Mathematical Physics
Generalized infinite-dimensional Fresnel integrals
Comptes Rendus. Mathématique, Volume 338 (2004) no. 3, pp. 255-259.

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:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.11.022
Albeverio, Sergio 1, 2; Mazzucchi, Sonia 2

1 Institut für Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany
2 Dipartimento di Matematica, Università di Trento, 38050 Povo, Italy
@article{CRMATH_2004__338_3_255_0,
     author = {Albeverio, Sergio and Mazzucchi, Sonia},
     title = {Generalized infinite-dimensional {Fresnel} integrals},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {255--259},
     publisher = {Elsevier},
     volume = {338},
     number = {3},
     year = {2004},
     doi = {10.1016/j.crma.2003.11.022},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2003.11.022/}
}
TY  - JOUR
AU  - Albeverio, Sergio
AU  - Mazzucchi, Sonia
TI  - Generalized infinite-dimensional Fresnel integrals
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 255
EP  - 259
VL  - 338
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2003.11.022/
DO  - 10.1016/j.crma.2003.11.022
LA  - en
ID  - CRMATH_2004__338_3_255_0
ER  - 
%0 Journal Article
%A Albeverio, Sergio
%A Mazzucchi, Sonia
%T Generalized infinite-dimensional Fresnel integrals
%J Comptes Rendus. Mathématique
%D 2004
%P 255-259
%V 338
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2003.11.022/
%R 10.1016/j.crma.2003.11.022
%G en
%F CRMATH_2004__338_3_255_0
Albeverio, Sergio; Mazzucchi, Sonia. Generalized infinite-dimensional Fresnel integrals. Comptes Rendus. Mathématique, Volume 338 (2004) no. 3, pp. 255-259. doi : 10.1016/j.crma.2003.11.022. http://www.numdam.org/articles/10.1016/j.crma.2003.11.022/

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

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

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

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

Cited by Sources: