Advertisement

The Trace Formula for the Heat Semigroup with Polynomial Potential

  • Sergio Albeverio
  • Sonia Mazzucchi
Conference paper
Part of the Progress in Probability book series (PRPR, volume 63)

Abstract

We consider the heat semigroup \(e^{-\frac{t}{\hbar}H}, t > 0,\,\, {\rm on}\,\, \mathbb{R}^d\) with generator H corresponding to a potential growing polynomially at infinity. Its trace for positive times is represented as an analytically continued infinite-dimensional oscillatory integral. The asymptotics in the small parameter _ is exhibited by using Laplace’s method in infinite dimensions in the case of a degenerate phase (this corresponds to the limit from quantum mechanics to classical mechanics, in a situation where the Euclidean action functional has a degenerate critical point).

Keywords

Heat kernels polynomial potential infinite-dimensional oscillatory integrals Laplace method degenerate phase asymptotics semiclassical limit 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Albeverio, Wiener and Feynman path integrals and their applications, Proceedings of the Norbert Wiener Centenary Congress, East Lansing, MI, 1994, 163194, Proc. Sympos. Appl. Math., 52 (1997), Amer. Math. Soc., Providence, RI.Google Scholar
  2. 2.
    S. Albeverio, T. Arede, and M. de Faria, Remarks on nonlinear filtering problems: white noise representation and asymptotic expansions, In: Stochastic Processes, Physics and Geometry (Ascona and Locarno, 1988), World Sci. Publ., Teaneck, NJ, (1990), 7786.Google Scholar
  3. 3.
    S. Albeverio, Ph. Blanchard, and R. Høegh-Krohn, Feynman path integrals and the trace formula for the Schrödinger operators, Comm. Math. Phys., 83 (1) (1982), 4976.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    S. Albeverio, A. Boutet de Monvel-Berthier, and Z. Brze´zniak, The trace formula for Schr ¨ odinger operators from infinite-dimensional oscillatory integrals, Math. Nachr., 182 (1996), 2165.Google Scholar
  5. 5.
    S. Albeverio, A. Boutet de Monvel-Berthier, and Z. Brze´zniak, Stationary phase method in infinite dimensions by finite-dimensional approximations: applications to the Schr ¨ odinger equation, Potential Anal., 4 (5) (1995), 469502.Google Scholar
  6. 6.
    S. Albeverio and Z. Brze´zniak, Finite-dimensional approximation approach to oscillatory integrals and stationary phase in infinite dimensions, J. Funct. Anal., 113 (1) (1993), 177244.Google Scholar
  7. 7.
    S. Albeverio and Z. Brze´zniak, Feynman path integrals as infinite-dimensional oscillatory integrals: some new developments, In: White Noise Models and Stochastic Systems (Enschede, 1992), Acta Appl. Math., 35 (1-2) (1994), 526.Google Scholar
  8. 8.
    S. Albeverio and R. Høegh-Krohn, Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics I, Invent. Math., 40 (1) (1977), 59106.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    S. Albeverio, R. Høegh-Krohn, and S. Mazzucchi, Mathematical Theory of Feynman Path Integrals. An Introduction, 2nd edition, Lecture Notes in Mathematics, vol. 523, Springer, Berlin, 2008.Google Scholar
  10. 10.
    S. Albeverio and S. Liang, Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables, Ann. Probab., 33 (1) (2005), 300 336.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    S. Albeverio and S. Mazzucchi, Infinite dimensional oscillatory integrals with polynomial phase function and the trace formula for the heat semigroup, Ast´erisque No. 327 (2009), 1745 (2010).Google Scholar
  12. 12.
    S. Albeverio, H. R¨ockle, and V. Steblovskaya, Asymptotic expansions for Ornstein- Uhlenbeck semigroups perturbed by potentials over Banach spaces, Stochastics Rep., 69 (3-4) (2000), 195238.Google Scholar
  13. 13.
    S. Albeverio and V. Steblovskaya, Asymptotics of infinite-dimensional integrals with respect to smooth measures. I., Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2 (4) (1999), 529556.Google Scholar
  14. 14.
