Generalized first boundary value problem for Schrödinger equation

Ren, Yan Xia

doi:10.1090/S0002-9939-1992-1092930-0

Generalized first boundary value problem for Schrödinger equation
HTML articles powered by AMS MathViewer

by Yan Xia Ren PDF

Proc. Amer. Math. Soc. 115 (1992), 1101-1109 Request permission

Abstract:

In this paper, we have obtained two main results by using probabilistic methods: (i) For a domain, we obtained a representation formula of the bounded solution to the first boundary value problem for Schrödinger equation; (ii) For $\alpha \in {R^1}$, under certain conditions, we proved that the bounded solution having limit $\alpha$ at infinity to the generalized first boundary value problem for Schrödinger equation exists and is unique, and it is represented in explicit formula. The results of this paper are generalizations of Chung and Rao.

References

M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), no. 2, 209–273. MR 644024, DOI 10.1002/cpa.3160350206
K. L. Chung, Notes on the inhomogeneous Schrödinger equation, Seminar on stochastic processes, 1984 (Evanston, Ill., 1984) Progr. Probab. Statist., vol. 9, Birkhäuser Boston, Boston, MA, 1986, pp. 55–62. MR 896721
K. L. Chung and K. M. Rao, Feynman-Kac functional and the Schrödinger equation, Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981) Progr. Prob. Statist., vol. 1, Birkhäuser, Boston, Mass., 1981, pp. 1–29. MR 647779
Richard Durrett, Brownian motion and martingales in analysis, Wadsworth Mathematics Series, Wadsworth International Group, Belmont, CA, 1984. MR 750829
Neil Falkner, Feynman-Kac functionals and positive solutions of ${1\over 2}\Delta u+qu=0$, Z. Wahrsch. Verw. Gebiete 65 (1983), no. 1, 19–33. MR 717930, DOI 10.1007/BF00534991
Zhi Chan Li, An extension of the Dirichlet problem, Acta Math. Sci. (Chinese) 4 (1984), no. 3, 287–295 (Chinese). MR 812916
Sidney C. Port and Charles J. Stone, Brownian motion and classical potential theory, Probability and Mathematical Statistics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0492329
Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. MR 670130, DOI 10.1090/S0273-0979-1982-15041-8