Multiparticle quasiexactly solvable difference equations
Several explicit examples of multiparticle quasiexactly solvable “discrete” quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable multiparticle Hamiltoni...
Several explicit examples of multiparticle quasiexactly solvable “discrete” quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable multiparticle Hamiltoni...
Parametric-time coherent states for Smorodinsky-Winternitz potentials
In this study, we construct the coherent states for a particle in the Smorodinsky-Winternitz potentials, which are the generalizations of the two-dimensional Kepler problem. In the third case, the sys...
In this study, we construct the coherent states for a particle in the Smorodinsky-Winternitz potentials, which are the generalizations of the two-dimensional Kepler problem. In the third case, the sys...
Stochastic representations of Feynman integration
J. Math. Phys. 48, 122106 (2007); doi:10.1063/1.2812416
Published 17 December 2007
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For polynomially bounded potentials V such that H=H0+V is essentially self-adjoint on S(Rd)![[subset, equals]](http://scitation.aip.org/stockgif3/sube.gif)
D(H0)![[intersection]](http://scitation.aip.org/stockgif3/cap.gif)
D(V), this essay offers two reconstructions of Feynman's sum over histories as the unitary image of a genuine integral with respect to Wiener measure µ of a functional 
(
) defined on the space W of Brownian paths into momentum space Rd. The first representation, based on Feynman's original argument, “lifts” () from a “convolutional Trotter product formula” for the Fourier-transformed image 
t(p) of the time-evolved wave function 
t(x)=exp(−itH)(x) in L2(Rd). The second—which varies and extends a construction introduced in a slightly different context by Albeverio and Høegh-Krohn [Mathematical Theory of Feynman Integrals, Springer Lecture Notes in Mathematics Vol. 523 (Springer, New York, 1976)]—lifts the functional () from a “convolutional Dyson expansion” of the time-evolved momentum-space function t(p).
©2007 American Institute of Physics
D(H0)D(V), this essay offers two reconstructions of Feynman's sum over histories as the unitary image of a genuine integral with respect to Wiener measure µ of a functional ) defined on the space W of Brownian paths into momentum space Rd. The first representation, based on Feynman's original argument, “lifts” ) from a “convolutional Trotter product formula” for the Fourier-transformed image t(x)=exp(−itH)(x) in L2(Rd). The second—which varies and extends a construction introduced in a slightly different context by Albeverio and Høegh-Krohn [Mathematical Theory of Feynman Integrals, Springer Lecture Notes in Mathematics Vol. 523 (Springer, New York, 1976)]—lifts the functional ) from a “convolutional Dyson expansion” of the time-evolved momentum-space function | History: | Received 15 March 2007; accepted 11 October 2007; published 17 December 2007 |
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REFERENCES (15)
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- Albeverio, S. and Høegh-Krohn, R., Mathematical Theory of Feynman Integrals, Springer Lecture Notes in Mathematics Vol. 523 (Springer, Berlin, 1976).
- Blank, J., Exner, P., and Havlíček, M., Hilbert Space Operators in Quantum Physics (AIP, New York, 1994).
- Exner, P., Open Quantum Systems and Feynman Integrals (Reidel, Dordrecht, 1985).
- Hida, T., Brownian Motion (Springer, New York, 1980).
- Ikeda, N., and Watanabe, S., Stochastic Differential Equations and Diffusion Processes (North-Holland, Amsterdam, 1981).
- Janson, S., Gaussian Hilbert Spaces (Cambridge University Press, Cambridge, 1997).
- Kallenberg, O., Foundations of Modern Probability, 2nd ed., (Springer, New York, 2000).
- Karatzas, J. and Shreve, S., Brownian Motion and Stochastic Calculus 2nd ed., (Springer, New York, 1991).
- Malliavin, P., Stochastic Analysis (Springer, New York, 2002).
- Reed, M. and Siman, B., Methods of Modern Analysis (Academic, New York, 1975), Vol. I.
- Reed, M. and Simon, B., Methods of Modern Analysis, 2nd ed. (Academic, New York, 1980) Vol. II.
- Rudin, W., Real and Complex Analysis (McGraw-Hill, New York, 1974).
- Simon, B., Functional Integration and Quantum Physics (Academic Press, New York, 1979).
- Weidmann, J., Linear Operators in Hilbert Spaces (Springer, New York, 1980).
- Yosida, K., Functional Analysis (Springer, New York, 1974).
