Abstract
We study questions of existence and membership to a given functional class of unbounded solutions of the stationary Schrödinger equation Δu − cu = 0, where c is a smooth non-negative function on a non-compact Riemannian manifold M without boundary. We establish some interrelation between problems of existence of solutions of this equation on M and off some compact B ⊂ M with the same growth “at infinity”.
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M. T. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. Math. (2) 121 (1985), 429–461.
W. Ballmann, On the Dirichlet problem at infinity for manifolds of nonpositive cuvature, Forum Math. 1 (1989), 201–213.
W. Ballmann, The Martin boundary of certain Hadamard manifolds, in: S. K. Vodop’yanov (ed.), Proceedings on Analysis and Geometry, International conference in honor of the 70th birthday of Professor Yu. G. Reshetnyak, Novosibirsk, August 30-September 3, 1999, Novosibirsk, Izdatel’stvo Instituta Matematiki Im. S. L. Soboleva SO RAN, 2000, 36–46.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math Wiss., Bd. 224, Springer-Verlag, Berlin-New York, 1983.
A. Grigor’yan, Analytic and geometric background of recurence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135–249.
A. A. Grigor’yan and N. S. Nadirashvili, Liouville theorems and exterior boundary value problems, Isv. Vyssh. Uchebn. Zaved. Mat. 5 (1987), 25–33 (in Russian); English transl. in: Soviet Math. 31 (1987) No.5, 31–42.
A. G. Losev, Elliptic partial differential equation on the warped products of Riemannian manifolds, Appl. Anal. 71 (1999) No.1–4, 325–339.
A. G. Losev and E. A. Mazepa, Bounded solutions of the Schrödinger equation on Riemannian products, St. Petersbg. Math. J. 13 (2002) No.1, 57–73.
E. A. Mazepa, Boundary value problems for the stationary Schrödinger equation on Riemannian manifolds, Sib. Math. J. 43 (2002) No.3, 473–479.
V. M. Miklyukov, Parabolic and hyperbolic type of boundary sets of surfaces, Russian Mat. Izv., Ser. Mat. 60 (1996) No.4, 111–158.
L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Diff. Geom. 36 (1992), 417–450.
S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228.
S. T. Yau, Nonlinear analysis in geometry, Enseign. Math. II Sér. 33 (1987), 109–158.
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Research supported by the RFBR (grant no. 03-01-00304).
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Losev, A.G., Mazepa, E.A. & Chebanenko, V.Y. Unbounded Solutions of the Stationary Schrödinger Equation on Riemannian Manifolds. Comput. Methods Funct. Theory 3, 443–451 (2004). https://doi.org/10.1007/BF03321048
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DOI: https://doi.org/10.1007/BF03321048