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Journal of the London Mathematical Society 1987 s2-36(2):370-384; doi:10.1112/jlms/s2-36.2.370
© 1987 by London Mathematical Society
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© Oxford University Press

On the Poisson Kernel for the Neumann Problem of Schrödinger Operators

Pei Hsu

Courant Institute of Mathematical Sciences 251 Mercer Street, New York, New York 10012, USA

Let D be a bounded domain in Rd (d ≥ 3) and let b{t, x, y) be the kernel of the Feynamn-Kac semigroup associated with the reflecting Brownian motion {Xt: t ≥ 0} and potential V, namely


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We assume that V is in the Kato class Kd [1]. The Poisson kernel studied in this paper is


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In general Nv may be infinite. We show that if Nv(x, y) is finite for one pair of points then it is finite for all x != y and there exist two constants c1, c2 (depending on D and V) such that


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This happens precisely when the spectrum of Hv = {Delta}/2 + V under the Neumann boundary condition lies in the negative half-axis. This result is used to discuss the Neumann boundary value problem of Hv. We prove that for any boundary function f{varepsilon}L{alpha}({partial}D), {alpha} ≥ 1, the problem has a unique weak solution


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and its growth rate near the boundary can be estimated by ||f{alpha},{partial}D.


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