Bounded solutions for the Schrodinger operator on Riemannian manifolds
Bull. Korean Math. Soc. 2007 Vol. 44, No. 3, 507-516
Printed September 1, 2007
Seok Woo Kim and Yong Hah Lee
Konkuk University, Ewha Womans University
Abstract : Let $M$ be a complete Riemannian manifold and ${\mathcal L}$ be a Schr\"o\-d\-inger operator on $M$. We prove that if $M$ has finitely many $\mathcal L$-nonpara\-bol\-ic ends, then the space of bounded $\mathcal L$-harmonic functions on $M$ has the same dimension as the sum of dimensions of the spaces of bounded $\mathcal L$-harmonic functions on each $\mathcal L$-nonparabolic end, which vanish at the boundary of the end.
Keywords : Schrodinger operator, $\mathcal L$-harmonic function, $\mathcal L$-massive set, end
MSC numbers : 58J05, 35J10
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