Proceedings of the Japan Academy, Series A, Mathematical Sciences
- Proc. Japan Acad. Ser. A Math. Sci.
- Volume 89, Number 6 (2013), 69-73.
Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun’s differential equation
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.
Proc. Japan Acad. Ser. A Math. Sci. Volume 89, Number 6 (2013), 69-73.
First available in Project Euclid: 31 May 2013
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.)
Secondary: 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 34M05: Entire and meromorphic solutions 81S05: Canonical quantization, commutation relations and statistics
Wakayama, Masato. 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 (2013), no. 6, 69--73. doi:10.3792/pjaa.89.69. https://projecteuclid.org/euclid.pja/1370004861