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## Studia Mathematica

1999 | 133 | 1 | 53-92
Tytuł artykułu

### Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The purpose of the paper is to extend results of the potential theory of the classical Schrödinger operator to the α-stable case. To obtain this we analyze a weak version of the Schrödinger operator based on the fractional Laplacian and we prove the Conditional Gauge Theorem.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
53-92
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-03-23
poprawiono
1998-06-29
Twórcy
autor
• Institute of Mathematics, Technical University of Wrocław, 50-370 Wrocław, Poland, bogdan@im.pwr.wroc.pl
autor
• Institute of Mathematics, Technical University of Wrocław, 50-370 Wrocław, Poland
Bibliografia
• [BC] R. F. Bass and M. Cranston, Exit times for symmetric stable processes in $ℝ^n$, Ann. Probab. 11 (1983), 578-588.
• [BG] R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Springer, New York, 1968.
• [BGR] R. M. Blumenthal, R. K. Getoor and D. B. Ray, On the distribution of first hits for the symmetric stable processes, Trans. Amer. Math. Soc. 99 (1961), 540-554.
• [B1] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math. 123 (1997), 43-80.
• [B2] K. Bogdan, Representation of α-harmonic functions in Lipschitz domains, Hiroshima Math. J. (1999), to appear.
• [BB] K. Bogdan and T. Byczkowski, Probabilistic proof of the boundary Harnack principle for symmetric stable processes, Potential Anal. (1999), to appear.
• [CMS] R. Carmona, W. C. Masters and B. Simon, Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions, J. Funct. Anal. 91 (1990), 117-142.
• [CS] Z. Q. Chen and R. Song, Intrinsic ultracontractivity and Conditional Gauge for symmetric stable processes, ibid. 150 (1997), 204-239.
• [ChZ] K. L. Chung and Z. Zhao, From Brownian Motion to Schrödinger's Equation, Springer, New York, 1995.
• [CFZ] M. Cranston, E. Fabes and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc. 307 (1988), 174-194.
• [D] E. B. Dynkin, Markov Processes, Vols. I, II, Academic Press, New York, 1965.
• [Fe] C. Fefferman, The N-body problem in quantum mechanics, Comm. Pure Appl. Math. 39 (1986), S67-S109.
• [Fo] G. B. Folland, Introduction to Partial Differential Equations, Princeton Univ. Press, Princeton, 1976.
• [IW] N. Ikeda and S. Watanabe, On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes, J. Math. Kyoto Univ. 2 (1962), 79-95.
• [K] T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist. 17 (1997), 339-364.
• [La] N. S. Landkof, Foundations of Modern Potential Theory, Springer, New York, 1972.
• [Li] E. H. Lieb, The stability of matter: from atoms to stars, Bull. Amer. Math. Soc. 22 (1990), 1-49.
• [MS] K. Michalik and K. Samotij, Martin representation for α-harmonic functions, preprint, 1997.
• [PS] S. C. Port and C. J. Stone, Infinitely divisible processes and their potential theory, Ann. Inst. Fourier (Grenoble) 21 (1971), no. 2, 157-275, no. 4, 179-265.
• [R] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973.
• [S] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970.
• [W] R. A. Weder, Spectral analysis of pseudodifferential operators, J. Funct. Anal. 20 (1975), 319-337.
• [Z] Z. Zhao, A probabilistic principle and generalized Schrödinger perturbation, J. Funct. Anal. 101 (1991), 162-176.
Typ dokumentu
Bibliografia
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