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

Silvestro Fassari,Luis Miguel Nieto,Fabio Rinaldi,Manuel Gadella

doi:10.3390/sym13091561

Silvestro Fassari, Luis Miguel Nieto + Show 2 more

Open Access

https://doi.org/10.3390/sym13091561

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Journal: Symmetry	Publication Date: Aug 25, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: Marconi University, University of Valladolid

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.

Highlights

The study of solvable models in quantum physics has drawn a great deal of interest over the last four decades
Resonances may be defined in various ways, not always equivalent, their equivalence may be shown for a wide range of solvable models
As attested in [28], the same spectral features appear even more spectacularly when the harmonic confinement gets replaced by the square pyramidal one or by a mixture of the type 12 ( x2 + |y|). They have been studied to a far lesser extent than quantum models with point interactions, other potentials/interactions leading to solvable or quasi-solvable models have been considered in the relevant literature

Summary

Introduction

The study of solvable models in quantum physics has drawn a great deal of interest over the last four decades. As attested in [28], the same spectral features appear even more spectacularly when the harmonic confinement gets replaced by the square pyramidal one or by a mixture of the type 12 ( x2 + |y|) They have been studied to a far lesser extent than quantum models with point interactions, other potentials/interactions leading to solvable or quasi-solvable models have been considered in the relevant literature. We have made an attempt to go beyond those results by providing an accurate approximation of the energy value of the ground state up to the fourth order on the coupling constant To accomplish this objective, we have taken advantage of some advanced mathematical machinery, and bound states are obtained as the zeroes of the modified Fredholm determinant (to be defined in Section 3) of the operator I − λBE , where I is the identity, λ is the coupling constant and BE is the Birman–Schwinger operator for the total Hamiltonian. For the sake of clarity, we have collected all the relevant mathematical results in Appendix A

Preliminaries

Calculation of the Ground State Energy

Final Remarks