, Volume 70, Issue 2, pp 133-139

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

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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.

The research of the author is supported in part by a Grant-in-Aid for Scientific Research (B) (No. 15340005) from the Ministry of Education, Culture, Sports, Science and Technology.
Mathematics Subject classifications (2000). primary, 34M35, secondary, 33E20.
This revised version was published online in March 2005 with corrections to the cover date.