×

zbMATH — the first resource for mathematics

Schrödinger semigroups. (English) Zbl 0524.35002

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J10 Schrödinger operator, Schrödinger equation
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
47F05 General theory of partial differential operators
35P05 General topics in linear spectral theory for PDEs
47D03 Groups and semigroups of linear operators
35K05 Heat equation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Shmuel Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151 – 218. · Zbl 0315.47007
[2] S. Agmon, On exponential decay of solutions of second order elliptic equations in unbounded domains, Proc. Conf. A. Pleijel. · Zbl 0503.35001
[3] S. Agmon, private communication (in preparation).
[4] Shmuel Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of \?-body Schrödinger operators, Mathematical Notes, vol. 29, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982. · Zbl 0503.35001
[5] J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22 (1971), 269 – 279. · Zbl 0219.47011
[6] Reinhart Ahlrichs, Asymptotic behavior of atomic bound state wave functions, J. Mathematical Phys. 14 (1973), 1860 – 1863.
[7] R. Ahlrichs, M. Hoffman-Ostenhof and T. Hoffman-Ostenhof, Bounds on the long-range behavior of atomic wave functions, J. Chem. Phys. 68 (1978), 1402-1410.
[8] Reinhart Ahlrichs, Maria Hoffmann-Ostenhof, Thomas Hoffmann-Ostenhof, and John D. Morgan III, Bounds on the decay of electron densities with screening, Phys. Rev. A (3) 23 (1981), no. 5, 2106 – 2117. · Zbl 0446.35017
[9] M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), no. 2, 209 – 273. · Zbl 0459.60069
[10] Sergio Albeverio, Raphael Høegh-Krohn, and Ludwig Streit, Energy forms, Hamiltonians, and distorted Brownian paths, J. Mathematical Phys. 18 (1977), no. 5, 907 – 917. · Zbl 0368.60091
[11] S. Albeverio, R. Høegh-Krohn, and L. Streit, Regularization of Hamiltonians and processes, J. Math. Phys. 21 (1980), no. 7, 1636 – 1642. · Zbl 0455.60088
[12] V. de Alfaro and T. Regge, Potential scattering, North-Holland, Amsterdam, 1965. · Zbl 0141.23202
[13] W. Allegretto, On the equivalence of two types of oscillation for elliptic operators, Pacific J. Math. 55 (1974), 319 – 328. · Zbl 0279.35036
[14] Walter Allegretto, Spectral estimates and oscillations of singular differential operators, Proc. Amer. Math. Soc. 73 (1979), no. 1, 51 – 56. · Zbl 0427.35029
[15] W. Allegretto, Positive solutions and spectral properties of second order elliptic operators, Pacific J. Math. 92 (1981), no. 1, 15 – 25. · Zbl 0421.35063
[16] W. O. Amrein, A.-M. Berthier, and V. Georgescu, \?^{\?}-inequalities for the Laplacian and unique continuation, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 3, vii, 153 – 168 (English, with French summary). · Zbl 0468.35017
[17] J. E. Avron and I. W. Herbst, Spectral and scattering theory of Schrödinger operators related to the Stark effect, Comm. Math. Phys. 52 (1977), no. 3, 239 – 254. · Zbl 0351.47007
[18] Joseph Avron and Barry Simon, Singular continuous spectrum for a class of almost periodic Jacobi matrices, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 81 – 85. · Zbl 0491.47014
[19] J. Avron and B. Simon, Almost periodic Schrödinger operators. II, The density of states (preprint) · Zbl 0544.35030
[20] N. Bazley and D. Fox, Bounds for eigenfunctions of one-electron molecular systems, Internat. J. Quant. Chem. 3 (1969), 581-586.
[21] M. M. Benderskiĭ and L. A. Pastur, The spectrum of the one-dimensional Schrödinger equation with random potential, Mat. Sb. (N.S.) 82 (124) (1970), 273 – 284 (Russian).
[22] Yu. M. Berezanskiĭ, On expansion according to eigenfunctions of general self-adjoint differential operators, Dokl. Akad. Nauk SSSR (N.S.) 108 (1956), 379 – 382 (Russian).
[23] Ju. M. Berezans\(^{\prime}\)kiĭ, Expansions in eigenfunctions of selfadjoint operators, Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968.
[24] F. Berezin, Wick and anti-Wick symbols, Math. USSR Sb. 15 (1971), 577-606. · Zbl 0247.47018
[25] Anne-Marie Berthier, Sur le spectre ponctuel de l’opérateur de Schrödinger, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 8, A393 – A395 (French, with English summary). · Zbl 0454.35070
[26] A. M. Berthier, On the point spectrum of Schrödinger operators, Ann. Sci. École Norm. Sup. (to appear). · Zbl 0487.35071
[27] Anne-Marie Berthier and Bernard Gaveau, Critère de convergence des fonctionnelles de Kac et application en mécanique quantique et en géométrie, J. Funct. Anal. 29 (1978), no. 3, 416 – 424 (French). · Zbl 0398.60076
[28] M. Š. Birman and M. G. Kreĭn, On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR 144 (1962), 475 – 478 (Russian).
