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Probabilistic representation of solutions of Schrödinger equations. (Chinese. English summary) Zbl 0595.60076

This paper is about the probabilistic representation for a solution u of the Schrödinger equation \((\Delta +2q)u=0\) in a domain D as studied by K. L. Chung and K. M. Rao [Stochastic processes, Semin. Evanston Ill. 1981, Progr. Probab. Stat. 1, 1-29 (1981; Zbl 0492.60073)]. It is proved that a condition for the solution u to be expressed in the form \[ u(x)=E_ x[\exp (\int^{\tau_ D}_{0}q(x(s))ds)\phi (x(\tau_ D))],\quad x\in D, \] is as follows: for every \(x\in D,\) there exists \[ \lim_{t\uparrow \tau_ D}u(x(t))\quad (a.s.\quad P_ x), \] and \[ u(x)=E_ x[\exp (\int^{\tau_ D}_{0}q(x(s))ds)\lim_{t\uparrow \tau_ D}u(x(t))]. \] The existence and uniqueness of solution of the stochastic Dirichlet problem for the Schrödinger equation is also proved.
Reviewer: Chengxun Wu

MSC:

60J45 Probabilistic potential theory
60H25 Random operators and equations (aspects of stochastic analysis)
35J10 Schrödinger operator, Schrödinger equation
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Citations:

Zbl 0492.60073
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