### Abstract

We study the eigenvalue problem of the formally self-adjoint operator Q_{(αβ)} ≡ I_{(αβ)} (-1/2 d^{2}/dx^{2} + x^{2}/2) + J (x d/dx + 1/2), x ∈ R, where I_{(αβ)} ≡ equation omitted, J ≡ equation omitted ∈ Mat_{2} (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 language | English |
---|---|

Pages (from-to) | 633-650 |

Number of pages | 18 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 23 |

Issue number | 5-6 |

DOIs | |

Publication status | Published - 2002 Aug |

Externally published | Yes |

### Fingerprint

### Keywords

- Elliptic eigenvalue problems
- Harmonic oscillators
- Numerical enclosure

### ASJC Scopus subject areas

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

### Cite this

*Numerical Functional Analysis and Optimization*,

*23*(5-6), 633-650. https://doi.org/10.1081/NFA-120014756

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

Research output: Contribution to journal › Article

*Numerical Functional Analysis and Optimization*, vol. 23, no. 5-6, pp. 633-650. https://doi.org/10.1081/NFA-120014756

}

TY - JOUR

T1 - Verified numerical computations for eigenvalues of non-commutative harmonic oscillators

AU - Nagatou, K.

AU - Nakao, M. T.

AU - Wakayama, M.

PY - 2002/8

Y1 - 2002/8

N2 - 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.

AB - 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.

KW - Elliptic eigenvalue problems

KW - Harmonic oscillators

KW - Numerical enclosure

UR - http://www.scopus.com/inward/record.url?scp=0036697655&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036697655&partnerID=8YFLogxK

U2 - 10.1081/NFA-120014756

DO - 10.1081/NFA-120014756

M3 - Article

VL - 23

SP - 633

EP - 650

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 5-6

ER -