[29] Herm Jan Brascamp and Elliott H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Advances in Math. 20 (1976), no. 2, 151 – 173. · Zbl 0339.26020
[30] H. J. Brascamp, Elliott H. Lieb, and J. M. Luttinger, A general rearrangement inequality for multiple integrals, J. Functional Analysis 17 (1974), 227 – 237. · Zbl 0286.26005
[31] 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. · Zbl 0408.35025
[32] Felix E. Browder, Eigenfunction expansions for formally self-adjoint partial differential operators. I, II, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 769 – 771, 870 – 872. · Zbl 0071.09801
[33] T. Carleman, Sur un problème d’unicité pour les systèmes d’équations aux dérivés partielles à deux variables indépendantes, Ark. Mat. 26B (1939), 1-9. · Zbl 0022.34201
[34] René Carmona, Regularity properties of Schrödinger and Dirichlet semigroups, J. Funct. Anal. 33 (1979), no. 3, 259 – 296. · Zbl 0419.60075
[35] René Carmona, Pointwise bounds for Schrödinger eigenstates, Comm. Math. Phys. 62 (1978), no. 2, 97 – 106. · Zbl 0403.47016
[36] René Carmona, Exponential localization in one-dimensional disordered systems, Duke Math. J. 49 (1982), no. 1, 191 – 213. · Zbl 0491.60058
[37] R. Carmona and B. Simon, Pointwise bounds on eigenfunctions and wave packets in \?-body quantum systems. V. Lower bounds and path integrals, Comm. Math. Phys. 80 (1981), no. 1, 59 – 98. Elliott H. Lieb and Barry Simon, Pointwise bounds on eigenfunctions and wave packets in \?-body quantum systems. VI. Asymptotics in the two-cluster region, Adv. in Appl. Math. 1 (1980), no. 3, 324 – 343. · Zbl 0482.35065
[38] Kai Lai Chung and S. R. S. Varadhan, Kac functional and Schrödinger equation, Studia Math. 68 (1980), no. 3, 249 – 260. · Zbl 0448.60054
[39] Kai Lai Chung and K. Murali Rao, Sur la théorie du potentiel avec la fonctionnelle de Feynman-Kac, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 13, A629 – A631 (French, with English summary). · Zbl 0439.31005
[40] J. Combes, Time dependent approach to multichannel scattering, Nuovo Cimento A 64 (1969), 111-144. · Zbl 0181.27604
[41] J.-M. Combes, M. Hoffmann-Ostenhof, and T. Hoffmann-Ostenhof, Asymptotics of atomic ground states: the relation between the ground state of helium and the ground state of \?\?\(^{+}\), J. Math. Phys. 22 (1981), no. 6, 1299 – 1305. · Zbl 0471.35011
[42] J. M. Combes, R. Schrader, and R. Seiler, Classical bounds and limits for energy distributions of Hamilton operators in electromagnetic fields, Ann Physics 111 (1978), no. 1, 1 – 18.
[43] J. M. Combes and L. Thomas, Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators, Comm. Math. Phys. 34 (1973), 251 – 270. · Zbl 0271.35062
[44] Michael Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. of Math. (2) 106 (1977), no. 1, 93 – 100. · Zbl 0362.47006
[45] E. B. Davies, Properties of Green’s functions of some Schrödinger operators, J. London Math. Soc. 7 (1973), 473-491. · Zbl 0271.47003
[46] E. B. Davies, Eigenfunction expansions for singular Schrödinger operators, Arch. Rational Mech. Anal. 63 (1976), no. 3, 261 – 272. · Zbl 0352.35068
[47] Barry Simon, Hardy and Rellich inequalities in nonintegral dimension, J. Operator Theory 9 (1983), no. 1, 143 – 146. E. B. Davies, Some norm bounds and quadratic form inequalities for Schrödinger operators, J. Operator Theory 9 (1983), no. 1, 147 – 162.
[48] E. B. Davies, JWKB and related bounds on Schrödinger eigenfunctions, Bull. London Math. Soc. 14 (1982), no. 4, 273 – 284. · Zbl 0525.35026
[49] E. B. Davies and B. Simon, Scattering theory for systems with different spatial asymptotics on the left and right, Comm. Math. Phys. 63 (1978), no. 3, 277 – 301. · Zbl 0393.34015
[50] P. Deift, W. Hunziker, B. Simon, and E. Vock, Pointwise bounds on eigenfunctions and wave packets in \?-body quantum systems. IV, Comm. Math. Phys. 64 (1978/79), no. 1, 1 – 34. · Zbl 0419.35079
[51] M. Eastham, The spectral theory of periodic differential equations, Scottish Academic Press, 1973. · Zbl 0287.34016
[52] J.-P. Eckmann, Hypercontractivity for anharmonic oscillators, J. Functional Analysis 16 (1974), 388 – 404. With an appendix by D. Pearson. · Zbl 0285.47032
[53] Volker Enss, A note on Hunziker’s theorem, Comm. Math. Phys. 52 (1977), no. 3, 233 – 238.
