The two lowest eigenvalues of the harmonic oscillator in the presence of a Gaussian perturbation
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The two lowest eigenvalues of the harmonic oscillator in the presence of a Gaussian perturbation S. Fassari1,2,3,a
, L. M. Nieto4,b
, F. Rinaldi2,3,c
1 Department of Higher Mathematics, ITMO University, St. Petersburg, Russian Federation 2 CERFIM, PO Box 1132, 6601 Locarno, Switzerland 3 Dipartimento di Fisica Nucleare, Subnucleare e delle Radiazioni, Univ. degli Studi Guglielmo Marconi,
Via Plinio 44, 00193 Rome, Italy
4 Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, 47011 Valladolid,
Spain Received: 15 May 2020 / Accepted: 7 September 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this note, we consider a one-dimensional quantum mechanical particle constrained by a parabolic well perturbed by a Gaussian potential. As the related Birman– Schwinger operator is trace class, the Fredholm determinant can be exploited in order to compute the modified eigenenergies which differ from those of the harmonic oscillator due to the presence of the Gaussian perturbation. By taking advantage of Wang’s results on scalar products of four eigenfunctions of the harmonic oscillator, it is possible to evaluate quite accurately the two lowest lying eigenvalues as functions of the coupling constant λ.
1 Introduction As is well known, the harmonic oscillator is one of the very few solvable quantum models; that is to say, its eigenfunctions and eigenvalues can be expressed analytically. As a consequence, any quantum mechanics textbook such as [1] contains a chapter devoted to its detailed description. This remarkable property has stimulated researchers to study various types of models involving perturbations of the harmonic oscillator over many decades. The interested reader can find a brief review of the literature on time-independent perturbations of the harmonic oscillator in [2] (see also [3] and [4]). Although the Birman–Schwinger principle was used in [5] and [6] to investigate the spectral effects of a particular type of short-range perturbation of the one-dimensional harmonic oscillator, namely a Lorentzian perturbation, the method is clearly applicable to any absolutely summable potential. Given our recent interest in various quantum models involving Gaussian perturbations (see [2,7–9]), we have decided to make use of the abovementioned principle in order to investigate the modifications of the discrete spectrum of the
a e-mail: [email protected] (corresponding author) b e-mail: [email protected] c e-mail: [email protected]
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Eur. Phys. J. Plus
(2020) 135:728
one-dimensional harmonic oscillator once a Gaussian perturbation is added to the Hamiltod2 2 ≥ 1. nian H0 = 21 − dx + x 2 2 Furthermore, we have been motivated to study such a model due to the lack of relevant contributions to the existing literature in theoretical/mathematical physics. As a matter of fact, to the best of our knowledge, the most relevant work on this model is to be foun
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