = √ 2 0 .…”

“…In recent papers [10,11], we have considered a two-dimensional quasi-solvable model that combines several properties such as simplicity in the Hamiltonian with a mathematical complexity to solve the eigenvalue problem associated to this Hamiltonian, even for the calculation of the energy of the ground state, which is already a real challenge. In addition, this model serves as a two-dimensional limit for a quantum wire.…”

“…Then, γ 0 ( 0 ) = √ 2 0 . For this value of E, the integral kernel, denoted by B 0 , can be divided for the sake of convenience into the sum of three terms [11]:…”