  We propose a new solvable class of multidimensional quantum harmonic
oscillators for a linear diffusive particle and a quadratic energy absorbing
well associated with a semi-definite positive matrix force. Under natural and
easily checked controllability conditions, the ground state and the zero-point
energy are explicitly computed in terms of a positive fixed point of a
continuous time algebraic Riccati matrix equation. We also present an explicit
solution of normalized and time dependent Feynman-Kac measures in terms of a
time varying linear dynamical system coupled with a differential Riccati matrix
equation. A refined non asymptotic analysis of the stability of these models is
developed based on a recently developed Floquet-type representation of time
varying exponential semigroups of Riccati matrices. We provide explicit and non
asymptotic estimates of the exponential decays to equilibrium of Feynman-Kac
semigroups in terms of Wasserstein distances or Boltzmann-relative entropy. For
reversible models we develop a series of functional inequalities including de
Bruijn identity, Fisher's information decays, log-Sobolev inequalities, and
entropy contraction estimates. In this context, we also provide a complete and
explicit description of all the spectrum and the excited states of the
Hamiltonian, yielding what seems to be the first result of this type for this
class of models. We illustrate these formulae with the traditional harmonic
oscillator associated with real time Brownian particles and Mehler's formula.
The analysis developed in this article can also be extended to solve time
dependent Schrodinger equations equipped with time varying linear diffusions
and quadratic potential functions.
