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June 2013 Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun’s differential equation
Masato Wakayama
Proc. Japan Acad. Ser. A Math. Sci. 89(6): 69-73 (June 2013). DOI: 10.3792/pjaa.89.69

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.

Citation

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Masato Wakayama. "Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun’s differential equation." Proc. Japan Acad. Ser. A Math. Sci. 89 (6) 69 - 73, June 2013. https://doi.org/10.3792/pjaa.89.69

Information

Published: June 2013
First available in Project Euclid: 31 May 2013

zbMATH: 1278.34099
MathSciNet: MR3079292
Digital Object Identifier: 10.3792/pjaa.89.69

Subjects:
Primary: 34L40
Secondary: 34M05, 81Q10, 81S05

Rights: Copyright © 2013 The Japan Academy

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Vol.89 • No. 6 • June 2013
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