The initial aim of the present paper is to provide a complete description of the eigenvalue problem for the non-commutative harmonic oscillator (NcHO), which is defined by a (two-by-two) matrix-valued self-adjoint parity-preserving ordinary differential operator , in terms of Heun's ordinary differential equations, the second-order Fuchsian differential equations with four regular singularities in a complex domain. This description has been achieved for odd eigenfunctions in Ochiai  nicely but missing up to now for the even parity which is more important from the viewpoint of determination of the ground state of the NcHO. As a by-product of this study examining the monodromy data (characteristic exponents, etc.) of the Heun equation, we prove that the multiplicity of the eigenvalue of the NcHO is at most two. Moreover, we give a condition for the existence of a finite-type eigenfunction (i.e., given by essentially a finite sum of Hermite functions) for the eigenvalue problem and an explicit example of such eigenvalues, from which one finds that doubly degenerate eigenstates of the NcHO actually exist even in the same parity Also, we determine the possible shape of (so-called) Heun polynomial solutions of the Heun equations, which are obtained by the eigenvalue problem.
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