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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Nagoya University, Chikusa, Nagoya, 464-8602, Japan

    Hiroyuki Ochiai

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  1. Hiroyuki Ochiai
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Correspondence to Hiroyuki Ochiai.

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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.

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Ochiai, H. Non-commutative Harmonic Oscillators and the Connection Problem for the Heun Differential Equation. Lett Math Phys 70, 133–139 (2004). https://doi.org/10.1007/s11005-004-4292-5

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  • DOI: https://doi.org/10.1007/s11005-004-4292-5

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