Generalized infinite-dimensional Fresnel integrals

Sergio Albeverio; Sonia Mazzucchi

doi:10.1016/j.crma.2003.11.022

Mathematical Physics

Generalized infinite-dimensional Fresnel integrals

Presented by: Paul Malliavin

Sergio Albeverio ^1,²; Sonia Mazzucchi ²

¹ Institut für Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany
² Dipartimento di Matematica, Università di Trento, 38050 Povo, Italy

Comptes Rendus. Mathématique, Volume 338 (2004) no. 3, pp. 255-259.

Abstract
Résumé

A generalized infinite dimensional oscillatory integral with a polynomially growing phase function is defined and explicitly computed in terms of an absolutely convergent Gaussian integral. The results are applied to the Feynman path integral representation for the solution of the Schrödinger equation with an anharmonic oscillator potential.

Un concept d'intégrale oscillatoire généralisée en dimension infinie, avec une fonction de phase de croissance, polynomiale à l'infini, est introduit. L'intégrale est calculée explicitement en termes d'intégrales gaussiennes absolument convergentes. Les résultats sont appliqués à une representation de type « intégrale sur les chemins de Feynman » de la solution de l'équation de Schrödinger à potentiel anharmonique.

Received: 2003-10-28
Accepted: 2003-11-07
Published online: 2003-12-30

DOI: 10.1016/j.crma.2003.11.022

Author's affiliations:

Sergio Albeverio ^{1, 2}; Sonia Mazzucchi ²

¹ Institut für Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany
² Dipartimento di Matematica, Università di Trento, 38050 Povo, Italy

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     title = {Generalized infinite-dimensional {Fresnel} integrals},
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Sergio Albeverio; Sonia Mazzucchi. Generalized infinite-dimensional Fresnel integrals. Comptes Rendus. Mathématique, Volume 338 (2004) no. 3, pp. 255-259. doi : 10.1016/j.crma.2003.11.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.11.022/

References
Cited by

[1] S. Albeverio; Z. Brzeźniak Finite-dimensional approximation approach to oscillatory integrals and stationary phase in infinite dimensions, J. Funct. Anal., Volume 113 (1993) no. 1, pp. 177-244

[2] S. Albeverio; Z. Brzeźniak; Z. Haba On the Schrödinger equation with potentials which are Laplace transform of measures, Potential Anal., Volume 9 (1998) no. 1, pp. 65-82

[3] S. Albeverio; R. Høegh-Krohn Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics, Invent. Math., Volume 40 (1977) no. 1, pp. 59-106

[4] S. Albeverio, S. Mazzucchi, Generalized Fresnel integrals, SFB 611, Preprint No. 59, Bonn, 2003

[5] S. Albeverio, S. Mazzucchi, Feynman path integrals for polynomially growing potentials, Preprint of the University of Trento, 2003

[6] D. Elworthy; A. Truman Feynman maps, Cameron–Martin formulae and anharmonic oscillators, Ann. Inst. H. Poincaré Phys. Théor., Volume 41 (1984) no. 2, pp. 115-142

[7] G.W. Johnson; M.L. Lapidus The Feynman Integral and Feynman's Operational Calculus, Oxford University Press, New York, 2000

[8] T. Kuna; L. Streit; W. Westerkamp Feynman integrals for a class of exponentially growing potentials, J. Math. Phys., Volume 39 (1998) no. 9, pp. 4476-4491

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