zbMATH — the first resource for mathematics

Dirichlet problem for the Schrödinger operator in a half space. (English) Zbl 1246.35072
Summary: For continuous boundary data, the modified Poisson integral is used to write solutions to the half space Dirichlet problem for the Schrödinger operator. Meanwhile, a solution of the Poisson integral for any continuous boundary function is also given explicitly by the Poisson integral with the generalized Poisson kernel depending on this boundary function.

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J10 Schrödinger operator, Schrödinger equation
PDF BibTeX Cite
Full Text: DOI
[1] A. I. Kheyfits, “Dirichlet problem for the Schrödinger operator in a half-space with boundary data of arbitrary growth at infinity,” Differential and Integral Equations, vol. 10, no. 1, pp. 153-164, 1997. · Zbl 0879.35039
[2] A. I. Kheyfits, “Liouville theorems for generalized harmonic functions,” Potential Analysis, vol. 16, no. 1, pp. 93-101, 2002. · Zbl 1078.35020
[3] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 3, Academic Press, New York, NY, USA, 1970. · Zbl 0401.47001
[4] G. Rosenblum, M. Solomyak, and M. Shubin, Spectral Theory of Differential Operators, Viniti, Moscow, Russia, 1989. · Zbl 0715.35057
[5] C. Müller, Spherical Harmonics, vol. 17 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1966. · Zbl 0138.05101
[6] B. Simon, “Schrödinger semigroups,” Bulletin of the American Mathematical Society, vol. 7, no. 3, pp. 447-526, 1982. · Zbl 0524.35002
[7] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1964. · Zbl 0125.32102
[8] A. Escassut, W. Tutschke, and C. C. Yang, Some Topics on Value Distribution and Differentiability in Complex and P-adic Analysis, vol. 11 of Mathematics Monograph Series, Science Press, Beijing, China, 2008. · Zbl 1233.30005
[9] A. I. Kheyfits, “Representation of the analytic functions of infinite order in a half-plane,” Izvestija Akademii Nauk Armjanskoĭ SSR, vol. 6, no. 6, pp. 472-476, 1971.
[10] M. Finkelstein and S. Scheinberg, “Kernels for solving problems of Dirichlet type in a half-plane,” Advances in Mathematics, vol. 18, no. 1, pp. 108-113, 1975. · Zbl 0309.31001
[11] D. Siegel and E. Talvila, “Sharp growth estimates for modified Poisson integrals in a half space,” Potential Analysis, vol. 15, no. 4, pp. 333-360, 2001. · Zbl 0987.31003
[12] W. K. Hayman and P. B. Kennedy, Subharmonic Functions, vol. 1, Academic Press, London, UK, 1976. · Zbl 0419.31001
[13] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, USA, 1970. · Zbl 0207.13501
[14] A. Ancona, “First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains,” Journal d’Analyse Mathématique, vol. 72, no. 1, pp. 45-92, 1997. · Zbl 0944.58016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.