Perturbation of translation invariant positivity preserving semigroups on 𝐿²(𝑅^{𝑁})

Herbst, Ira W.; Sloan, Alan D.

doi:10.1090/S0002-9947-1978-0470750-2

Perturbation of translation invariant positivity preserving semigroups on $L^{2}(\textbf {R}^{N})$
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by Ira W. Herbst and Alan D. Sloan PDF

Trans. Amer. Math. Soc. 236 (1978), 325-360 Request permission

Abstract:

The theory of singular local perturbations of translation invariant positivity preserving semigroups on ${L^2}({{\mathbf {R}}^N},{d^N}x)$ is developed. A powerful approximation theorem is proved which allows the treatment of a very general class of singular perturbations. Estimates on the local singularities of the kernels of the semigroups, ${e^{ - tH}}$, are given. Eigenfunction expansions are derived. The local singularities of the eigenfunction and generalized eigenfunctions are discussed. Results are illustrated with examples involving singular perturbations of —$\Delta$.

References

Ju. M. Berezans′kiĭ, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. MR 0222718, DOI 10.1090/mmono/017
A. Beurling and J. Deny, Espaces de Dirichlet. I. Le cas élémentaire, Acta Math. 99 (1958), 203–224 (French). MR 98924, DOI 10.1007/BF02392426
Felix E. Browder, The eigenfunction expansion theorem for the general self-adjoint singular elliptic partial differential operator. I. The analytical foundation, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 454–459. MR 63535, DOI 10.1073/pnas.40.6.454
T. Carleman, Sur la détermination d’un fonction analytique par certaines valeurs moyennes de ses dérivées, Ark. Mat. Astr. Fys. 32B (1945), no. 4, 7 (French). MR 0013806
K. M. Case, Singular potentials, Phys. Rev. (2) 80 (1950), 797–806. MR 40532, DOI 10.1103/PhysRev.80.797
M. Combescure and J. Ginibre, Spectral and scattering theory for the Schrödinger operator with strongly oscillating potentials, Ann. Inst. H. Poincaré Sect. A (N.S.) 24 (1976), no. 1, 17–30. MR 400976
P. Deift and B. Simon, On the decoupling of finite singularities from the question of asymptotic completeness in two body quantum systems, J. Functional Analysis 23 (1976), no. 3, 218–238. MR 0432051, DOI 10.1016/0022-1236(76)90049-5
P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford, at the Clarendon Press, 1947. 3d ed. MR 0023198
William G. Faris, Perturbations and non-normalizable eigenvectors, Helv. Phys. Acta 44 (1971), 930–936. MR 356751
William G. Faris, Self-adjoint operators, Lecture Notes in Mathematics, Vol. 433, Springer-Verlag, Berlin-New York, 1975. MR 0467348, DOI 10.1007/BFb0068567
William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
Masatoshi Fukushima, On the generation of Markov processes by symmetric forms, Proceedings of the Second Japan-USSR Symposium on Probability Theory (Kyoto, 1972) Lecture Notes in Math., Vol. 330, Springer, Berlin, 1973, pp. 46–79. MR 0494513
I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831

The theory of probability

Ira W. Herbst, Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$, Comm. Math. Phys. 53 (1977), no. 3, 285–294. MR 436854
Barry Simon and Raphael Høegh-Krohn, Hypercontractive semigroups and two dimensional self-coupled Bose fields, J. Functional Analysis 9 (1972), 121–180. MR 0293451, DOI 10.1016/0022-1236(72)90008-0
James S. Howland, A perturbation-theoretic approach to eigenfunction expansions, J. Functional Analysis 2 (1968), 1–23. MR 0227798, DOI 10.1016/0022-1236(68)90022-0
Teruo Ikebe, Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory, Arch. Rational Mech. Anal. 5 (1960), 1–34 (1960). MR 128355, DOI 10.1007/BF00252896
Tosio Kato, Fundamental properties of Hamiltonian operators of Schrödinger type, Trans. Amer. Math. Soc. 70 (1951), 195–211. MR 41010, DOI 10.1090/S0002-9947-1951-0041010-X
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473

Schrödinger operators with singular potentials

13

Tosio Kato and S. T. Kuroda, Theory of simple scattering and eigenfunction expansions, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 99–131. MR 0385603
Tosio Kato, On the eigenfunctions of many-particle systems in quantum mechanics, Comm. Pure Appl. Math. 10 (1957), 151–177. MR 88318, DOI 10.1002/cpa.3160100201
L. D. Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics, Vol. 1, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960. Translated from the Russian by J. B. Bell. MR 0120782
F. I. Mautner, On eigenfunction expansions, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 49–53. MR 52690, DOI 10.1073/pnas.39.1.49
Edward Nelson, Feynman integrals and the Schrödinger equation, J. Mathematical Phys. 5 (1964), 332–343. MR 161189, DOI 10.1063/1.1704124
G. Nenciu, Eigenfunction expansions for Schrödinger and Dirac operators with singular potentials, Comm. Math. Phys. 42 (1975), 221–229. MR 374705, DOI 10.1007/BF01608974

Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren

102

Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
Martin Schechter, Hamiltonians for singular potentials, Indiana Univ. Math. J. 22 (1972/73), 483–503. MR 305150, DOI 10.1512/iumj.1972.22.22042
B. Simon, Essential self-adjointness of Schrödinger operators with positive potentials, Math. Ann. 201 (1973), 211–220. MR 337215, DOI 10.1007/BF01427943
Barry Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Anal. 52 (1973), 44–48. MR 338548, DOI 10.1007/BF00249091
Barry Simon, Pointwise bounds on eigenfunctions and wave packets in $N$-body quantum systems. I, II, Proc. Amer. Math. Soc. 42 (1974), 395–401: ibid. 45 (1974), 454–456. MR 417596, DOI 10.1090/S0002-9939-1974-0417596-0
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Dale W. Thoe, Eigenfunction expansions associated with Schroedinger operators in $R_{n}$,$n\geq 4$, Arch. Rational Mech. Anal. 26 (1967), 335–356. MR 218772, DOI 10.1007/BF00281639
H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math. 8 (1958), 887–919. MR 103420, DOI 10.2140/pjm.1958.8.887
François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131