Proceedings of the Japan Academy, Series A, Mathematical Sciences

Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun’s differential equation

Masato Wakayama

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Abstract

The non-commutative harmonic oscillator (NcHO) is a special type of self-adjoint ordinary differential operator with non-commutative coefficients. In the present note, we aim to provide a reasonable criterion that derives the simplicity of the lowest eigenvalue of NcHO. It actually proves the simplicity of the lowest eigenvalue for a large class of structure parameters. Moreover, this note describes a certain equivalence between the spectral problem of the NcHO (for the even parity) and existence of holomorphic solutions of Heun’s ordinary differential equations in a complex domain. The corresponding Riemann scheme allows us to give another proof to the criterion.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 89, Number 6 (2013), 69-73.

Dates
First available: 31 May 2013

Permanent link to this document
http://projecteuclid.org/euclid.pja/1370004861

Digital Object Identifier
doi:10.3792/pjaa.89.69

Mathematical Reviews number (MathSciNet)
MR3079292

Subjects
Primary: 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.)
Secondary: 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 34M05: Entire and meromorphic solutions 81S05: Canonical quantization, commutation relations and statistics

Keywords
Non-commutative harmonic oscillators lowest eigenvalue multiplicity of eigenvalues oscillator representation Heun’s differential equation Riemann’s scheme

Citation

Wakayama, Masato. Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun’s differential equation. Proceedings of the Japan Academy, Series A, Mathematical Sciences 89 (2013), no. 6, 69--73. doi:10.3792/pjaa.89.69. http://projecteuclid.org/euclid.pja/1370004861.


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