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Green’s functions for the Schrödinger equation with short-range potentials. (English) Zbl 0728.35027

Let \(\Omega\) be an open set with compact boundary in \(R^ d\) (d\(\geq 2)\) and \(\Delta_{\Omega}\) be the Laplacian with Dirichlet conditions on \(\partial \Omega\). Let V be a complex-valued measurable function on \(\Omega\) with \(V=0\) on comp(\(\Omega\)) that satisfies the Kato regularity conditions.
This paper determines general conditions from \(| V|\) and \(\partial \Omega\) under which the Green’s function \(G_{\Omega}^{V,k}\) associated with the operator \((-\Delta_{\Omega}+V+k^ 2)^{-1}\), Re k\(>0\), is comparable to the free Green’s function \(G_{\Omega}^{0,k}\). These conditions are exploited to analyze precise short range conditions providing upper and lower bounds for \(| G_{\Omega}^{V,k}(x,y)/G_{\Omega}^{0,k}(x,y)|\). Continuity properties of this ratio are established and similar results are derived for open sets \(\Omega\) with compact Lipschitz boundaries. In many cases, bounds on the Poisson kernel follow. Information about this kernel allows investigation of the map from Dirichlet data on \(\partial \Omega\) to “data at infinity” for decaying solutions. Finally, the extreme points of the set of positive solutions are identified. Then the Martin boundary of \(\Omega\) is identified as \(\partial \Omega \cup S^{d-1}\) where \(S^{d-1}\) may be interpreted as a sphere at infinity.

MSC:

35J10 Schrödinger operator, Schrödinger equation
35A08 Fundamental solutions to PDEs
31C35 Martin boundary theory
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