[54] William G. Faris, Perturbations and non-normalizable eigenvectors, Helv. Phys. Acta 44 (1971), 930 – 936.
[55] William G. Faris, Quadratic forms and essential self-adjointness, Helv. Phys. Acta 45 (1972/73), 1074 – 1088.
[56] William G. Faris, Essential self-adjointness of operators in ordered Hilbert space, Comm. Math. Phys. 30 (1973), 23 – 34. · Zbl 0251.47020
[57] W. Faris, private communication.
[58] William G. Faris and Richard B. Lavine, Commutators and self-adjointness of Hamiltonian operators, Comm. Math. Phys. 35 (1974), 39 – 48. · Zbl 0287.47004
[59] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801
[60] R. Froese, I. Herbst, M. Hoffman-Ostenhof and T. Hoffman-Ostenhof, L, Comm. Math. Phys. (to appear).
[61] Daisuke Fujiwara, A construction of the fundamental solution for the Schrödinger equation, J. Analyse Math. 35 (1979), 41 – 96. · Zbl 0418.35032
[62] Daisuke Fujiwara, On a nature of convergence of some Feynman path integrals. I, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 6, 195 – 200. Daisuke Fujiwara, On a nature of convergence of some Feynman path integrals. II, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 8, 273 – 277. · Zbl 0432.35007
[63] Daisuke Fujiwara, Remarks on convergence of the Feynman path integrals, Duke Math. J. 47 (1980), no. 3, 559 – 600. · Zbl 0457.35026
[64] Masatoshi Fukushima, On the spectral distribution of a disordered system and the range of a random walk, Osaka J. Math. 11 (1974), 73 – 85. · Zbl 0366.60099
[65] Masatoshi Fukushima, On holomorphic diffusions and plurisubharmonic functions, Geometry of random motion (Ithaca, N.Y., 1987) Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 65 – 78.
[66] Masatoshi Fukushima and Shintaro Nakao, On spectra of the Schrödinger operator with a white Gaussian noise potential, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37 (1976/77), no. 3, 267 – 274. · Zbl 0361.60044
[67] Lars Gårding, Eigenfunction expansions connected with elliptic differential operators, Tolfte Skandinaviska Matematikerkongressen, Lund, 1953, Lunds Universitets Matematiska Institution, Lund, 1954, pp. 44 – 55. · Zbl 0053.39101
[68] Bernard Gaveau and Edmond Mazet, Divergence des fonctionnelles de Kac et diffusion quantique, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), no. 9, A559 – A562 (French, with English summary). · Zbl 0451.60072
[69] I. M. Gel\(^{\prime}\)fand, Expansion in characteristic functions of an equation with periodic coefficients, Doklady Akad. Nauk SSSR (N.S.) 73 (1950), 1117 – 1120 (Russian).
[70] I. M. Gel\(^{\prime}\)fand and A. G. Kostyučenko, Expansion in eigenfunctions of differential and other operators, Dokl. Akad. Nauk SSSR (N.S.) 103 (1955), 349 – 352 (Russian).
[71] V. Georgescu, On the unique continuation property for Schrödinger Hamiltonians, Helv. Phys. Acta 52 (1979), no. 5-6, 655 – 670 (1980).
[72] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. · Zbl 0361.35003
[73] James Glimm, Boson fields with the :\Phi \(^{4}\): interaction in three dimensions, Comm. Math. Phys. 10 (1968), 1 – 47. · Zbl 0175.24702
[74] James Glimm and Arthur Jaffe, A \?\?\(^{4}\) quantum field without cutoffs. I, Phys. Rev. (2) 176 (1968), 1945 – 1951. · Zbl 0177.28203
[75] James Glimm and Arthur Jaffe, The \?(\Pi \(^{4}\))\(_{2}\) quantum field theory without cutoffs. II. The field operators and the approximate vacuum, Ann. of Math. (2) 91 (1970), 362 – 401. · Zbl 0191.27005
[76] James Glimm and Arthur Jaffe, Quantum physics, Springer-Verlag, New York-Berlin, 1981. A functional integral point of view. · Zbl 0461.46051
[77] I. Ja. Gol\(^{\prime}\)dšeĭd, S. A. Molčanov, and L. A. Pastur, A random homogeneous Schrödinger operator has a pure point spectrum, Funkcional. Anal. i Priložen. 11 (1977), no. 1, 1 – 10, 96 (Russian).
