Probabilistic approach to boundary value problems for Schrödinger’s equation. (English) Zbl 0577.60066

The paper contains a short survey of recent results on the probabilistic representation of the boundary problem solutions for the Schrödinger equation (1) \((\Delta /2+q(x))\phi =0\), \(x\in {\mathcal D}\), where \({\mathcal D}\) is a bounded domain in \({\mathbb{R}}^ d\), \(d\geq 2\), and (2) \(\phi =f|_{x\in \partial {\mathcal D}}\) or (3) \((\partial \phi /\partial n)|_{x\in \partial {\mathcal D}}=f\). The gauge for the problem (1)-(2) is the function \[ u(x)=E^ x\{\exp \int_{0}^{\tau_{{\mathcal D}}}q(B_ t)dt\} \] where \(B_ t\) is the Brownian motion in \({\mathbb{R}}^ d\), \(\tau_{{\mathcal D}}\) is the first exit time, \(E^ x\) is the mathematical expectation starting at x.
The gauge theorem: If u is not identically \(+\infty\) in \({\mathcal D}\) then it is bounded there. In this case the unique solution of the Dirichlet problem (1)-(2) is given by \[ u_ f(x)=E^ x\{\exp [\int_{0}^{\tau_{{\mathcal D}}}q(B_ t)dt]f(B(\tau_{{\mathcal D}})\quad)\}. \] For a regular domain, bounded Hölder continuous q and continuous f the theorem was proved by the author and K. M. Rao [Stochastic processes, Semin. Evanston/Ill. 1981, Progr. Probab. Stat. 1, 1-29 (1981; Zbl 0492.60073)]. For the q(x) satisfying the condition \[ \lim_{t\downarrow 0}\sup_{x\in {\mathcal D}}E^ x\{\int^{t}_{0}1_ D| q(B_ t)| dt\}=\quad 0 \] (the condition was introduced by A. Aizenman and B. Simon in Commun. Pure Appl. Math. 35, 209- 273 (1982; Zbl 0459.60069)], the theorem was extended by Zhongxin Zhao [Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 13-18 (1983; Zbl 0521.60074)] and for the Neumann boundary problem (1), (3) by the author and Hsu (to appear in Semin. Stoch. Proc.).
Reviewer: L.Pastur


60H25 Random operators and equations (aspects of stochastic analysis)
60J65 Brownian motion
35J10 Schrödinger operator, Schrödinger equation
35J25 Boundary value problems for second-order elliptic equations