Bound States Energies of a Harmonic Oscillator Perturbed by Point Interactions

Abstract

We determine explicitly the exact transcendental bound states energies equation for a one-dimensional harmonic oscillator perturbed by a single and a double point interactions via Green’s function techniques using both momentum and position space representations. The even and odd solutions of the problem are discussed. The corresponding limiting cases are recovered. For the harmonic oscillator with a point interaction in more than one dimension, divergent series appear. We use to remove this divergence an exponential regulator and we obtain a transcendental equation for the energy bound states. The results obtained here are consistent with other investigations using different methods.

Time-independent Green’s function of a quantum simple harmonic oscillator system and solutions with additional generic delta-function potentials

“…From equations (55) and (56), we know that b = 1 for even function v n (x) and b = -1 for the odd function v n (x). We note that in passing the above transcendental equations are much simpler than those shown in [21].…”

Variational approach to the Schrödinger equation with a delta-function potential

“…A quantum-mechanical Hamiltonian operator H perturbed by a delta-function potential gδ(x) has received considerable attention [1][2][3][4][5][6][7][8][9][10][11][12][13]. In most cases H describes a free particle [1], a particle in a box [1][2][3][4] or the harmonic oscillator [1,[5][6][7][8][9][10][11][12][13].…”

“…A quantum-mechanical Hamiltonian operator H perturbed by a delta-function potential gδ(x) has received considerable attention [1][2][3][4][5][6][7][8][9][10][11][12][13]. In most cases H describes a free particle [1], a particle in a box [1][2][3][4] or the harmonic oscillator [1,[5][6][7][8][9][10][11][12][13]. Since in these cases the Schrödinger equation for H is exactly solvable one can obtain closed form expressions for the solutions to the Schrödinger equation for H g = H + gδ(x) in several different ways.…”

“…Since in these cases the Schrödinger equation for H is exactly solvable one can obtain closed form expressions for the solutions to the Schrödinger equation for H g = H + gδ(x) in several different ways. For example, from the eigenvalues and eigenfunctions of H [1,8], by solving the eigenvalue equation left and right of the origin and matching those solutions at x = 0 [2,3,5,9,10] or by means of the Green function [11,12]. In some cases the authors resorted to this kind of models to illustrate the application of approximate methods like perturbation theory [4,9], WKM method [9], or variational approaches [9,13].…”

Exact calculation of the trace of the Birman–Schwinger operator of the one-dimensional harmonic oscillator perturbed by an attractive Gaussian potential

“…The same model was subsequently investigated by means of the integral operator isospectral to the Birman-Schwinger operator in [27]. A renewed interest in this model, to a great extent motivated by its applications to Bose-Einstein condensates, has led to more recent contributions such as [28][29][30][31][32][33][34][35][36]. The case in which the Dirac distribution is not centered at the origin was analyzed in [37] (see also [28]) while the model with two identical deltas symmetrically situated about the origin was thoroughly investigated in [38] (see also [28]).…”