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 selfadjoint 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.