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Conditional gauge and potential theory for the Schrödinger operator. (English) Zbl 0652.60076
The note is concerned with the Schrödinger operator \(L=-A+q\) on a Euclidean domain \(D\subset R\) n(n\(\geq 3)\). Probabilistic methods are used in order to study the conditional gauge theorem and this is then applied again to obtain potential-theoretic results for L. The domain D is assumed to be bounded and Lipschitz. The operator A will be uniformly elliptic and of the form, \[ A=\sum^{d}_{i,j=1}(\partial /\partial x\quad i)(a_{ij}(x)\partial /\partial x\quad j)\quad with\quad a_{ij}\in L^{\infty}(D,m)\quad and\quad symmetric. \] The function q satisfies the Kato conditions, i.e., \[ \lim_{r\downarrow 0}\sup_{x\in D}\int_{| x-y| \leq r}| x-y|^{- d+2}q(y)dy=0\quad. \] Under these hypotheses the following conditional gauge theorem is proved:
Let \(P\) \(x_ y\) be the law for the diffusion X with infinitesimal generator A started at \(x\in \bar D\) but conditioned to converge to \(y\in \bar D\) at the path life-time \(\tau_ D\). Define \[ F(x,y)=E\quad x_ y(e_ q(\tau_ D))\quad where\quad e_ q(\tau_ D)=\exp \{- \int^{\tau_ D}_{0}q(X_ s)ds\} \] as the conditional gauge. If there exists a point \((x_ 0,y_ 0)\in \bar D\times \bar D\) with \(x_ 0\neq y_ 0\) and \(F(x_ 0,y_ 0)<\infty\), then \[ 0<\inf_{x,y\in \bar D}F(x,y)\leq \sup_{x,y\in \bar D}F(x,y)<\infty \quad (Th.\quad 4.2). \] Furthermore, assume that \(F\neq \infty\), then the Martin boundary for L on D is identical with its topological boundary \(\partial D\) and any positive solution of the equation L u\(=0\) admits a Martin representation (Th. 5.5). Under the same conditions all points on \(\partial D\) are regular for the Dirichlet problem for L.
Reviewer: Nguyen Xuan Loc

MSC:
60J45 Probabilistic potential theory
31B25 Boundary behavior of harmonic functions in higher dimensions
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