Abstract
Path integral methods for the investigation of the properties of Schrödinger operators go back to the pioneering works of Feynman and Kac. They have been successfully used for a long time but they have been restricted to nonrelativistic operators. We describe a class of stochastic processes, the so-called Levy processes (i.e. processes with stationary independent increments), and we explain how they can be used in the study of the relativistic operators and some of their abstract generalizations. We mimic the classical approach based on the use of Brownian motion and the Feynman-Kac's formula. We describe the class of potentials which can be handled by this approach, we discuss the regularity properties of the semigroups, the decay of the eigenfunctions and the existence of bound states. We also discuss some of the open problems which are naturally unravelled.
to appear in the Proceedings of the Nordic Summer School in Mathematics 1988partially supported by NSF Grant DMS-8701320.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Bibliography
Aizenman M. and Simon B. (1982): Brownian motion and Harnack's inequality for Schrödinger operators. Comm. Pure and Appl. Math. 35, 209–271.
Bakry D. (1989): La propriété de sous-harmonicité des diffusions dans les variétés. sl Séminaire de Probabilites XXIII, Lect. Notes in Math. (to appear). Springer Verlag, New-York, N.Y.
Blumenthal R.M. and Getoor R.K. (1960): Some Theorems on Stable Processes. Trans. Amer. Math. Soc. 95, 263–273.
Blumenthal R.M. and Getoor R.K. (1968): Markov Processes and Potential Theory. Academic Press, New York.
Carmona R. (1978): Pointwise Bounds for Schrödinger Eigenstates. Comm. Math. Phys. 62, 97–106.
Carmona R. (1981): Regularity Properties of Schrödinger and Dirichlet Semigroups. J. Functional Anal. 33, 59–98.
Carmona R. (1986): “Random Schrödinger Operators” in Ecole d'Eté de Probabilités de Saint Flour XIV-1984 ed. P.L. Hennequin, Lect. Notes in Math. vol. 1180 Springer Verlag, New York, N.Y.
Carmona R. (1989); Levy Processes and Relativistic Schrödinger Operators. (in preparation)
Carmona R., Masters W. C. and Simon B. (1989): Relativistic Schrödinger Operators: Asymptotic Behavior of the Eigenfunctions. J. Functional Anal., to appear.
Carmona R and Simon B. (1981): Pointwise Bounds on Eigenfunctions and Wave Packets in N-Body Quantum Systems: V Lower Bounds and Path Integrals. Comm. Math. Phys. 80, 59–98.
Daubechies I. (1983): An Uncertainty Principle for Fermions with Generalized Kinetic Energy. Comm. Math. Phys. 90, 311–320.
Daubechies I. (1984): One Electron Molecules with Relativistic Kinetic Energy: Properties of the discrete spectrum. Comm. Math. Phys. 94, 523–535.
Daubechies I. and Lieb E.H. (1983): One Electron Relativistic Molecules with Coulomb Interaction. Comm. Math. Phys. 90, 297–310.
Fefferman C. and de la Llave R. (1986): Relativistic Stability of Matter-I. Rev. Mat. Iberoamericana 2, 119–223.
Feferman C. (1986): The N-Body Problem in Quantum Mechanics. Comm. Pure and Appl. Math. 39, S67–S109.
Fridstedt B.. (1974): Sample Functions of Stochastic Processes with Stationary Independent Increments. in Advances in Probability and Related Fields vol.3, ed. P. Ney & S. Port, 241–396. Helffer B. and Sjöstrand(1989): Analyse semi-classique pour 1'equation de HarperII Comportement semi-classique près d'un rationnel. these proceedings
Herbst I.W. (1977): Spectral Theory of the Operator (p 2 + m 2)1/2-Ze 2/r. Comm. Math. Phys. 53, 285–294.
Herbst I.W. and Sloan A.D. (1978): Perturbation of translation invariant positivity preserving semigroups on L 2(Rn). Trans. Amer. Math. Soc. 236, 325–360.
Lieb E. (1976): The Stability of Matter. Rev. Modern Phys. 48, 553–569.
Lieb E. (1989): The Stability of Relativistic Matter. These proceedings.
Lieb E. and Yau H.T. (1988): Stability and Instability of Relativistic Matter. Comm. in Math. Phys. (in press)
Nardini F. (1986): Exponential decay for the eigenfunctions of the two body relativistic Hamiltonian. J. Analyse Mathématique 47, 87–109.
Nardini F. (1988): On the asymptotic behavior of the eigenfunctions of the relativistic N-body Schrödinger operator. preprint.
Okura H. (1979): On the spectral distribution of certain integro-differential operators with random potentials. Osaka. J. Math. 16, 633–666.
Port S. and Stone C. (1971): Infinitely divisible processes and their potential theory. Annales Inst. Fourier 21.2 157–275, 21.4 179–265.
Reed M. and Simon B. (1975): Methods of Modern Mathematical Physics. II Fourier Analysis, Self-Adjointness. Academic Press, New-York.
Reed M. and Simon B. (1978): Methods of Modern Mathematical Physics. IV Analysis of Operators. Academic Press, New-York.
Simon B. (1981): Functional integrals and quantum mechanics. Academic Press, New York.
Simon B. (1982): Schrödinger Semigroups. Bull. Amer. Math. Soc. 7, 447–526.
Simon B. (1984): Semiclassical analysis of low lying eigenvalues, II. Tunneling. Ann. Math. 120, 89–118.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag
About this paper
Cite this paper
Carmona, R. (1989). Path integrals for relativistic Schrodinger operators. In: Holden, H., Jensen, A. (eds) Schrödinger Operators. Lecture Notes in Physics, vol 345. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51783-9_17
Download citation
DOI: https://doi.org/10.1007/3-540-51783-9_17
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51783-2
Online ISBN: 978-3-540-46807-3
eBook Packages: Springer Book Archive