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

TL;DR

This paper introduces a new class of multidimensional quantum harmonic oscillators for linear diffusive particles, providing explicit solutions for ground states, energy, and Feynman-Kac measures, along with stability analysis and spectral properties.

Contribution

It develops explicit solutions and stability estimates for a novel class of solvable quantum harmonic oscillators with quadratic potentials, extending to time-dependent Schrödinger equations.

Findings

Explicit ground state and zero-point energy formulas.

Non-asymptotic decay estimates for Feynman-Kac semigroups.

Complete spectral characterization of the Hamiltonian.

Abstract

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.