The Wigner(–Poisson)–Fokker–Planck equation with singular potential

Abstract

This paper is concerned with the Wigner(–Poisson)–Fokker–Planck equation with coupling to the confining potential which is a singular perturbation of the harmonic oscillator potential. In this work, we study the global (local) existence and uniqueness of these solutions on different weighted- space, respectively. The main difficulties for establishing global (local) mild solution are to derive a-priori estimates on the appropriate potential terms and obtain dissipativity of the quantum Fokker–Planck operator. The proofs are based on semigroup theory and -weighted Hardy’s inequality.

Keywords:

• Wigner(–Poisson)–Fokker–Planck equation
• singular potential
• semigroup theory
• weighted Hardy’s inequality

AMS Subject Classifications:

• 35K55
• 47B44
• 35Q40

Notes

No potential conflict of interest was reported by the authors.