Abstract and Applied Analysis
Volume 2010 (2010), Article ID 902638, 18 pages
doi:10.1155/2010/902638
Research Article
Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space
Department of Mathematical Sciences, University of Oulu, P. O. Box 3000, 90014 Oulu, Finland
Received 29 August 2009; Accepted 4 February 2010
Academic Editor: Martin D. Schechter
Copyright © 2010 Valery Serov. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- R. A. Adams and J. F. Fournier, Sobolev Spaces, Academic Press, New York, NY, USA, 2nd edition, 2003.
- H. Triebel, Interpolation Theory Function Spaces Differential Operators, Mir, Moscow, Russia, 1980.
- M. Schechter, Spectra of Partial Differential Operators, vol. 1, North-Holland, Amsterdam, The Netherlands, 1971, North-Holland Series in Applied Mathematics and Mechanics. View at MathSciNet
- L. Hörmander, The Analysis of Linear Partial Differential Equations, Vols. 1-2, Springer, New York, NY, USA, 1983.
- L. Hörmander, The Analysis of Linear Partial Differential Equations, Vols. 3-4, Springer, New York, NY, USA, 1985.
- S. A. Alimov, “Fractional powers of elliptic operators and isomorphism of classes of differentiable functions,” Differentsial'nye Uravneniya, vol. 8, pp. 1609–1626, 1972 (Russian). View at MathSciNet
- S. A. Alimov, “Uniform convergence and summability of the spectral expansions of functions from ,” Differentsial'nye Uravneniya, vol. 9, no. 4, pp. 669–681, 1973. View at MathSciNet
- S. A. Alimov, “The spectral expansions of functions belonging to ,” Matematicheskii Sbornik, vol. 101(143), no. 1, pp. 3–20, 1976. View at MathSciNet
- S. A. Alimov, “On the absolute convergence of spectral expansions,” Doklady Akademii Nauk, vol. 342, no. 4, pp. 446–448, 1995 (Russian). View at MathSciNet
- L. Gårding, “Dirichlet's problem for linear elliptic partial differential equations,” Mathematica Scandinavica, vol. 1, pp. 55–72, 1953.
- J. P. Krasovskiĭ, “Isolation of the singularity in Green's function,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 31, pp. 977–1010, 1967 (Russian). View at MathSciNet
- J. P. Krasovskiĭ, “Properties of Green's functions, and generalized solutions of elliptic boundary value problems,” Doklady Akademii Nauk SSSR, vol. 184, no. 2, pp. 270–273, 1969. View at MathSciNet
- V. S. Serov, “On the fundamental solution of a differential operator with a singularity,” Differentsial'nye Uravneniya, vol. 23, no. 3, pp. 531–534, 1987. View at Zentralblatt MATH · View at MathSciNet
- V. S. Serov, “The absolute convergence of spectral expansions of operators with a singularity,” Differentsial'nye Uravneniya, vol. 28, no. 1, pp. 127–136, 1992. View at MathSciNet
- V. S. Serov, “On spectral expansions of functions in for a differential operator with a singularity on the surface,” Doklady Rossiĭskaya Akademiya Nauk, vol. 340, no. 1, pp. 26–28, 1995. View at MathSciNet
- S. A. Alimov and I. Joó, “On the Riesz summability of eigenfunction expansions,” Acta Scientiarum Mathematicarum, vol. 45, no. 1–4, pp. 5–18, 1983. View at Zentralblatt MATH · View at MathSciNet
- R. R. Ashurov, “Asymptotic behavior of a spectral function of the Schrödinger operator with potential ,” Differentsial'nye Uravneniya, vol. 23, no. 1, pp. 169–172, 1987. View at MathSciNet
- R. R. Ashurov and J. E. Faiziev, “On eigenfunction expansions associated with the Schrödinger operator with a singular potential,” Differentsial'nye Uravneniya, vol. 41, no. 2, pp. 241–249, 2005.
- A. R. Khalmukhamedov, “Eigenfunction expansions for the Schrödinger operator with singular potentials,” Differentsial'nye Uravneniya, vol. 20, no. 9, pp. 1642–1645, 1984.
- A. R. Khalmukhamedov, “Convergence of spectral expansion for a singular operator,” Differentsial'nye Uravneniya, vol. 22, no. 12, pp. 2107–2117, 1986.
- V. S. Serov, “On the convergence of Fourier series in eigenfunctions of the Schrödinger operator with Kato potential,” Matematicheskie Zametki, vol. 67, no. 5, pp. 755–763, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
- V. S. Serov, “Fundamental solution and Fourier series in eigenfunctions of degenerate elliptic operator,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 132–144, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
- N. S. Buzurnyuk and V. S. Serov, “On the convergence of Riesz means of spectral expansions that correspond to the Schrödinger operator with a singular potential,” Differentsial'nye Uravneniya, vol. 32, no. 1, pp. 83–89, 1996. View at MathSciNet
- E. B. Davies, Spectral Theory and Differential Operators, vol. 42 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1995. View at MathSciNet
- E. B. Davies, “ spectral theory of higher-order elliptic differential operators,” The Bulletin of the London Mathematical Society, vol. 29, no. 5, pp. 513–546, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. Peetre, “Absolute convergence of eigenfunction expansions,” Mathematische Annalen, vol. 169, pp. 307–314, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet