Letters in Mathematical Physics

, Volume 70, Issue 2, pp 133–139

Non-commutative Harmonic Oscillators and the Connection Problem for the Heun Differential Equation

Article

abstract

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.

Keywords

Heun’s equation monodromy algebraic solution accessory parameter apparent singularity 

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Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  1. 1.Department of MathematicsNagoya UniversityChikusa, NagoyaJapan

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