On a positivity preservation property for Schrödinger operators on Riemannian manifolds

Milatovic, Ognjen

doi:10.1090/proc/12701

On a positivity preservation property for Schrödinger operators on Riemannian manifolds
HTML articles powered by AMS MathViewer

by Ognjen Milatovic PDF

Proc. Amer. Math. Soc. 144 (2016), 301-313 Request permission

Abstract:

We study a positivity preservation property for Schrödinger operators with singular potential on geodesically complete Riemannian manifolds with non-negative Ricci curvature. We apply this property to the question of self-adjointness of the maximal realization of the corresponding operator.

References

Lashi Bandara, Density problems on vector bundles and manifolds, Proc. Amer. Math. Soc. 142 (2014), no. 8, 2683–2695. MR 3209324, DOI 10.1090/S0002-9939-2014-12284-2
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
Haïm Brézis and Tosio Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9) 58 (1979), no. 2, 137–151. MR 539217
Jochen Brüning, Vladimir Geyler, and Konstantin Pankrashkin, Continuity properties of integral kernels associated with Schrödinger operators on manifolds, Ann. Henri Poincaré 8 (2007), no. 4, 781–816. MR 2333782, DOI 10.1007/s00023-006-0322-z
H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 883643
Allen Devinatz, Schrödinger operators with singular potentials, J. Operator Theory 4 (1980), no. 1, 25–35. MR 587366
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
Alexander Grigor’yan, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, vol. 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. MR 2569498, DOI 10.1090/amsip/047
Robert Grummt and Martin Kolb, Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds, J. Math. Anal. Appl. 388 (2012), no. 1, 480–489. MR 2869762, DOI 10.1016/j.jmaa.2011.09.060
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, Kato’s inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds, Proc. Amer. Math. Soc. 142 (2014), no. 4, 1289–1300. MR 3162250, DOI 10.1090/S0002-9939-2014-11859-4
B. Güneysu, Sequences of Laplacian cut-off functions. J. Geom. Anal. DOI 10.1007/s12220-014-9543-9
Batu Güneysu and Olaf Post, Path integrals and the essential self-adjointness of differential operators on noncompact manifolds, Math. Z. 275 (2013), no. 1-2, 331–348. MR 3101810, DOI 10.1007/s00209-012-1137-2
Robert Jensen, Uniqueness of solutions to ${}-{}\Delta u-qu=0$, Comm. Partial Differential Equations 3 (1978), no. 11, 1053–1076. MR 507786, DOI 10.1080/03605307808820086
H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (1977), no. 5, 509–541. MR 442975, DOI 10.1002/cpa.3160300502
Tosio Kato, Schrödinger operators with singular potentials, Israel J. Math. 13 (1972), 135–148 (1973). MR 333833, DOI 10.1007/BF02760233
Tosio Kato, A second look at the essential selfadjointness of the Schrödinger operators, Physical reality and mathematical description, Reidel, Dordrecht, 1974, pp. 193–201. MR 0477431
T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, New York, 1980.
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
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
József Lőrinczi, Fumio Hiroshima, and Volker Betz, Feynman-Kac-type theorems and Gibbs measures on path space, De Gruyter Studies in Mathematics, vol. 34, Walter de Gruyter & Co., Berlin, 2011. With applications to rigorous quantum field theory. MR 2848339, DOI 10.1515/9783110203738
Ognjen Milatovic, Self-adjoint realizations of Schrödinger operators on vector bundles over Riemannian manifolds, Recent advances in harmonic analysis and partial differential equations, Contemp. Math., vol. 581, Amer. Math. Soc., Providence, RI, 2012, pp. 175–197. MR 3013059, DOI 10.1090/conm/581/11488
Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420
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