Verified numerical computations for eigenvalues of non-commutative harmonic oscillators

K. Nagatou, M. T. Nakao, M. Wakayama

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

We study the eigenvalue problem of the formally self-adjoint operator Q(αβ) ≡ I(αβ) (-1/2 d2/dx2 + x2/2) + J (x d/dx + 1/2), x ∈ R, where I(αβ) ≡ equation omitted, J ≡ equation omitted ∈ Mat2 (R), and α and β are positive real constants satisfying αβ > 1. This latter condition makes the operator Q(αβ) to be elliptic. The operator Q(αβ) defines a sort of non-trivial couple of the usual harmonic oscillators and was first introduced and studied by Parmeggiani et al. (Parmeggiani, A.; Wakayama, M. Non-Commutative Harmonic Oscillators-I. Forum Mathematicum 2002, 14, 539-604). However, since one has only a limited understanding of its explicit value of eigenvalues, a behavior of eigenfunctions and an information about the multiplicity of each eigenvalue, here we try to make a numerical approach to this system. More precisely, applying a numerical enclosure method for elliptic eigenvalue problems which is based on the verification procedure for nonlinear elliptic equations established by Nagatou (Nagatou, K. A Numerical Method to Verify the Elliptic Eigenvalue Problems Including a Uniqueness Property. Computing 1999, 63, 109-130; Nakao, M.T.; Yamamoto, N.; Nagatou, K. Numerical Verifications for Eigenvalues of Second-Order Elliptic Operators. Japan Journal of Industrial and Applied Mathematics 1999, 16, 307-320) to such coupling type eigenvalue problems in the unbounded domain, we develop a verified numerical computation of the eigenvalue of Q(αβ) and its multiplicity.

Original languageEnglish
Pages (from-to)633-650
Number of pages18
JournalNumerical Functional Analysis and Optimization
Volume23
Issue number5-6
DOIs
Publication statusPublished - 2002 Aug
Externally publishedYes

Fingerprint

Harmonic Oscillator
Numerical Computation
Eigenvalue Problem
Eigenvalue
Enclosures
Eigenvalues and eigenfunctions
Elliptic Problems
Mathematical operators
Numerical methods
Multiplicity
Numerical Verification
Nonlinear Elliptic Equations
Mathematica
Enclosure
Unbounded Domain
Operator
Self-adjoint Operator
Applied mathematics
Elliptic Operator
Japan

Keywords

  • Elliptic eigenvalue problems
  • Harmonic oscillators
  • Numerical enclosure

ASJC Scopus subject areas

  • Analysis
  • Signal Processing
  • Computer Science Applications
  • Control and Optimization

Cite this

Verified numerical computations for eigenvalues of non-commutative harmonic oscillators. / Nagatou, K.; Nakao, M. T.; Wakayama, M.

In: Numerical Functional Analysis and Optimization, Vol. 23, No. 5-6, 08.2002, p. 633-650.

Research output: Contribution to journalArticle

Nagatou, K. ; Nakao, M. T. ; Wakayama, M. / Verified numerical computations for eigenvalues of non-commutative harmonic oscillators. In: Numerical Functional Analysis and Optimization. 2002 ; Vol. 23, No. 5-6. pp. 633-650.
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