[78] Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061 – 1083. · Zbl 0318.46049
[79] Erhard Heinz, Über die Eindeutigkeit beim Cauchyschen Anfangswertproblem einer elliptischen Differentialgleichung zweiter Ordnung, Nachr. Akad. Wiss. Göttingen. IIa. 1955 (1955), 1 – 12 (German). · Zbl 0067.07503
[80] Ira W. Herbst, Dilation analyticity in constant electric field. I. The two body problem, Comm. Math. Phys. 64 (1979), no. 3, 279 – 298. · Zbl 0447.47028
[81] I. W. Herbst and J. S. Howland, The Stark ladder and other one-dimensional external field problems, Comm. Math. Phys. 80 (1981), no. 1, 23 – 42. · Zbl 0473.47037
[82] Ira W. Herbst and B. Simon, Dilation analyticity in constant electric field. II. \?-body problem, Borel summability, Comm. Math. Phys. 80 (1981), no. 2, 181 – 216. · Zbl 0473.47038
[83] Ira W. Herbst and Alan D. Sloan, Perturbation of translation invariant positivity preserving semigroups on \?²(\?^{\?}), Trans. Amer. Math. Soc. 236 (1978), 325 – 360. · Zbl 0388.47022
[84] H. Hess, R. Schrader, and D. A. Uhlenbrock, Domination of semigroups and generalization of Kato’s inequality, Duke Math. J. 44 (1977), no. 4, 893 – 904. · Zbl 0379.47028
[85] Raphael Høegh-Krohn, A general class of quantum fields without cut-offs in two space-time dimensions, Comm. Math. Phys. 21 (1971), 244 – 255.
[86] Maria Hoffmann-Ostenhof and Thomas Hoffmann-Ostenhof, ”Schrödinger inequalities” and asymptotic behavior of the electron density of atoms and molecules, Phys. Rev. A (3) 16 (1977), no. 5, 1782 – 1785.
[87] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and B. Simon, On the nodal structure of atomic eigenfunctions, J. Phys. A 13 (1980), no. 4, 1131 – 1133. · Zbl 0428.35078
[88] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and B. Simon, Brownian motion and a consequence of Harnack’s inequality: nodes of quantum wave functions, Proc. Amer. Math. Soc. 80 (1980), no. 2, 301 – 305. · Zbl 0444.35024
[89] M. Hoffmann-Ostenhof and R. Seiler, Cusp conditions for eigenfunctions of \?-electron systems, Phys. Rev. A (3) 23 (1981), no. 1, 21 – 23.
[90] Thomas Hoffmann-Ostenhof, A comparison theorem for differential inequalities with applications in quantum mechanics, J. Phys. A 13 (1980), no. 2, 417 – 424. · Zbl 0432.35080
[91] Thomas Hoffmann-Ostenhof, Lower and upper bounds to the decay of the ground state one-electron density of helium-like systems, J. Phys. A 12 (1979), no. 8, 1181 – 1187.
[92] T. Hoffman-Ostenhof, A lower bound to the decay of ground states of two electron atoms, Phys. Lett. A 77 (1980), 140-142.
[93] T. Hoffman-Ostenhof and M. Hoffman-Ostenhof, Bounds to expectation values and exponentially decreasing upper bounds to the one electron density of atoms, J. Phys. B 11 (1978), 17-24.
[94] T. Hoffman-Ostenhof and M. Hoffman-Ostenhof, Exponentially decreasing upper bounds to the electron density of atoms, Phys. Lett. A 59 (1976), 373-374.
[95] T. Hoffman-Ostenhof, M. Hoffman-Ostenhof and R. Ahlrichs, Schrödinger inequalities and asymptotic behavior of many electron densities, Phys. Rev. A 18 (1978), 328-334.
[96] James G. Hooton, Dirichlet forms associated with hypercontractive semigroups, Trans. Amer. Math. Soc. 253 (1979), 237 – 256. · Zbl 0424.47028
[97] James G. Hooton, Dirichlet semigroups on bounded domains, Rocky Mountain J. Math. 12 (1982), no. 2, 283 – 297. · Zbl 0521.47021
[98] L. Hörmander, Seminar Schwarz, 1981.
