Lp-independence of spectral bounds of Schrödinger-type operators with non-local potentials
Yoshihiro TAWARA
Source: J. Math. Soc. Japan Volume 62, Number 3
(2010), 767-788.
Abstract
We establish a necessary and sufficient condition for spectral bounds of a non-local Feynman-Kac semigroup being Lp-independent. This result is an extension of that in [24] to more general symmetric Markov processes; in [24], we only treated a symmetric stable process on Rd. For example, we consider a symmetric stable process on the hyperbolic space, the jump process generated by the fractional power of the Laplace-Beltrami operator, and prove that by adding a non-local potential, the associated Feynman-Kac semigroup satisfies the Lp-independence.
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Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1280496819
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References
S. Albeverio, P. Blanchard and Z.-M. Ma, Feynman-Kac semigroups in terms of signed smooth measures, In: Random Partial Differential Equations, Internat. Ser. Numer. Math., 102, Birkhäuser, Boston, 1991, pp.,1–31.
Mathematical Reviews (MathSciNet): MR1185735
Zentralblatt MATH: 0737.60065
S. Albeverio and Z.-M. Ma, Perturbation of Dirichlet forms – lower semiboundedness, closability, and form cores, J. Funct. Anal., 99 (1991), 332–356.
Mathematical Reviews (MathSciNet): MR1121617
Zentralblatt MATH: 0743.60071
Digital Object Identifier: doi:10.1016/0022-1236(91)90044-6
S. Albeverio and Z.-M. Ma, Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms, Osaka J. Math., 29 (1992), 247–265.
Mathematical Reviews (MathSciNet): MR1173989
Zentralblatt MATH: 0786.60104
Project Euclid: euclid.ojm/1200783756
R. Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France, 102 (1974), 193–240.
Mathematical Reviews (MathSciNet): MR356254
Zentralblatt MATH: 0293.60071
R. Bass and D. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc., 354 (2002), 2933–2953.
Mathematical Reviews (MathSciNet): MR1895210
Zentralblatt MATH: 0993.60070
Digital Object Identifier: doi:10.1090/S0002-9947-02-02998-7
JSTOR: links.jstor.org
Z.-Q. Chen, Gaugeability and conditional gaugeability, Trans. Amer. Math. Soc., 354 (2002), 4639–4679.
Mathematical Reviews (MathSciNet): MR1926893
Zentralblatt MATH: 1006.60072
Digital Object Identifier: doi:10.1090/S0002-9947-02-03059-3
JSTOR: links.jstor.org
Z.-Q. Chen and R. Song, Conditional gauge theorem for non-local Feynman-Kac transforms, Probab. Theory Relat. Fields, 125 (2003), 45–72.
Mathematical Reviews (MathSciNet): MR1952456
Zentralblatt MATH: 1025.60032
Digital Object Identifier: doi:10.1007/s004400200219
K. L. Chung, Doubly Feller process with multiplicative functional, Seminar on Stochastic Processes, 1985, Birkhäuser, Boston, 1986, pp.,63–78.
Mathematical Reviews (MathSciNet): MR896735
Zentralblatt MATH: 0603.60066
E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989.
Mathematical Reviews (MathSciNet): MR990239
Zentralblatt MATH: 0699.35006
M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 1994.
Mathematical Reviews (MathSciNet): MR1303354
A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 135–249.
Mathematical Reviews (MathSciNet): MR1659871
Digital Object Identifier: doi:10.1090/S0273-0979-99-00776-4
A. Grigor'yan and M. Noguchi, The heat kernel on hyperbolic space, Bull. London Math. Soc., 30 (1998), 643–650.
Mathematical Reviews (MathSciNet): MR1642767
Digital Object Identifier: doi:10.1112/S0024609398004780
S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992.
Mathematical Reviews (MathSciNet): MR1219534
Zentralblatt MATH: 0781.60002
D. Kim, Asymptotic properties for continuous and jump type's Feynman-Kac functionals, Osaka J. Math., 37 (2000), 147–173.
Mathematical Reviews (MathSciNet): MR1750274
Zentralblatt MATH: 0957.60034
Project Euclid: euclid.ojm/1200789042
I. McGillivray, A recurrence condition for some subordinated strongly local Dirichlet forms, Forum Math., 9 (1997), 229–246.
Mathematical Reviews (MathSciNet): MR1431122
Zentralblatt MATH: 0972.60504
Digital Object Identifier: doi:10.1515/form.1997.9.229
H. Ôkura, Recurrence and transience criteria for subordinated symmetric Markov processes, Forum Math., 14 (2002), 121–146.
Mathematical Reviews (MathSciNet): MR1880197
Zentralblatt MATH: 0990.60084
Digital Object Identifier: doi:10.1515/form.2002.001
M. Sharpe, General Theory of Markov Processes, Pure and Applied Mathematics, 133, Academic Press, 1988.
Mathematical Reviews (MathSciNet): MR958914
Zentralblatt MATH: 0649.60079
B. Simon, Brownian motion, ${L}^{p}$ properties of Schrödinger semigroups and the localization of binding, J. Funct. Anal., 35 (1982), 215–229.
Mathematical Reviews (MathSciNet): MR561987
Digital Object Identifier: doi:10.1016/0022-1236(80)90006-3
P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potential Anal., 5 (1996), 109–138.
Mathematical Reviews (MathSciNet): MR1378151
Digital Object Identifier: doi:10.1007/BF00396775
K.-Th. Sturm, On the ${L}^{p}$-spectrum of uniformly elliptic operators on Riemannian manifolds, J. Funct. Anal., 118 (1993), 442–453.
Mathematical Reviews (MathSciNet): MR1250269
Zentralblatt MATH: 0795.58050
Digital Object Identifier: doi:10.1006/jfan.1993.1150
K.-Th. Sturm, Schrödinger semigroups on manifolds, J. Funct. Anal., 118 (1993), 309–350.
Mathematical Reviews (MathSciNet): MR1250266
Digital Object Identifier: doi:10.1006/jfan.1993.1147
M. Takeda, Asymptotic properties of generalized Feynman-Kac functionals, Potential Anal., 9 (1998), 261–291.
Mathematical Reviews (MathSciNet): MR1666887
Digital Object Identifier: doi:10.1023/A:1008656907265
M. Takeda, ${L}^{p}$-independence of spectral bounds of Schrödinger type semigroups, J. Funct. Anal., 252 (2007), 550–565.
Mathematical Reviews (MathSciNet): MR2360927
Digital Object Identifier: doi:10.1016/j.jfa.2007.08.003
M. Takeda and Y. Tawara, ${L}^{p}$-independence of spectral bounds of non-local Feynman-Kac semigroups, Forum Math., 21 (2009), 1067–1080.
Mathematical Reviews (MathSciNet): MR2574148
Zentralblatt MATH: 1187.60058
Digital Object Identifier: doi:10.1515/FORUM.2009.053
J. Ying, Dirichlet forms perturbated by additive functionals of extended Kato class, Osaka J. Math., 34 (1997), 933–952.
Mathematical Reviews (MathSciNet): MR1618693
Zentralblatt MATH: 0903.60062
Project Euclid: euclid.ojm/1200787790
K. Yosida, Functinal Analysis, 6th ed., Springer-Verlag, Berlin, 1980.
Mathematical Reviews (MathSciNet): MR617913
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