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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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
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Abstract 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.
AbstractList In this paper we study the perturbation L=H+V, where H=-d2mdx2m+x2m on R, m∈N∗ and V is a decreasing scalar potential. Let λk be the kth eigenvalue of H. We suppose that the eigenvalues of L around λk can be written in the form λk+μk. The main result of the paper is an asymptotic formula for fluctuation {μk} which is given by a transformation of V. In the case m=1 we recover a result on the harmonic oscillator.
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.
Author Aarab, Ilias
Tagmouti, Mohamed Ali
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Keywords Perturbation theory
Eigenvalue asymptotics
Averaging method
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Pseudo-differential operator
Spectrum
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D Gurarie (284_CR4) 1987; 112
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Snippet 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...
In this paper we study the perturbation L=H+V, where H=-d2mdx2m+x2m on R, m∈N∗ and V is a decreasing scalar potential. Let λk be the kth eigenvalue of H. We...
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StartPage 141
SubjectTerms Algebra
Analysis
Applications of Mathematics
Eigenvalues
Functional Analysis
Harmonic oscillators
Mathematics
Mathematics and Statistics
Operator Theory
Partial Differential Equations
Perturbation
Variations
Title Harmonic oscillator perturbed by a decreasing scalar potential
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