×

zbMATH — the first resource for mathematics

Product formula for resolvents of normal operators and the modified Feynman integral. (English) Zbl 0718.47023
In the series of papers [see for example, J. Funct. Anal. 63, 261-275 (1985; Zbl 0601.47025)], M. L. Lapidus developed the theory of the “modified Feynman integral” for Schrödinger operators in terms of Trotter-like product formula using the imaginary resolvents of the intervening operators. The aim of the reviewed paper is to extend this theory to the case of Schrödinger operators \(H=-(\nabla -ia)^ 2+V\) (a is a real vector potential) with highly singular complex potentials V. Both semigroups \(e^{-itH}\) and \(e^{-tH}\) corresponding to the dissipative systems or Nelson approach to Feynman integral respectively are studied.
Reviewer: R.Alicki (Gdańsk)

MSC:
47D06 One-parameter semigroups and linear evolution equations
47L90 Applications of operator algebras to the sciences
81S40 Path integrals in quantum mechanics
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
47B44 Linear accretive operators, dissipative operators, etc.
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] António de Bivar-Weinholtz and Rémi Piraux, Formule de Trotter pour l’opérateur -\Delta +\?\(^{+}\)-\?\(^{-}\)+\?\?\(^{\prime}\), Ann. Fac. Sci. Toulouse Math. (5) 5 (1983), no. 1, 15 – 37 (French, with English summary).
[2] António de Bivar-Weinholtz, Sur l’opérateur de Schrödinger avec potentiel singulier magnétique, dans un ouvert arbitraire de \?^\?, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 6, 213 – 216 (French, with English summary). · Zbl 0493.35035
[3] António de Bivar-Weinholtz, Opérateurs de Schrödinger avec potentiel singulier magnétique, dans un ouvert arbitraire de \?^\?, Portugal. Math. 41 (1982), no. 1-4, 1 – 12 (1984) (French, with English summary). · Zbl 0565.35024
[4] -, Operadores de Schrödinger com potenciais singulares, Ph.D. dissertation, University of Lisbon, Portugal, 1982.
[5] 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
[6] R. H. Cameron, A family of integrals serving to connect the Wiener and Feynman integrals, J. Math. and Phys. 39 (1960/1961), 126 – 140. · Zbl 0096.06901
[7] Pavel Exner, Open quantum systems and Feynman integrals, Fundamental Theories of Physics, D. Reidel Publishing Co., Dordrecht, 1985. · Zbl 0638.46051
[8] B. O. Haugsby, An operator valued integral in a function space of continuous vector valued functions, Ph.D. dissertation, University of Minnesota, Minneapolis, MN, 1972.
[9] G. W. Johnson, Existence theorems for the analytic operator-valued Feynman integral, Séminaire d’Analyse Moderne [Seminar on Modern Analysis], vol. 20, Université de Sherbrooke, Département de Mathématiques, Sherbrooke, QC, 1988. · Zbl 0693.46050
[10] Tosio Kato, On some Schrödinger operators with a singular complex potential, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 1, 105 – 114. · Zbl 0376.47021
[11] M. L. Lapidus, Formules de moyenne et de produit pour les résolvantes imaginaires d’opérateurs auto-adjoints, C. R. Acad. Sci. Paris Sér. A 291 (1980), 451-454. · Zbl 0446.47010
[12] -, The problem of the Trotter-Lie formula for unitary groups of operators, Séminaire Choquet: Initiation à l’Analyse, Publ. Math. Université Pierre et Marie Curie 46, 1980-81, 20ème année (1982), 1701-1745.
[13] Michel L. Lapidus, Modification de l’intégrale de Feynman pour un potentiel positif singulier: approche séquentielle, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 1, 1 – 3 (French, with English summary). · Zbl 0493.35038
[14] Michel L. Lapidus, Intégrale de Feynman modifiée et formule du produit pour un potentiel singulier négatif, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 13, 719 – 722 (French, with English summary). · Zbl 0508.35027
[15] Michel L. Lapidus, Product formula for imaginary resolvents with application to a modified Feynman integral, J. Funct. Anal. 63 (1985), no. 3, 261 – 275. · Zbl 0601.47025 · doi:10.1016/0022-1236(85)90088-6 · doi.org
[16] Michel L. Lapidus, Perturbation theory and a dominated convergence theorem for Feynman integrals, Integral Equations Operator Theory 8 (1985), no. 1, 36 – 62. · Zbl 0567.47015 · doi:10.1007/BF01199981 · doi.org
[17] -, Formules de Trotter et calcul opérationnel de Feynman, Thèse de Doctorat d’Etat ès Sciences, Mathématiques, Université Pierre et Marie Curie (Paris VI), France, June 1986. (Part I: Formules de Trotter et intégrales de Feynman)
[18] Herbert Leinfelder and Christian G. Simader, Schrödinger operators with singular magnetic vector potentials, Math. Z. 176 (1981), no. 1, 1 – 19. · Zbl 0468.35038 · doi:10.1007/BF01258900 · doi.org
[19] Edward Nelson, Feynman integrals and the Schrödinger equation, J. Mathematical Phys. 5 (1964), 332 – 343. · Zbl 0133.22905 · doi:10.1063/1.1704124 · doi.org
[20] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. · Zbl 0516.47023
[21] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. · Zbl 0867.46001
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.