Proc. Japan Acad. Ser. A Math. Sci. 89 (6), 69-73, (June 2013) DOI: 10.3792/pjaa.89.69
KEYWORDS: Non-commutative harmonic oscillators, lowest eigenvalue, multiplicity of eigenvalues, oscillator representation, Heun’s differential equation, Riemann’s scheme, 34L40, 81Q10, 34M05, 81S05
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.