Decay and regularity for harmonic oscillator-type equations

We prove super-exponential decay estimates and holomorphic extensions for the solutions of harmonic oscillator type-equations. The functional setting for our estimates is given by the Gelfand-Shilov classes $S^{\mu}_{\nu}(\R^n),$ cf. Introduction. In the one-dimensional case, explicit solutions are given in terms of special functions of hypergeometric confluent type.