Conditional gauge and potential theory for the Schrödinger operator

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

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

Conditional gauge and potential theory for the Schrödinger operator
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by M. Cranston, E. Fabes and Z. Zhao

Trans. Amer. Math. Soc. 307 (1988), 171-194

DOI: https://doi.org/10.1090/S0002-9947-1988-0936811-2

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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.

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
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, DOI 10.2307/1971181
D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 607–694. MR 435594

On an estimate of Cranston and McConnell concerning the lifetime of certain diffusions

R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR 0264757
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

Le problème de Dirichlet pour l’opérateur de Schrödinger

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, DOI 10.1512/iumj.1981.30.30049
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, DOI 10.1090/S0002-9939-1986-0857933-4
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, DOI 10.1007/BFb0075868
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
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

From Brownian motion to the Schrödinger equation

M. Cranston, Lifetime of conditioned Brownian motion in Lipschitz domains, Z. Wahrsch. Verw. Gebiete 70 (1985), no. 3, 335–340. MR 803674, DOI 10.1007/BF00534865
Björn E. J. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), no. 5, 1119–1138. MR 859772, DOI 10.2307/2374598

Sulla differentiabilitá l’analizitá degli integrali multipli regolari

1

J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431–458. MR 109961, DOI 10.24033/bsmf.1494
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, DOI 10.2307/2006966

Feynman-Kac functionals and positive solutions of

65

Conditional Brownian motion in rapidly exhaustible domains

Masatoshi Fukushima, Dirichlet forms and Markov processes, North-Holland Mathematical Library, vol. 23, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1980. MR 569058
Peter W. Jones, A geometric localization theorem, Adv. in Math. 46 (1982), no. 1, 71–79. MR 676987, DOI 10.1016/0001-8708(82)90054-8
W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43–77. MR 161019

On Harnack’s inequality for elliptic differential equations

80

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. MR 100158, DOI 10.2307/2372841
Thomas S. Salisbury, A Martin boundary in the plane, Trans. Amer. Math. Soc. 293 (1986), no. 2, 623–642. MR 816315, DOI 10.1090/S0002-9947-1986-0816315-6
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
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
Zhong Xin Zhao, Uniform boundedness of conditional gauge and Schrödinger equations, Comm. Math. Phys. 93 (1984), no. 1, 19–31. MR 737462, DOI 10.1007/BF01218637
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, DOI 10.1016/S0022-247X(86)80001-4