Non-commutative Harmonic Oscillators and the Connection Problem for the Heun Differential Equation
- Hiroyuki Ochiai
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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.
- Heun’s differential equations. With contributions by Arscott, F. M., Slavyanov, S. Yu., Schmidt, D., Wolf, G., Maroni, P. and Duval, A.: Edited by Ronveaux, A., Oxford University Press, New York, 1995.
- Iwasaki, K., Kimura, H., Shimomura, S. and Yoshida, M.: From Gauss to Painlev\’e, a modern theory of special functions, Vieweg, Aspects Math. 16 (1991).
- Nagatou, K., Nakao, M.T., Wakayama, K. (2002) Verified numerical computations for eigenvalues of non-commutative harmonic oscillators. Numer. Funct. Anal. Optim. 23: pp. 633-650
- Ochiai, H. (2001) Non-commutative harmonic oscillators and Fuchsian ordinary differential operators. Comm. Math. Phys. 217: pp. 357-373
- Parmeggiani, A., Wakayama, M. (2001) Oscillator representations and systems of ordinary differential equations. Proc. Nat. Acad. Sci. U.S.A. 98: pp. 26-30
- Parmeggiani, A. and Wakayama, M.: Non-commutative harmonic oscillators. I, II. Forum Math. 14 (2002), 539–604, 669--690.
- Non-commutative Harmonic Oscillators and the Connection Problem for the Heun Differential Equation
Letters in Mathematical Physics
Volume 70, Issue 2 , pp 133-139
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- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
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- Heun’s equation
- algebraic solution
- accessory parameter
- apparent singularity
- Hiroyuki Ochiai (1)
- Author Affiliations
- 1. Department of Mathematics, Nagoya University, Chikusa, Nagoya, 464-8602, Japan