Kato’s inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds
HTML articles powered by AMS MathViewer
- by Batu Güneysu PDF
- Proc. Amer. Math. Soc. 142 (2014), 1289-1300 Request permission
Abstract:
Let $M$ be a Riemannian manifold and let $E\to M$ be a Hermitian vector bundle with a Hermitian covariant derivative $\nabla$. Furthermore, let $H(0)$ denote the Friedrichs extension of $\nabla ^*\nabla /2$ and let $V:M\to \mathrm {End}(E)$ be a potential. We prove that if $V$ has a decomposition of the form $V=V_1-V_2$ with $V_j\geq 0$, $V_1$ locally integrable and $\left | V_2 \right |$ in the Kato class of $M$, then one can define the form sum $H(V):=H(0)\dotplus V$ in $\Gamma _{\mathsf {L}^2}(M,E)$ without any further assumptions on $M$. Applications to quantum physics are discussed.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
- M. Braverman, O. Milatovich, and M. Shubin, Essential selfadjointness of Schrödinger-type operators on manifolds, Uspekhi Mat. Nauk 57 (2002), no. 4(346), 3–58 (Russian, with Russian summary); English transl., Russian Math. Surveys 57 (2002), no. 4, 641–692. MR 1942115, DOI 10.1070/RM2002v057n04ABEH000532
- Jozef Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), no. 5, 703–716. MR 711862, DOI 10.1512/iumj.1983.32.32046
- Bruce K. Driver and Anton Thalmaier, Heat equation derivative formulas for vector bundles, J. Funct. Anal. 183 (2001), no. 1, 42–108. MR 1837533, DOI 10.1006/jfan.2001.3746
- Alberto Enciso, Coulomb systems on Riemannian manifolds and stability of matter, Ann. Henri Poincaré 12 (2011), no. 4, 723–741. MR 2787767, DOI 10.1007/s00023-011-0084-5
- László Erdős and Jan Philip Solovej, The kernel of Dirac operators on $\Bbb S^3$ and $\Bbb R^3$, Rev. Math. Phys. 13 (2001), no. 10, 1247–1280. MR 1860416, DOI 10.1142/S0129055X01000983
- Batu Güneysu, The Feynman-Kac formula for Schrödinger operators on vector bundles over complete manifolds, J. Geom. Phys. 60 (2010), no. 12, 1997–2010. MR 2735286, DOI 10.1016/j.geomphys.2010.08.007
- Batu Güneysu, On generalized Schrödinger semigroups, J. Funct. Anal. 262 (2012), no. 11, 4639–4674. MR 2913682, DOI 10.1016/j.jfa.2011.11.030
- Batu Güneysu, Nonrelativistic hydrogen type stability problems on nonparabolic 3-manifolds, Ann. Henri Poincaré 13 (2012), no. 7, 1557–1573. MR 2982633, DOI 10.1007/s00023-012-0167-y
- Elton P. Hsu, Stochastic analysis on manifolds, Graduate Studies in Mathematics, vol. 38, American Mathematical Society, Providence, RI, 2002. MR 1882015, DOI 10.1090/gsm/038
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
- Tosio Kato, Schrödinger operators with singular potentials, Israel J. Math. 13 (1972), 135–148 (1973). MR 333833, DOI 10.1007/BF02760233
- Kazuhiro Kuwae and Masayuki Takahashi, Kato class measures of symmetric Markov processes under heat kernel estimates, J. Funct. Anal. 250 (2007), no. 1, 86–113. MR 2345907, DOI 10.1016/j.jfa.2006.10.010
- Kazuhiro Kuwae and Masayuki Takahashi, Kato class functions of Markov processes under ultracontractivity, Potential theory in Matsue, Adv. Stud. Pure Math., vol. 44, Math. Soc. Japan, Tokyo, 2006, pp. 193–202. MR 2277833, DOI 10.2969/aspm/04410193
- Ognjen Milatovic, “Localized” self-adjointness of Schrödinger type operators on Riemannian manifolds, J. Math. Anal. Appl. 283 (2003), no. 1, 304–318. MR 1994192, DOI 10.1016/S0022-247X(03)00296-8
- Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
- 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
Additional Information
- Batu Güneysu
- Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany
- Email: gueneysu@math.hu-berlin.de
- Received by editor(s): September 19, 2011
- Received by editor(s) in revised form: May 10, 2012
- Published electronically: January 27, 2014
- Communicated by: Varghese Mathai
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 1289-1300
- MSC (2010): Primary 47B25, 58J35; Secondary 60H30
- DOI: https://doi.org/10.1090/S0002-9939-2014-11859-4
- MathSciNet review: 3162250