×
Albeverio, S.; Mazzucchi, S.

An asymptotic functional-integral solution for the Schrödinger equation with polynomial potential. (English) Zbl 1172.35068

J. Funct. Anal. 257, No. 4, 1030-1052 (2009).
Summary: A functional integral representation for the weak solution of the Schrödinger equation with a polynomially growing potential is proposed in terms of an analytically continued Wiener integral. The asymptotic expansion in powers of the coupling constant \(\lambda \) of the matrix elements of the Schrödinger group is studied and its Borel summability is proved.

Cited in 4 Documents

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J10 Schrödinger operator, Schrödinger equation
35C15 Integral representations of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

Feynman path integrals; Schrödinger equation; analytic continuation of Wiener integrals; polynomial potential; asymptotic expansions
Cite Review PDF
Full Text: DOI

References:

[1] Albeverio, S.; Høegh-Krohn, R.; Mazzucchi, S., Mathematical theory of Feynman path integrals, (An Introduction. An Introduction, Lecture Notes in Math., vol. 523 (2008), Springer-Verlag: Springer-Verlag Berlin/Heidelberg) · Zbl 1222.46001
[2] Albeverio, S.; Mazzucchi, S., Generalized infinite-dimensional Fresnel integrals, C. R. Math. Acad. Sci. Paris, 338, 3, 255-259 (2004) · Zbl 1038.58007
[3] Albeverio, S.; Mazzucchi, S., Some new developments in the theory of path integrals, with applications to quantum theory, J. Stat. Phys., 115, 112, 191-215 (2004) · Zbl 1157.81324
[4] Albeverio, S.; Mazzucchi, S., Generalized Fresnel integrals, Bull. Sci. Math., 129, 1, 1-23 (2005) · Zbl 1096.28010
[5] Albeverio, S.; Mazzucchi, S., Feynman path integrals for polynomially growing potentials, J. Funct. Anal., 221, 1, 83-121 (2005) · Zbl 1095.81041
[6] Albeverio, S.; Mazzucchi, S., Feynman path integrals for the time dependent quartic oscillator, C. R. Math. Acad. Sci. Paris, 341, 10, 647-650 (2005) · Zbl 1078.81049
[7] Albeverio, S.; Mazzucchi, S., The time dependent quartic oscillator—A Feynman path integral approach, J. Funct. Anal., 238, 2, 471-488 (2006) · Zbl 1130.81032
[8] S. Albeverio, S. Mazzucchi, The trace formula for the heat semigroup with polynomial potential, SFB-611-Preprint No. 332, Bonn, 2007; S. Albeverio, S. Mazzucchi, The trace formula for the heat semigroup with polynomial potential, SFB-611-Preprint No. 332, Bonn, 2007
[9] Bender, C.; Wu, T., Anharmonic oscillator, Phys. Rev. (2), 184, 1231-1260 (1969)
[10] Cameron, R. H., A family of integrals serving to connect the Wiener and Feynman integrals, J. Math. Phys., 39, 126-140 (1960) · Zbl 0096.06901
[11] Davies, I. M.; Truman, A., On the Laplace asymptotic expansion of conditional Wiener integrals and the Bender-Wu formula for \(x^{2 N}\)-anharmonic oscillators, J. Math. Phys., 24, 2, 255-266 (1983) · Zbl 0517.28014
[12] Doss, H., Sur une Résolution Stochastique de l’Equation de Schrödinger à Coefficients Analytiques, Comm. Math. Phys., 73, 247-264 (1980) · Zbl 0427.60099
[13] Engel, K.-J.; Nagel, R., One-parameter semigroups for linear evolution equations, (Brendle, S.; Campiti, M.; Hahn, T.; Metafune, G.; Nickel, G.; Pallara, D.; Perazzoli, C.; Rhandi, A.; Romanelli, S.; Schnaubelt, R., Grad. Texts in Math., vol. 194 (2000), Springer-Verlag: Springer-Verlag New York) · Zbl 0616.34060
[14] Hardy, G. H., Divergent Series (1963), Oxford University Press: Oxford University Press London · Zbl 0897.01044
[15] Hida, T.; Kuo, H. H.; Potthoff, J.; Streit, L., White Noise (1995), Kluwer: Kluwer Dordrecht
[16] Johnson, G. W.; Lapidus, M. L., The Feynman Integral and Feynman’s Operational Calculus (2000), Oxford University Press: Oxford University Press New York · Zbl 0952.46044
[17] Kallianpur, G.; Kannan, D.; Karandikar, R. L., Analytic and sequential Feynman integrals on abstract Wiener and Hilbert spaces, and a Cameron Martin formula, Ann. Inst. H. Poincaré Probab. Th., 21, 323-361 (1985) · Zbl 0583.60049
[18] Kato, T., On some Schrödinger operators with a singular complex potential, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 5, 1, 105-114 (1978) · Zbl 0376.47021
[19] Mazzucchi, S., Feynman path integrals for the inverse quartic oscillator, J. Math. Phys., 49, 9 (2008), 093502 (15 pp.) · Zbl 1152.81555
[20] Mazzucchi, S., Mathematical Feynman Path Integrals and Applications (2009), World Scientific Publishing: World Scientific Publishing Singapore · Zbl 1221.58010
[21] Nelson, E., Feynman integrals and the Schrödinger equation, J. Math. Phys., 5, 332-343 (1964) · Zbl 0133.22905
[22] Nevanlinna, F., Zur Theorie der asymptotischen Potenzreihen, Ann. Acad. Sci. Fenn. (A), 12, 3, 1-81 (1919) · JFM 46.1463.01
[23] Reed, M.; Simon, B., Methods of modern mathematical physics, II, (Fourier Analysis, Self-Adjointness (1975), Academic Press/Harcourt Brace Jovanovich Publishers: Academic Press/Harcourt Brace Jovanovich Publishers New York/London) · Zbl 0517.47006
[24] Reed, M.; Simon, B., Methods of modern mathematical physics, IV, (Analysis of Operators (1978), Academic Press/Harcourt Brace Jovanovich Publishers: Academic Press/Harcourt Brace Jovanovich Publishers New York/London) · Zbl 0517.47006
[25] Simon, B., Coupling constant analyticity for the anharmonic oscillator, Ann. Physics, 58, 76-136 (1970)
[26] Simon, B., Functional Integration and Quantum Physics (2005), Amer. Math. Soc. Chelsea Publishing: Amer. Math. Soc. Chelsea Publishing Providence, RI · Zbl 1061.28010
[27] Sokal, A., An improvement of Watson’s theorem on Borel summability, J. Math. Phys., 21, 261-263 (1980) · Zbl 0441.40012
[28] Yajima, K., Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations, Comm. Math. Phys., 181, 605-629 (1996) · Zbl 0883.35022
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.