Abstract
A perturbation theorem for unitary groups generated by form sums of operators is established. This result is used to derive a general dominated convergence theorem for Feynman path integrals. It is then applied to the modified Feynman integral which is shown to enjoy optimal stability properties. The fact that the modified Feynman integral — previously introduced by the author — converges in a rather general setting, is used in an essential manner. Other results concerning less general definitions of the Feynman integral are also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. BREZIS, “Analyse Fonctionelle. Théorie et Applications,” Masson, Paris, 1983.
H. BREZIS and T. KATO, Remarks on the Schrödinger operator with singular complex potentials, J. Math Pures et Appl.,58(1979), 137–151.
R. H. CAMERON and D. A. STORVICK, An operator valued function space integral and a related integral equation, J. Math. and Mech.,18(1968), 517–552.
E. B. DAVIES, “One-Parameter Semigroups,” Academic Press, London, 1980.
R. DESCOMBES, “Intégration,” Hermann, Paris, 1972.
E. HILLE and R. S. PHILLIPS, “Functional Analysis and Semigroups,” American Mathematical Society Colloquium Publications, revised ed., Vol. 31, Providence, Rhode Island, 1957.
G. W. JOHNSON, A bounded convergence theorem for the Feynman integral, preprint, The University of Nebraska, Lincoln; to appear in J. Math. Phys.
G. W. JOHNSON and D. L. SKOUG, A Banach algebra of Feynman integrable functionals with application to an integral equation formally equivalent to Schroedinger's equation, J. Functional Analysis,12(1973), 129–152.
G. W. JOHNSON and D. L. SKOUG, The Cameron-Storwick function space integral: The L1 theory, J. Math. Anal. Appl.,50(1975), 647–667.
G. W. JOHNSON and D. L. SKOUG, The Cameron-Storwick function space integral: an L(Lp, Lp′) theory, Nagoya Math. J.,60(1976), 93–137.
T. KATO, “Perturbation Theory for Linear Operators,” 2nd ed., Springer, Berlin, 1976.
T. KATO, Remarks on Schrödinger operators with vector potentials, Integral Equations and Operator Theory,1(1978), 103–113.
R. A. KUNZE and I. E. SEGAL, “Integrals and Operators,” 2nd rev. and enl. ed., Springer, Berlin, 1978.
M. L. LAPIDUS, The problem of the Trotter-Lie formula for unitary groups of operators, Séminaire Choquet, Publ. Math. Univ. Pierre-et-Marie-Curie, 1980/81,46(1982), 1701–1745.
M. L. LPIDUS, Modification de l'intégrale de Feynman pour un potentiel positif singulier: approche séquentielle, C. R. Acad. Sci. Paris Sér. I,295(1982), 1–3.
M. L. LAPIDUS, Intégrale de Feynman modifiée et formule du produit pour un potentiel singulier négatif, C. R. Acad. Sci. Paris Sér. I,295(1982), 719–722.
M. L. LAPIDUS, Product formula, imaginary resolvents and modified Feynman integral, preprint, The University of Southern California, Los Angeles 1983; submitted for publication. (Communicated at the American Mathematical Society Summer Research Institute on Nonlinear Functional Analysis and Applications, Univ. of California, Berkeley, July 1983. Lecture Notes No. L23.)
M. L. LAPIDUS, Product formula for imaginary resolvents with application to a modified Feynman integral, preprint, The University of Southern California, Los Angeles, 1983; submitted for publication.
E. NELSON, Feynman integrals and the Schrödinger equation, J. Math. Phys.,5(1964), 332–343.
M. REED and B. SIMON, “Methods of Modern Mathematical Physics,” Vol. II, “Fourier Analysis, Self-Adjointness”, Academic Press, New York, 1975.
M. REED and B. SIMON, “Methods of Modern Mathematical Physics,” Vol. I, “Functional Analysis,” rev. and enl. ed., Academic Press, New York, 1980.
C. G. SIMADER, Bemerkungen über Schrödinger-operatoren mit stark singularen potentialen, Math. Z.,138(1974), 53–70.
B. SIMON, Maximal and minimal Schrödinger forms, J. Operator Theory,1(1979), 37–47.
B. SIMON, Schrödinger semigroups, Bull. Amer. Math. Soc.,7(1982), 447–526.
Author information
Authors and Affiliations
Additional information
This research was supported in part by a grant from the Faculty Research and Innovation Fund at the University of Southern California.
Rights and permissions
About this article
Cite this article
Lapidus, M.L. Perturbation theory and a dominated convergence theorem for Feynman integrals. Integr equ oper theory 8, 36–62 (1985). https://doi.org/10.1007/BF01199981
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01199981