[99] Lars Hörmander, Linear partial differential operators, Third revised printing. Die Grundlehren der mathematischen Wissenschaften, Band 116, Springer-Verlag New York Inc., New York, 1969. · Zbl 0175.39201
[100] W. Hunziker, On the space-time behavior of Schrödinger wavefunctions, J. Mathematical Phys. 7 (1966), 300 – 304. · Zbl 0151.43801
[101] Teruo Ikebe, Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory, Arch. Rational Mech. Anal. 5 (1960), 1 – 34 (1960). · Zbl 0145.36902
[102] Teruo Ikebe, Spectral representation for Schrödinger operators with long-range potentials. II. Perturbation by short-range potentials, Publ. Res. Inst. Math. Sci. 11 (1975/76), no. 2, 551 – 558. · Zbl 0345.35032
[103] Teruo Ikebe and Hiroshi Isozaki, Completeness of modified wave operators for long-range potentials, Publ. Res. Inst. Math. Sci. 15 (1979), no. 3, 679 – 718. · Zbl 0432.35061
[104] Hiroshi Isozaki, On the long-range stationary wave operator, Publ. Res. Inst. Math. Sci. 13 (1977/78), no. 3, 589 – 626. · Zbl 0435.47011
[105] Arne Jensen and Tosio Kato, Asymptotic behavior of the scattering phase for exterior domains, Comm. Partial Differential Equations 3 (1978), no. 12, 1165 – 1195. · Zbl 0419.35067
[106] R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), no. 3, 403 – 438. · Zbl 0497.35026
[107] G. I. Kac, Expansion in characteristic functions of self-adjoint operators, Dokl. Akad. Nauk SSSR (N.S.) 119 (1958), 19 – 22 (Russian). · Zbl 0080.10602
[108] H. Kalf, U.-W. Schmincke, J. Walter, and R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Springer, Berlin, 1975, pp. 182 – 226. Lecture Notes in Math., Vol. 448. · Zbl 0311.47021
[109] B. Karlerson, Self-adjointness of Schrödinger operators, Institut Mittag-Leffler Rep. no. 6, 1976.
[110] Tosio Kato, On the eigenfunctions of many-particle systems in quantum mechanics, Comm. Pure Appl. Math. 10 (1957), 151 – 177. · Zbl 0077.20904
[111] Tosio Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162 (1965/1966), 258 – 279. · Zbl 0139.31203
[112] Tosio Kato, Schrödinger operators with singular potentials, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 135 – 148 (1973). · Zbl 0246.35025
[113] Tosio Kato, Remarks on Schrödinger operators with vector potentials, Integral Equations Operator Theory 1 (1978), no. 1, 103 – 113. · Zbl 0395.47023
[114] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. · Zbl 0148.12601
[115] R. Z. Has\(^{\prime}\)minskiĭ, On positive solutions of the equation \cal\?+\?\?=0, Theor. Probability Appl. 4 (1959), 309 – 318.
[116] Werner Kirsch and Fabio Martinelli, On the density of states of Schrödinger operators with a random potential, J. Phys. A 15 (1982), no. 7, 2139 – 2156. · Zbl 0492.60055
[117] Hitoshi Kitada, Scattering theory for Schrödinger operators with long-range potentials. I. Abstract theory, J. Math. Soc. Japan 29 (1977), no. 4, 665 – 691. , https://doi.org/10.2969/jmsj/02940665 Hitoshi Kitada, Scattering theory for Schrödinger operators with long-range potentials. II. Spectral and scattering theory, J. Math. Soc. Japan 30 (1978), no. 4, 603 – 632. · Zbl 0388.35055
[118] Hitoshi Kitada, On a construction of the fundamental solution for Schrödinger equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 1, 193 – 226. · Zbl 0433.35018
[119] Hitoshi Kitada and Hitoshi Kumano-go, A family of Fourier integral operators and the fundamental solution for a Schrödinger equation, Osaka J. Math. 18 (1981), no. 2, 291 – 360. · Zbl 0472.35034
[120] V. B. Korotkov, On integral operators with Carleman kernels, Dokl. Akad. Nauk SSSR 165 (1965), 748 – 751 (Russian). · Zbl 0186.20601
[121] Shinichi Kotani, On asymptotic behaviour of the spectra of a one-dimensional Hamiltonian with a certain random coefficient, Publ. Res. Inst. Math. Sci. 12 (1976/77), no. 2, 447 – 492. · Zbl 0362.34043
[122] V. F. Kovalenko, M. A. Perel\(^{\prime}\)muter, and Ya. A. Semenov, Schrödinger operators with \?^{1/2}_{\?}(\?^{\?})-potentials, J. Math. Phys. 22 (1981), no. 5, 1033 – 1044. · Zbl 0463.47027
[123] V. F. Kovalenko and Ju. A. Semenov, Some questions on expansions in generalized eigenfunctions of a Schrödinger operator with strongly singular potentials, Uspekhi Mat. Nauk 33 (1978), no. 4(202), 107 – 140, 255 (Russian). · Zbl 0383.47010
[124] M. G. Kreĭn, On the trace formula in perturbation theory, Mat. Sbornik N.S. 33(75) (1953), 597 – 626 (Russian). · Zbl 0052.12303
[125] M. G. Kreĭn, On perturbation determinants and a trace formula for unitary and self-adjoint operators, Dokl. Akad. Nauk SSSR 144 (1962), 268 – 271 (Russian).
[126] S. Kuroda, Scattering theory for differential operators. I, II, J. Math. Soc. Japan 25 (1973), 75-104; 222-234. · Zbl 0245.47006
[127] R. Lavine, The local spectral density and its classical limit (preprint).
