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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.

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