Harmonic oscillator perturbed by a decreasing scalar potential

In this paper we study the perturbation L = H + V , where H = - d 2 m d x 2 m + x 2 m on R , m ∈ N ∗ and V is a decreasing scalar potential. Let λ k be the k th eigenvalue of H . We suppose that the eigenvalues of L around λ k can be written in the form λ k + μ k . The main result of the paper is an...

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Bibliographic Details
Published inJournal of pseudo-differential operators and applications Vol. 11; no. 1; pp. 141 - 157
Main Authors Aarab, Ilias, Tagmouti, Mohamed Ali
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.03.2020
Springer Nature B.V
Subjects
Algebra
Analysis
Applications of Mathematics
Eigenvalues
Functional Analysis
Harmonic oscillators
Mathematics
Mathematics and Statistics
Operator Theory
Partial Differential Equations
Perturbation
Variations
Perturbation theory
Eigenvalue asymptotics
Averaging method
Secondary 00A00
Primary 99Z99
Pseudo-differential operator
Spectrum
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Summary:In this paper we study the perturbation L = H + V , where H = - d 2 m d x 2 m + x 2 m on R , m ∈ N ∗ and V is a decreasing scalar potential. Let λ k be the k th eigenvalue of H . We suppose that the eigenvalues of L around λ k can be written in the form λ k + μ k . The main result of the paper is an asymptotic formula for fluctuation { μ k } which is given by a transformation of V . In the case m = 1 we recover a result on the harmonic oscillator.
ISSN:1662-9981
1662-999X
DOI:10.1007/s11868-019-00284-4