[128] Herbert Leinfelder, Gauge invariance of Schrödinger operators and related spectral properties, J. Operator Theory 9 (1983), no. 1, 163 – 179. · Zbl 0528.35024
[129] Herbert Leinfelder and Christian G. Simader, Schrödinger operators with singular magnetic vector potentials, Math. Z. 176 (1981), no. 1, 1 – 19. · Zbl 0468.35038
[130] R. Carmona and B. Simon, Pointwise bounds on eigenfunctions and wave packets in \?-body quantum systems. V. Lower bounds and path integrals, Comm. Math. Phys. 80 (1981), no. 1, 59 – 98. Elliott H. Lieb and Barry Simon, Pointwise bounds on eigenfunctions and wave packets in \?-body quantum systems. VI. Asymptotics in the two-cluster region, Adv. in Appl. Math. 1 (1980), no. 3, 324 – 343. · Zbl 0482.35065
[131] Lars Lithner, A theorem of the Phragmén-Lindelöf type for second-order elliptic operators, Ark. Mat. 5 (1964), 281 – 285 (1964). · Zbl 0161.30901
[132] V. Maslov, On the asymptotics of generalized functions of the Schrödinger equation, Uspehi Mat. Nauk. 16 (1961), 253-254. · Zbl 0114.04203
[133] H. P. McKean, -\? plus a bad potential, J. Mathematical Phys. 18 (1977), no. 6, 1277 – 1279. · Zbl 0357.47025
[134] S. Mercuriev, On the asymptotic form of three body wave functions for the discrete spectrum, Soviet J. Nuclear Phys. 19 (1974), 222-229.
[135] S. A. Molčanov, Structure of the eigenfunctions of one-dimensional unordered structures, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 1, 70 – 103, 214 (Russian).
[136] J. Morgan, The exponential decay of subcontinuum wave functions of two electron atoms, J. Phys. A 10 (1977), 291.
[137] Jürgen Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577 – 591. · Zbl 0111.09302
[138] William F. Moss and John Piepenbrink, Positive solutions of elliptic equations, Pacific J. Math. 75 (1978), no. 1, 219 – 226. · Zbl 0381.35026
[139] Claus Müller, On the behavior of the solutions of the differential equation \Delta \?=\?(\?,\?) in the neighborhood of a point, Comm. Pure Appl. Math. 7 (1954), 505 – 515. · Zbl 0056.32201
[140] Shintaro Nakao, On the spectral distribution of the Schrödinger operator with random potential, Japan. J. Math. (N.S.) 3 (1977), no. 1, 111 – 139. · Zbl 0375.60067
[141] Edward Nelson, A quartic interaction in two dimensions, Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965) M.I.T. Press, Cambridge, Mass., 1966, pp. 69 – 73.
[142] Edward Nelson, The free Markoff field, J. Functional Analysis 12 (1973), 211 – 227. · Zbl 0273.60079
[143] A. J. O’Connor, Exponential decay of bound state wave functions, Comm. Math. Phys. 32 (1973), 319 – 340.
[144] L. A. Pastur, The Schrödinger equation with random potential, Teoret. Mat. Fiz. 6 (1971), no. 3, 415 – 424 (Russian, with English summary). · Zbl 0208.12903
[145] L. A. Pastur, The distribution of eigenvalues of the Schrödinger equation with a random potential, Funkcional. Anal. i Priložen. 6 (1972), no. 2, 93 – 94 (Russian). · Zbl 0252.35051
[146] L. A. Pastur, Spectra of random selfadjoint operators, Uspehi Mat. Nauk 28 (1973), no. 1(169), 3 – 64 (Russian).
[147] L. A. Pastur, The behavior of certain Wiener integrals as \?\to \infty and the density of states of Schrödinger equations with random potential, Teoret. Mat. Fiz. 32 (1977), no. 1, 88 – 95 (Russian, with English summary). · Zbl 0353.60053
[148] L. A. Pastur, Spectral properties of disordered systems in the one-body approximation, Comm. Math. Phys. 75 (1980), no. 2, 179 – 196. · Zbl 0429.60099
[149] D. B. Pearson, Singular continuous measures in scattering theory, Comm. Math. Phys. 60 (1978), no. 1, 13 – 36. · Zbl 0451.47013
[150] M. A. Perel\(^{\prime}\)muter, Perturbation of operators with an integral resolvent, Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 75 – 76 (Russian).