    G. Ben Arous, M ´ ethode de Laplace et de la phase stationnaire sur l espace de Wiener. (French) [The Laplace and stationary phase methods on Wiener space], Stochastics, 25 (3) (1988), 125153.Google Scholar
  15. 15.
    G. Ben Arous and R. L´eandre, D ´ ecroissance exponentielle du noyau de la chaleur sur la diagonale. II. (French) [Exponential decay of the heat kernel on the diagonal. II], Probab. Theory Related Fields, 90 (3) (1991), 377402.Google Scholar
  16. 16.
    D. Elworthy and A. Truman, Feynman maps, Cameron-Martin formulae and anharmonic oscillators, Ann. Inst. H. Poincar´e Phys. Th´eor., 41 (2) (1984), 115142.Google Scholar
  17. 17.
    A.J. Ellis and J.S. Rosen, Asymptotic analysis of Gaussian integrals II: Manifold of minimum points, Commun. Math. Phys., 82 (1981), 153181.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    A.J. Ellis and J.S. Rosen, Asymptotic analysis of Gaussian integrals I: Isolated minimum points, Trans. Amer. Math. Soc., 273 (1982), 447481.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    L. Gross, Abstract Wiener spaces, Proc. 5th Berkeley Symp. Math. Stat. Prob. 2, (1965), 3142.Google Scholar
  20. 20.
    L. H¨ormander, The Analysis of Linear Partial Differential Operators, I. Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin/Heidelberg/New York/Tokyo, 1983.Google Scholar
  21. 21.
    G. Kallianpur and H. Oodaira, Fre ˘ı dlin-Wentzell type estimates for abstract Wiener spaces, Sankhy¯a Ser. A, 40 (2) (1978), 116137.MathSciNetzbMATHGoogle Scholar
  22. 22.
    V.N. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes, Lecture Notes in Mathematics, 1724, Springer-Verlag, Berlin, 2000.Google Scholar
  23. 23.
    H.H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math., Springer- Verlag Berlin-Heidelberg-New York, 1975.Google Scholar
  24. 24.
    S. Mazzucchi, Mathematical Feynman Path Integrals and Applications, World Scientific Publishing Co. Singapore, 2009.Google Scholar
  25. 25.
    D. Nualart and V. Steblovskaya, Asymptotics of oscillatory integrals with quadratic phase function on Wiener space, Stochastics Rep., 66 (3-4) (1999), 293309.MathSciNetzbMATHGoogle Scholar
  26. 26.
    M. Pincus, Gaussian processes and Hammerstein integral equations, Trans. Amer. Math. Soc., 134 (1968), 193214.MathSciNetzbMATHGoogle Scholar
  27. 27.
    V.I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields, Translations of Mathematical Monographs, 148. American Mathematical Society, Providence, RI, 1996.Google Scholar
  28. 28.
    V.I. Piterbarg and V.R. Fatalov, The Laplace method for probability measures in Banach spaces, Russian Math. Surveys, 50 (6) (1995), 11511239.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.Google Scholar
  30. 30.
    S. Rossignol, D ´ eveloppements asymptotiques d int ´ egrales de Laplace sur l espace de Wiener dans le cas d ´ eg ´ en ´ er ´ e. (French) [Asymptotic expansions of Laplace integrals on Wiener space in the degenerate case], C. R. Acad. Sci. Paris S´er. I Math., 317 (10) (1993), 971974.Google Scholar
  31. 31.
    M. Schilder, Some asymptotic formulas for Wiener integrals, Trans. Amer. Math. Soc., 125 (1966), 6385.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    B. Simon, Functional Integration and Quantum Physics, Second edition, AMS Chelsea Publishing, Providence, RI, 2005.Google Scholar
  33. 33.
    B. Simon, Trace Ideals and Their Applications, London Mathematical Society Lecture Note Series, 35, Cambridge University Press, Cambridge-New York, 1979.Google Scholar
  34. 34.
    E.C. Tichmarsch, The Theory of Functions, Oxford University Press, London, 1939.Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Sergio Albeverio
    • 1
    • 2
    • 3
    • 4
  • Sonia Mazzucchi
    • 2
    • 5
    • 6
  1. 1.Institut für Angewandte MathematikHCM, SFB 611, BiBoS, IZKSBonnGermany
  2. 2.Dipartimento di MatematicaUniversità di TrentoPovo-TrentoItalia
  3. 3.CerfimLocarnoSwitzerland
  4. 4.Acc. Arch. (USI)MendrisioSwitzerland
  5. 5.Institut für Angewandte MathematikHCMBonnGermany
  6. 6.Alexander von Humboldt fellowBonnGermany

Personalised recommendations