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Positive solutions of Schrödinger equations and Martin boundaries for skew product elliptic operators. (English) Zbl 1376.35015
Summary: We consider positive solutions of elliptic partial differential equations on non-compact domains of Riemannian manifolds. We establish general theorems which determine Martin compactifications and Martin kernels for a wide class of elliptic equations in skew product form, by thoroughly exploiting parabolic Martin kernels for associated parabolic equations developed in [M. Murata, J. Funct. Anal. 245, No. 1, 177–212 (2007; Zbl 1172.58007); P. J. Mendez-Hernandez and M. Murata, J. Funct. Anal. 257, No. 6, 1799–1827 (2009; Zbl 1172.58007)]. As their applications, we explicitly determine the structure of all positive solutions to a Schrödinger equation and the Martin boundary of the product of Riemannian manifolds. For their sharpness, we show that the Martin compactification of \({\mathbb R}^2\) for some Schrödinger equation is so much distorted near infinity that no product structures remain.

MSC:
35J10 Schrödinger operator, Schrödinger equation
31C35 Martin boundary theory
35B09 Positive solutions to PDEs
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Full Text: Euclid