[151] P. Perry, I. M. Sigal, and B. Simon, Spectral analysis of \?-body Schrödinger operators, Ann. of Math. (2) 114 (1981), no. 3, 519 – 567. · Zbl 0477.35069
[152] Arne Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator, Math. Scand. 8 (1960), 143 – 153. · Zbl 0145.14901
[153] John Piepenbrink, Nonoscillatory elliptic equations, J. Differential Equations 15 (1974), 541 – 550. · Zbl 0289.35026
[154] John Piepenbrink, A conjecture of Glazman, J. Differential Equations 24 (1977), no. 2, 173 – 177. · Zbl 0353.35071
[155] Sidney C. Port and Charles J. Stone, Brownian motion and classical potential theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Probability and Mathematical Statistics. · Zbl 0413.60067
[156] N. I. Portenko, Diffusion processes with an unbounded drift coefficient, Teor. Verojatnost. i Primenen. 20 (1975), 29 – 39 (Russian, with English summary). · Zbl 0335.60050
[157] A. Ya. Povzner, On the expansion of arbitrary functions in characteristic functions of the operator -\Delta \?+\?\?, Mat. Sbornik N.S. 32(74) (1953), 109 – 156 (Russian). · Zbl 0050.32201
[158] A. Ya. Povzner, On expansions in functions which are solutions of a scattering problem, Dokl. Akad. Nauk SSSR (N.S.) 104 (1955), 360 – 363 (Russian).
[159] J. Rauch, private communication.
[160] Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. · Zbl 0459.46001
[161] Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. · Zbl 0459.46001
[162] Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. · Zbl 0459.46001
[163] Jay Rosen, Sobolev inequalities for weight spaces and supercontractivity, Trans. Amer. Math. Soc. 222 (1976), 367 – 376. · Zbl 0344.46072
[164] O. S. Rothaus, Lower bounds for eigenvalues of regular Sturm-Liouville operators and the logarithmic Sobolev inequality, Duke Math. J. 45 (1978), no. 2, 351 – 362. · Zbl 0435.47049
[165] O. S. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Sturm-Liouville operators, J. Funct. Anal. 39 (1980), no. 1, 42 – 56. · Zbl 0472.47024
[166] O. S. Rothaus, Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities, J. Funct. Anal. 42 (1981), no. 1, 102 – 109. , https://doi.org/10.1016/0022-1236(81)90049-5 O. S. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Funct. Anal. 42 (1981), no. 1, 110 – 120. · Zbl 0471.58025
[167] O. S. Rothaus, Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities, J. Funct. Anal. 42 (1981), no. 1, 102 – 109. , https://doi.org/10.1016/0022-1236(81)90049-5 O. S. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Funct. Anal. 42 (1981), no. 1, 110 – 120. · Zbl 0471.58025
[168] Y. Saito, Eigenfunctions for the Schrödinger operators with long range potentials Q(y) = O(|y|-\epsilon (\epsilon > 0), Osaka J. Math. 14 (1977), 11-35.
[169] Jean-Claude Saut and Bruno Scheurer, Un théorème de prolongement unique pour des opérateurs elliptiques dont les coefficients ne sont pas localement bornés, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 13, A595 – A598 (French, with English summary). · Zbl 0429.35020
[170] Martin Schechter, Hamiltonians for singular potentials, Indiana Univ. Math. J. 22 (1972/73), 483 – 503. · Zbl 0263.47009
[171] Martin Schechter, Essential self-adjointness of the Schrödinger operator with magnetic vector potential, J. Functional Analysis 20 (1975), no. 2, 93 – 104. · Zbl 0323.35022
[172] Martin Schechter, Spectra of partial differential operators, 2nd ed., North-Holland Series in Applied Mathematics and Mechanics, vol. 14, North-Holland Publishing Co., Amsterdam, 1986. · Zbl 0607.35005
[173] M. Schechter and B. Simon, Unique continuation for Schrödinger operators with unbounded potentials, J. Math. Anal. Appl. 77 (1980), no. 2, 482 – 492. · Zbl 0458.35024
[174] I. Sch’nol, On the behavior of the Schrödinger equation, Mat. Sb. 42 (1957), 273-286. (Russian)
[175] Irving Segal, Construction of non-linear local quantum processes. I, Ann. of Math. (2) 92 (1970), 462 – 481. · Zbl 0213.40904
[176] Irving Segal, Construction of non-linear local quantum processes. II, Invent. Math. 14 (1971), 211 – 241. · Zbl 0221.47023
[177] I. Sigal, Geometricparametrices in the QM N-body problem, Duke Math. J. (to appear).
