Verified numerical computations for eigenvalues of non-commutative harmonic oscillators

We study the eigenvalue problem of the formally self-adjoint operator Q_(αβ) ≡ I_(αβ) (-1/2 d²/dx² + x²/2) + J (x d/dx + 1/2), x ∈ R, where I_(αβ) ≡ equation omitted, J ≡ equation omitted ∈ Mat₂ (R), and α and β are positive real constants satisfying αβ > 1. This latter condition makes the operator Q_(αβ) to be elliptic. The operator Q_(αβ) defines a sort of non-trivial couple of the usual harmonic oscillators and was first introduced and studied by Parmeggiani et al. (Parmeggiani, A.; Wakayama, M. Non-Commutative Harmonic Oscillators-I. Forum Mathematicum 2002, 14, 539-604). However, since one has only a limited understanding of its explicit value of eigenvalues, a behavior of eigenfunctions and an information about the multiplicity of each eigenvalue, here we try to make a numerical approach to this system. More precisely, applying a numerical enclosure method for elliptic eigenvalue problems which is based on the verification procedure for nonlinear elliptic equations established by Nagatou (Nagatou, K. A Numerical Method to Verify the Elliptic Eigenvalue Problems Including a Uniqueness Property. Computing 1999, 63, 109-130; Nakao, M.T.; Yamamoto, N.; Nagatou, K. Numerical Verifications for Eigenvalues of Second-Order Elliptic Operators. Japan Journal of Industrial and Applied Mathematics 1999, 16, 307-320) to such coupling type eigenvalue problems in the unbounded domain, we develop a verified numerical computation of the eigenvalue of Q_(αβ) and its multiplicity.