Abstract
We develop perturbation theory of generators of sub-markovian semigroups by relatively form-bounded perturbations. The L p-smoothing properties of semigroups and the uniqueness problem are considered. Applications to operators of mathematical physics are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beliy, A. G. and Semenov, Yu. A.: On the L p-theory of the Schrödinger semigroups, II, Siberian Math. J. 31 (1990), 166–26.
Brezis, H.: Localized self-adjointness of Schrödinger operators, J. Operator Theory 1 (1979), 287–290.
Carlen, E. A., Kusuoka, S. and Stroock, D. W.: Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré 23 (1978), 245–287.
Combescure-Moulin, M. and Ginibre, J.. Essential self-adjointness of many particle Schrödinger Hamiltonians with singular two-body potentials, Ann. Inst. H. Poincaré, Sect. A 23 (1975), 211–234.
Davies, E. B.: One-Parameter Semigroups, Academic Press, London, 1980.
Fefferman, C.. The N-body problem in quantum mechanics, Comm. Pure Appl. Math. 39, Suppl. 1 (1986), S67-S109.
Fefferman, C. and de la, Llave, R.: Relativistic stability of matter, I, Rev. Math. Iberoamericana 1 (1985), 119–213.
Goldstein, J.. Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985.
Herbst, I. W.: Spectral theory of the operator (p 2 + m 2 1)/2 - Ze 2/r, Comm. Math. Phys. 53 (1977), 285–294.
Herbst, I. W. and Sloan, A. D.: Perturbation of translation invariant positivity preserving semigroups on L 2(ℝN), Trans. Amer. Math. Soc. 233 (1978), 325–360.
Kato, T.: Perturbation Theory for Linear Operators, 2nd edn, Springer-Verlag, Berlin, Heidelberg, New York, 1980.
Kato, T.: Schrödinger operators with singular potentials, Israel J. Math. 13 (1972), 135–148.
Kato, T.: L p-theory of Schrödinger operators, in: R., Nagel, U., Schlotterbeck and M, Wolf (eds), Aspects of Positivity in Functional Analysis, North-Holland, Amsterdam, 1986, pp. 63–78.
Kipnis, C.: Majoration des semi-groupes de contractions de et applications, Ann. Inst. H. Poincaré, Sect. B 10 (1974), 369–389.
Kovalenko, V. F., Perelmuter, M. A. and Semenov, Yu. A.: Schrödinger operators with L w l/2(ℝl)-potentials, J. Math. Phys. 22 (1981), 1033–1044.
Kovalenko, V. F. and Semenov, Yu. A.. Essential self-adjointness of many-particle Hamiltonian operators of Schrödinger type with singular two-particle potentials, Ann. Inst. H. Poincaré, Sect. A 24 (1977), 325–334.
Kovalenko, V. F. and Semenov, Yu. A.: On L p-theory of Schrödinger semigroups, I, Ukrain. Math. J. 41 (1989), 273–278.
Kubokawa, Y.: Ergodic theorems for contraction semigroups, J. Math. Soc. Japan 27 (1975), 184–193.
Lieb, E. H. and Thirring, W. E.: Gravitational collapse in quantum mechanics with relativistic kinetic energy, Ann. Phys. (USA) 155 (1984), 494–512.
Lieb, E. H. and Yau, H.-T.. The Chandrasekar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys. 112 (1987), 147–174.
Lieb, E. H. and Yau, H.-T., The stability and instability of relativistic matter, Comm. Math. Phys. 118 (1988), 177–213.
Liskevich, V. A. and Semenov, Yu. A.: One criterion of the essential self-adjointness of the Schrödinger operator, Weizmann Preprint, 1991.
Liskevich, V. A. and Semenov, Yu. A.: Some inequalities for sub-Markovian generators and their applications to the perturbation theory, Proc. Amer. Math. Soc. 119 (1993), 1171–1177.
Liskevich, V. A. and Semenov, Yu. A.: Dirichlet operators: Apriori estimates and uniqueness problems, J. Funct. Anal. 109 (1992), 199–213.
Oinarov, R.: On the separability of the Schrödinger operator in the space of summability functions, Dokl. Akad. Nauk SSSR 285 (1985), 1062–1064.
Perelmuter, M. A.: Schrödinger operators with form-bounded potentials in L p-spaces, Ann. Inst. H. Poincaré, Sect. A. 52 (1990), 151–161.
Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, Vol. 2, Academic Press, Orlando, 1975.
Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, Vol. 4, Academic Press, Orlando, 1978.
Semenov, Yu. A.: On the spectral theory of second-order elliptic differential operators, Math. USSR Sbornik 56 (1987), 221–247.
Simon, B.: Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526
Simader, C. G.: Essential self-adjointness of Schrödinger operators bounded from below, Math. Z. 159 (1978), 47–50.
Stein, E. M.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton Univ. Press, Princeton, N.J., 1970.
Stroock, D. W.: An Introduction to the Theory of Large Deviations, Springer-Verlag, New York, 1984.
Varopoulos, N. Th.: Hardy-Littlewood theory for semigroups, J. Funct. Anal. 63 (1985), 240–260.
Voigt, J.: Absorption semigroups, their generators, and Schrödinger semigroups, J. Funct. Anal. 67 (1986), 167–205.
Voigt, J.: Absorption semigroups, J. Operator Theory 20 (1988), 117–131.
Yosida, K.: Functional Analysis, Springer-Verlag, Berlin, New York, Heidelberg, 1980.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Liskevich, V.A., Perelmuter, M.A. & Semenov, Y.A. Form-bounded perturbations of generators of sub-Markovian semigroups. Acta Appl Math 44, 353–377 (1996). https://doi.org/10.1007/BF00047397
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00047397