Journal of Soviet Mathematics

, Volume 51, Issue 3, pp 2340–2349 | Cite as

Bounded solutions of the Schrödinger equation on noncompact Riemannian manifolds

  • A. A. Grigor'yan
Article

Abstract

Necessary and sufficient geometric conditions are proved for the equation Δu−Q(x)u=0, Q(x)≥0, to have a bounded nontrivial solution on a noncompact Riemannian manifold. The results imply as corollaries conditions for parabolicity and stochastic completeness of a manifold, previously established by other methods.

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Literature cited

  1. 1.
    S. Y. Cheng and S. T. Yau, “Differential equations on Riemannian manifolds and their geometric applications,” Comm. Pure Appl. Math.,28, No. 3, 333–354 (1975).Google Scholar
  2. 2.
    A. A. Grigor'yan, “On stochastically complete manifolds,” Dokl. Akad. Nauk SSSR,290, No. 3, 534–537 (1986).Google Scholar
  3. 3.
    A. A. Grigor'yan, “On the existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds,” Mat. Sb.,128, No. 3, 354–363 (1985).Google Scholar
  4. 4.
    E. B. Davies, “L1 properties of second order elliptic operators,” Bull. London Math. Soc.,17, No. 5, 417–436 (1985).Google Scholar
  5. 5.
    L. Sario, M. Nakai, C. Wang and L. O. Chung, Classification Theory of Riemannian Manifolds, Lecture Notes Math.,605 (1977).Google Scholar
  6. 6.
    N. T. Varopoulos, “Potential theory and diffusion on Riemannian manifolds,” in: Conferencec on Harmonic Analysis in Honor of Antoni Zygmund, I, II, Chicago, Ill., 1981, Wadsworth Math. Ser. (1983), pp. 821–837.Google Scholar
  7. 7.
    H. Federer, Geometric Measure Theory, Springer, Berlin (1967).Google Scholar
  8. 8.
    R. Greene and W. Wu, Function Theory of Manifolds which Possess a Pole, Lecture Notes Math.,699 (1979).Google Scholar
  9. 9.
    K. Ichihara, “Curvature, geodesics and the Brownian motion on a Riemannian manifold,” Nagoya Math. J.,87, 101–125 (1982).Google Scholar
  10. 10.
    R. Azencott, “Behavior of diffusion semi-groups at infinity,” Bull. Soc. Math. (France),102, 193–240 (1974).Google Scholar
  11. 11.
    S. T. Yau, “On the heat kernel of complete Riemannian manifold,” J. Math. Pure. Appl. Ser. 9,57, No. 2, 191–201 (1978).Google Scholar
  12. 12.
    P. Li and R. Schoen, “LP and mean value properties of subharmonic functions on Riemannian manifolds,” Acta Math.,153, Nos. 3–4, 279–301 (1984).Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • A. A. Grigor'yan

There are no affiliations available

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