[178] B. Simon, Essential self-adjointness of Schrödinger operators with positive potentials, Math. Ann. 201 (1973), 211 – 220. · Zbl 0234.47027
[179] Barry Simon, Schrödinger operators with singular magnetic vector potentials, Math. Z. 131 (1973), 361 – 370. · Zbl 0277.47006
[180] Barry Simon, Pointwise bounds on eigenfunctions and wave packets in \?-body quantum systems. I, II, Proc. Amer. Math. Soc. 42 (1974), 395 – 401: ibid. 45 (1974), 454 – 456. · Zbl 0285.35008
[181] Barry Simon, Pointwise bounds on eigenfunctions and wave packets in \?-body quantum systems. I, II, Proc. Amer. Math. Soc. 42 (1974), 395 – 401: ibid. 45 (1974), 454 – 456. · Zbl 0285.35008
[182] Barry Simon, Pointwise bounds on eigenfunctions and wave packets in \?-body quantum systems. III, Trans. Amer. Math. Soc. 208 (1975), 317 – 329. · Zbl 0305.35078
[183] Barry Simon, A remark on Nelson’s best hypercontractive estimates, Proc. Amer. Math. Soc. 55 (1976), no. 2, 376 – 378. · Zbl 0441.46026
[184] Barry Simon, An abstract Kato’s inequality for generators of positivity preserving semigroups, Indiana Univ. Math. J. 26 (1977), no. 6, 1067 – 1073. · Zbl 0389.47021
[185] Barry Simon, Kato’s inequality and the comparison of semigroups, J. Funct. Anal. 32 (1979), no. 1, 97 – 101. · Zbl 0413.47037
[186] Barry Simon, Phase space analysis of simple scattering systems: extensions of some work of Enss, Duke Math. J. 46 (1979), no. 1, 119 – 168. · Zbl 0402.35076
[187] Barry Simon, Maximal and minimal Schrödinger forms, J. Operator Theory 1 (1979), no. 1, 37 – 47. · Zbl 0446.35035
[188] Barry Simon, The classical limit of quantum partition functions, Comm. Math. Phys. 71 (1980), no. 3, 247 – 276. · Zbl 0436.22012
[189] Barry Simon, Brownian motion, \?^{\?} properties of Schrödinger operators and the localization of binding, J. Funct. Anal. 35 (1980), no. 2, 215 – 229. · Zbl 0446.47041
[190] Barry Simon, Spectrum and continuum eigenfunctions of Schrödinger operators, J. Funct. Anal. 42 (1981), no. 3, 347 – 355. · Zbl 0471.47028
[191] Barry Simon, Large time behavior of the \?^{\?} norm of Schrödinger semigroups, J. Funct. Anal. 40 (1981), no. 1, 66 – 83. · Zbl 0478.47024
[192] Barry Simon, Almost periodic Schrödinger operators: a review, Adv. in Appl. Math. 3 (1982), no. 4, 463 – 490. · Zbl 0545.34023
[193] Barry Simon, The \?(\?)\(_{2}\) Euclidean (quantum) field theory, Princeton University Press, Princeton, N.J., 1974. Princeton Series in Physics. · Zbl 1175.81146
[194] Barry Simon, Functional integration and quantum physics, Pure and Applied Mathematics, vol. 86, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. · Zbl 0434.28013
[195] Barry Simon, Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge-New York, 1979. · Zbl 0423.47001
[196] Barry Simon and Raphael Høegh-Krohn, Hypercontractive semigroups and two dimensional self-coupled Bose fields, J. Functional Analysis 9 (1972), 121 – 180. · Zbl 0241.47029
[197] E. Leo Slaggie and Eyvind H. Wichmann, Asymptotic properties of the wave function for a bound nonrelativistic three-body system, J. Mathematical Phys. 3 (1962), 946 – 968.
[198] I. M. Slivnjak, Spectrum of the Schrödinger operator with a random potential, Ž. Vyčisl. Mat. i Mat. Fiz. 6 (1966), 1104 – 1108 (Russian).
[199] Guido Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 189 – 258 (French). · Zbl 0151.15401
[200] Friedrich Stummel, Singuläre elliptische Differential-operatoren in Hilbertschen Räumen, Math. Ann. 132 (1956), 150 – 176 (German). · Zbl 0070.34603
[201] Walter Thirring, Lehrbuch der mathematischen Physik. 4, Springer-Verlag, Vienna, 1980 (German). Quantenmechanik grosser Systeme. [Quantum mechanics of large systems]. Walter Thirring, A course in mathematical physics. Vol. 4, Springer-Verlag, New York-Vienna, 1983. Quantum mechanics of large systems; Translated from the German by Evans M. Harrell.
[202] François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. · Zbl 0171.10402
[203] Neil S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa (3) 27 (1973), 265 – 308. · Zbl 0279.35025
[204] G. Velo and A. Wightman , Constructive quantum field theory, Springer-Verlag, Berlin-New York, 1973. The 1973 ”Ettore Majorana” International School of Mathematical Physics, Erice (Sicily), 26 July – 5 August 1973; Lecture Notes in Physics, Vol. 25. · Zbl 0325.00006
[205] Fred B. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle, J. Funct. Anal. 37 (1980), no. 2, 218 – 234. · Zbl 0463.46024
[206] Calvin H. Wilcox, Theory of Bloch waves, J. Analyse Math. 33 (1978), 146 – 167. · Zbl 0408.35067
[207] Steven Zelditch, Reconstruction of singularities for solutions of Schrödinger’s equation, Comm. Math. Phys. 90 (1983), no. 1, 1 – 26. · Zbl 0554.35031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.