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We propose a new solvable class of multidimensional quantum harmonicoscillators for a linear diffusive particle and a quadratic energy absorbingwell associated with a semi-definite positive matrix force. Under natural andeasily checked controllability conditions, the ground state and the zero-pointenergy are explicitly computed in terms of a positive fixed point of acontinuous time algebraic Riccati matrix equation. We also present an explicitsolution of normalized and time dependent Feynman-Kac measures in terms of atime varying linear dynamical system coupled with a differential Riccati matrixequation. A refined non asymptotic analysis of the stability of these models isdeveloped based on a recently developed Floquet-type representation of timevarying exponential semigroups of Riccati matrices. We provide explicit and nonasymptotic estimates of the exponential decays to equilibrium of Feynman-Kacsemigroups in terms of Wasserstein distances or Boltzmann-relative entropy. Forreversible models we develop a series of functional inequalities including deBruijn identity, Fisher's information decays, log-Sobolev inequalities, andentropy contraction estimates. In this context, we also provide a complete andexplicit description of all the spectrum and the excited states of theHamiltonian, yielding what seems to be the first result of this type for thisclass of models. We illustrate these formulae with the traditional harmonicoscillator associated with real time Brownian particles and Mehler's formula.The analysis developed in this article can also be extended to solve timedependent Schrodinger equations equipped with time varying linear diffusionsand quadratic potential functions.

Quantum harmonic oscillators and Feynman-Kac path integrals for linear diffusive particles