Conditional gauge and potential theory for the Schrödinger operator

Cranston, M.; Fabes, E.; Zhao, Z.

doi:10.1090/S0002-9947-1988-0936811-2

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

Conditional gauge and potential theory for the Schrödinger operator

Authors: M. Cranston, E. Fabes and Z. Zhao
Journal: Trans. Amer. Math. Soc. 307 (1988), 171-194
MSC: Primary 60J60; Secondary 31C35, 35J10, 60J45
MathSciNet review: 936811
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper extends the Conditional Gauge Theorem to more general operators and less regular domains than in previous works. We use this to obtain potential-theoretic results for the Schrödinger equation.

[1] 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 (84a:35062), http://dx.doi.org/10.1002/cpa.3160350206
[2] Michael T. Anderson and Richard Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. (2) 121 (1985), no. 3, 429–461. MR 794369 (87a:58151), http://dx.doi.org/10.2307/1971181
[3] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607–694. MR 0435594 (55 #8553)
[4] R. Bañuelos, On an estimate of Cranston and McConnell concerning the lifetime of certain diffusions, preprint.
[5] R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York, 1968. MR 0264757 (41 #9348)
[6] A. Boukricha, W. Hansen, and H. Hueber, Continuous solutions of the generalized Schrödinger equation and perturbation of harmonic spaces, Exposition. Math. 5 (1987), no. 2, 97–135. MR 887788 (88g:31019)
[7] J. Brossard, Le problème de Dirichlet pour l'opérateur de Schrödinger, preprint.
[8] L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30 (1981), no. 4, 621–640. MR 620271 (83c:35040), http://dx.doi.org/10.1512/iumj.1981.30.30049
[9] F. Chiarenza, E. Fabes, and N. Garofalo, Harnack’s inequality for Schrödinger operators and the continuity of solutions, Proc. Amer. Math. Soc. 98 (1986), no. 3, 415–425. MR 857933 (88a:35037), http://dx.doi.org/10.1090/S0002-9939-1986-0857933-4
[10] K. L. Chung, The gauge and conditional gauge theorem, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 496–503. MR 889497 (90h:60071), http://dx.doi.org/10.1007/BFb0075868
[11] K. L. Chung, P. Li, and R. J. Williams, Comparison of probability and classical methods for the Schrödinger equation, Exposition. Math. 4 (1986), no. 3, 271–278. MR 880127 (89c:60095)
[12] 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 (83g:60089)
[13] K. L. Chung and Z. Zhao, From Brownian motion to the Schrödinger equation, preprint.
[14] M. Cranston, Lifetime of conditioned Brownian motion in Lipschitz domains, Z. Wahrsch. Verw. Gebiete 70 (1985), no. 3, 335–340. MR 803674 (87a:60088), http://dx.doi.org/10.1007/BF00534865
[15] Björn E. J. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), no. 5, 1119–1138. MR 859772 (88i:35061), http://dx.doi.org/10.2307/2374598
[16] E. DiGiorgi, Sulla differentiabilitá l'analizitá degli integrali multipli regolari, Mem. Acad. Sci. Torino, S. III 1 (1957), 25-43.
[17] J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431–458. MR 0109961 (22 #844)
[18] Eugene B. Fabes, David S. Jerison, and Carlos E. Kenig, Necessary and sufficient conditions for absolute continuity of elliptic-harmonic measure, Ann. of Math. (2) 119 (1984), no. 1, 121–141. MR 736563 (85h:35069), http://dx.doi.org/10.2307/2006966
[19] N. Falkner, Feynman-Kac functionals and positive solutions of $\tfrac{1} {2}\Delta u + qu = 0$ , Z. Wahrsch. Verw. Gebiete. 65 (1983), 19-31.
[20] -, Conditional Brownian motion in rapidly exhaustible domains, preprint.
[21] Masatoshi Fukushima, Dirichlet forms and Markov processes, North-Holland Mathematical Library, vol. 23, North-Holland Publishing Co., Amsterdam, 1980. MR 569058 (81f:60105)
[22] David S. Jerison and Carlos E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 (1982), no. 1, 80–147. MR 676988 (84d:31005b), http://dx.doi.org/10.1016/0001-8708(82)90055-X
[23] W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 43–77. MR 0161019 (28 #4228)
[24] J. Moser, On Harnack's inequality for elliptic differential equations, Amer. J. Math. 80 (1958), 931-954.
[25] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. MR 0100158 (20 #6592)
[26] Thomas S. Salisbury, A Martin boundary in the plane, Trans. Amer. Math. Soc. 293 (1986), no. 2, 623–642. MR 816315 (87b:60114), http://dx.doi.org/10.1090/S0002-9947-1986-0816315-6
[27] Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. MR 670130 (86b:81001a), http://dx.doi.org/10.1090/S0273-0979-1982-15041-8
[28] Zhong Xin Zhao, Conditional gauge with unbounded potential, Z. Wahrsch. Verw. Gebiete 65 (1983), no. 1, 13–18. MR 717929 (86m:60188b), http://dx.doi.org/10.1007/BF00534990
[29] Zhong Xin Zhao, Uniform boundedness of conditional gauge and Schrödinger equations, Comm. Math. Phys. 93 (1984), no. 1, 19–31. MR 737462 (85i:35041)
[30] Zhong Xin Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl. 116 (1986), no. 2, 309–334. MR 842803 (88f:60142), http://dx.doi.org/10.1016/S0022-247X(86)80001-4

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|