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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
DOI: https://doi.org/10.1090/S0002-9947-1988-0936811-2
MathSciNet review: 936811
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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.


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  • [1] M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), 209-273. MR 644024 (84a:35062)
  • [2] M. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, preprint. MR 794369 (87a:58151)
  • [3] D. Aronson, Non-negative solutions of linear parabolic equations, Ann. Sci. Norm. Sup. Pisa 23 (1968). MR 0435594 (55:8553)
  • [4] R. Bañuelos, On an estimate of Cranston and McConnell concerning the lifetime of certain diffusions, preprint.
  • [5] R. Blumenthal and R. Getoor, Markov processes and potential theory, 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, preprint. 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 J. Math. 30 (1981), 621-640. MR 620271 (83c:35040)
  • [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), 415-425. MR 857933 (88a:35037)
  • [10] K. L. Chung, The gauge and conditional gauge theorem, Séminaire de Probabilitiés XIX, Lecture Notes in Math., vol. 1123, Springer, 1983/84, pp. 496-503. MR 889497 (90h:60071)
  • [11] K. L. Chung, P. Li and R. J. Williams, Comparison of probability and classical methods for the Schrödinger equation, Exposition. Math. 4 (1986), 271-278. MR 880127 (89c:60095)
  • [12] K. L. Chung and M. Rao, Feynman-Kac functional and Schrödinger equation, Sem. Stoch. Proc., Birkhäuser, Boston, Mass., 1981. 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), 335-340. MR 803674 (87a:60088)
  • [15] B. E. J. Dahlberg, On the absolute continuity of elliptic measures, preprint. MR 859772 (88i:35061)
  • [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, Conditioned Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431-458. MR 0109961 (22:844)
  • [18] E. Fabes, D. Jerison and C. Kenig, Necessary and sufficient conditions for absolute continuity of elliptic-harmonic measure, Ann. of Math. (2) 119 (1984), 121-141. MR 736563 (85h:35069)
  • [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] M. Fukushima, Dirichlet forms and Markov processes, North-Holland/Kodauzha, 1980. MR 569058 (81f:60105)
  • [22] D. Jerison and C. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Ann. of Math. (2) 113 (1981), 367-382. MR 676988 (84d:31005b)
  • [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), 45-79. 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 the solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931-954. MR 0100158 (20:6592)
  • [26] T. Salisbury, A Martin boundary in the plane, Trans. Amer. Math. Soc. 293 (1986), 623-642. MR 816315 (87b:60114)
  • [27] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447-526. MR 670130 (86b:81001a)
  • [28] Z. Zhao, Conditional gauge and unbounded potential, Z. Wahrsch. Verw. Gebiete. 65 (1983), 13-18. MR 717929 (86m:60188b)
  • [29] -, Uniform boundedness of conditional gauge and Schrödinger equations, Comm. Math. Phys. 93 (1984), 19-31. MR 737462 (85i:35041)
  • [30] -, Green function for Schrödinger operator and conditioned Feyman-Kac gauge, J. Math. Anal. Appl. 116 (1986), 309-334. MR 842803 (88f:60142)

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DOI: https://doi.org/10.1090/S0002-9947-1988-0936811-2
Article copyright: © Copyright 1988 American Mathematical Society

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