Criticality for Schrödinger type operators based on recurrent symmetric stable processes

Takeda, Masayoshi

doi:10.1090/tran/6319

Criticality for Schrödinger type operators based on recurrent symmetric stable processes
HTML articles powered by AMS MathViewer

by Masayoshi Takeda PDF

Trans. Amer. Math. Soc. 368 (2016), 149-167 Request permission

Abstract:

Let $\mu$ be a signed Radon measure on $\mathbb {R}^1$ in the Kato class and consider a Schrödinger type operator $\mathcal {H}^{\mu }=(-d^2/dx^2)^{\frac {\alpha }{2}} + \mu$ on $\mathbb {R}^1$. Let $1\leq \alpha <2$ and suppose the support of $\mu$ is compact. We then construct a bounded $\mathcal {H}^{\mu }$-harmonic function uniformly lower-bounded by a positive constant if $\mathcal {H}^{\mu }$ is critical. Moreover, we show that there exists no bounded positive $\mathcal {H}^{\mu }$-harmonic function if $\mathcal {H}^{\mu }$ is subcritical.

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
Sergio Albeverio, Philippe Blanchard, and Zhi Ming Ma, Feynman-Kac semigroups in terms of signed smooth measures, Random partial differential equations (Oberwolfach, 1989) Internat. Ser. Numer. Math., vol. 102, Birkhäuser, Basel, 1991, pp. 1–31. MR 1185735, DOI 10.1007/978-3-0348-6413-8_{1}
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
Krzysztof Bogdan and Tomasz Byczkowski, Potential theory of Schrödinger operator based on fractional Laplacian, Probab. Math. Statist. 20 (2000), no. 2, Acta Univ. Wratislav. No. 2256, 293–335. MR 1825645
K. J. Brown, C. Cosner, and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on $\textbf {R}^n$, Proc. Amer. Math. Soc. 109 (1990), no. 1, 147–155. MR 1007489, DOI 10.1090/S0002-9939-1990-1007489-1
K. J. Brown and A. Tertikas, The existence of principal eigenvalues for problems with indefinite weight function on $\textbf {R}^k$, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 3, 561–569. MR 1226617, DOI 10.1017/S0308210500025889
Zhen-Qing Chen, Gaugeability and conditional gaugeability, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4639–4679. MR 1926893, DOI 10.1090/S0002-9947-02-03059-3
Zhen-Qing Chen, $L^p$-independence of spectral bounds of generalized non-local Feynman-Kac semigroups, J. Funct. Anal. 262 (2012), no. 9, 4120–4139. MR 2899989, DOI 10.1016/j.jfa.2012.02.011
E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239, DOI 10.1017/CBO9780511566158
Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, Second revised and extended edition, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 2011. MR 2778606
R. K. Getoor, Transience and recurrence of Markov processes, Seminar on Probability, XIV (Paris, 1978/1979) Lecture Notes in Math., vol. 784, Springer, Berlin, 1980, pp. 397–409. MR 580144
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-New York, (1984).
D. Kim and K. Kuwae, On analytic characterizations of gaugeability for generalized Feynman-Kac functionals, to appear in Trans. Amer. Math. Soc.
Kazuhiro Kuwae, Stochastic calculus over symmetric Markov processes without time reversal, Ann. Probab. 38 (2010), no. 4, 1532–1569. MR 2663636, DOI 10.1214/09-AOP516
Minoru Murata, Positive solutions and large time behaviors of Schrödinger semigroups, Simon’s problem, J. Funct. Anal. 56 (1984), no. 3, 300–310. MR 743843, DOI 10.1016/0022-1236(84)90079-X
Y\B{o}ichi Ōshima, Potential of recurrent symmetric Markov processes and its associated Dirichlet spaces, Functional analysis in Markov processes (Katata/Kyoto, 1981) Lecture Notes in Math., vol. 923, Springer, Berlin, 1982, pp. 260–275. MR 661629, DOI 10.1007/BFb0093047
Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
Martin L. Silverstein, Symmetric Markov processes, Lecture Notes in Mathematics, Vol. 426, Springer-Verlag, Berlin-New York, 1974. MR 0386032, DOI 10.1007/BFb0073683
Barry Simon, Large time behavior of the $L^{p}$ norm of Schrödinger semigroups, J. Functional Analysis 40 (1981), no. 1, 66–83. MR 607592, DOI 10.1016/0022-1236(81)90073-2
Peter Stollmann and Jürgen Voigt, Perturbation of Dirichlet forms by measures, Potential Anal. 5 (1996), no. 2, 109–138. MR 1378151, DOI 10.1007/BF00396775
Masayoshi Takeda, Asymptotic properties of generalized Feynman-Kac functionals, Potential Anal. 9 (1998), no. 3, 261–291. MR 1666887, DOI 10.1023/A:1008656907265
Masayoshi Takeda, Exponential decay of lifetimes and a theorem of Kac on total occupation times, Potential Anal. 11 (1999), no. 3, 235–247. MR 1717103, DOI 10.1023/A:1008649623291
Masayoshi Takeda, $L^p$-independence of the spectral radius of symmetric Markov semigroups, Stochastic processes, physics and geometry: new interplays, II (Leipzig, 1999) CMS Conf. Proc., vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 613–623. MR 1803452
Masayoshi Takeda, Conditional gaugeability and subcriticality of generalized Schrödinger operators, J. Funct. Anal. 191 (2002), no. 2, 343–376. MR 1911190, DOI 10.1006/jfan.2001.3864
Masayoshi Takeda, Large deviation principle for additive functionals of Brownian motion corresponding to Kato measures, Potential Anal. 19 (2003), no. 1, 51–67. MR 1962951, DOI 10.1023/A:1022417815090
Masayoshi Takeda, $L^p$-independence of spectral bounds of Schrödinger type semigroups, J. Funct. Anal. 252 (2007), no. 2, 550–565. MR 2360927, DOI 10.1016/j.jfa.2007.08.003
Masayoshi Takeda, A large deviation principle for symmetric Markov processes with Feynman-Kac functional, J. Theoret. Probab. 24 (2011), no. 4, 1097–1129. MR 2851247, DOI 10.1007/s10959-010-0324-5
Masayoshi Takeda and Yoshihiro Tawara, A large deviation principle for symmetric Markov processes normalized by Feynman-Kac functionals, Osaka J. Math. 50 (2013), no. 2, 287–307. MR 3080801
Masayoshi Takeda and Kaneharu Tsuchida, Differentiability of spectral functions for symmetric $\alpha$-stable processes, Trans. Amer. Math. Soc. 359 (2007), no. 8, 4031–4054. MR 2302522, DOI 10.1090/S0002-9947-07-04149-9
Masayoshi Takeda and Toshihiro Uemura, Subcriticality and gaugeability for symmetric $\alpha$-stable processes, Forum Math. 16 (2004), no. 4, 505–517. MR 2044025, DOI 10.1515/form.2004.024
Z. Zhao, A probabilistic principle and generalized Schrödinger perturbation, J. Funct. Anal. 101 (1991), no. 1, 162–176. MR 1132313, DOI 10.1016/0022-1236(91)90153-V