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AN APPROACH TO SOLUTION OF THE SCHRÖDINGER EQUATION USING FOURIER-TYPE FUNCTIONALS

  • Received : 2011.12.06
  • Published : 2013.03.01

Abstract

In this paper, we consider the Fourier-type functionals on Wiener space. We then establish the analytic Feynman integrals involving the ${\diamond}$-convolutions. Further, we give an approach to solution of the Schr$\ddot{o}$dinger equation via Fourier-type functionals. Finally, we use this approach to obtain solutions of the Schr$\ddot{o}$dinger equations for harmonic oscillator and double-well potential. The Schr$\ddot{o}$dinger equations for harmonic oscillator and double-well potential are meaningful subjects in quantum mechanics.

Keywords

References

  1. S. J. Chang, J. G. Choi, and D. Skoug, Evaluation formulas for conditional function space integrals I, Stoch. Anal. Appl. 25 (2007), no. 1, 141-168. https://doi.org/10.1080/07362990601052185
  2. D. M. Chung and S. J. Kang, Evaluation formulas for conditional abstract Wiener integrals, Stoch. Anal. Appl. 7 (1989), no. 2, 125-144. https://doi.org/10.1080/07362998908809173
  3. D. M. Chung and S. J. Kang, Evaluation formulas for conditional abstract Wiener integrals II, J. Korean Math. Soc. 27 (1990), no. 2, 137-144.
  4. D. M. Chung and S. J. Kang, Evaluation of some conditional abstract Wiener integrals, Bull. Korean Math. Soc. 26 (1989), no. 2, 151-158.
  5. H. S. Chung and S. J. Chang, Some applications of the spectral theory for the integral transform involving the spectral representation, J. Funct. Space Appl. 2012 (2012), Article ID 573602, 17 pages.
  6. H. S. Chung and V. K. Tuan, Generalized integral transforms and convolution products on function space, Integral Transforms Spec. Funct. 22 (2011), no. 8, 573-586. https://doi.org/10.1080/10652469.2010.535798
  7. H. S. Chung and V. K. Tuan, Fourier-type functionals on Wiener space, Bull. Korean Math. Soc. 49 (2012), no. 3, 609-619. https://doi.org/10.4134/BKMS.2012.49.3.609
  8. H. S. Chung and V. K. Tuan, A sequential analytic Feynman integral of functionals in $L_2$(C0[0, T]), Integral Transforms Spec. Funct. 23 (2012), no. 7, 495-502. https://doi.org/10.1080/10652469.2011.606218
  9. R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Phys. 20 (1948), 367-387. https://doi.org/10.1103/RevModPhys.20.367
  10. G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman's Operational Calculus, Clarendon Press, Oxford, 2000.
  11. G. W. Johnson and D. L. Skoug, Scale-invariant measurability in Wiener space, Pacific J. Math. 83 (1979), no. 1, 157-176. https://doi.org/10.2140/pjm.1979.83.157
  12. M. Kac, On distributions of certain Wiener functionals, Trans. Amer. Math. Soc. 65 (1949), 1-13. https://doi.org/10.1090/S0002-9947-1949-0027960-X
  13. M. Kac, On Some connetions between probability theory and differential and integral equations, In: Proc. Second Berkeley Symposium on Mathematical Statistic and Probability( ed. J. Neyman), 189-215, Univ. of California Press, Berkeley, 1951.
  14. M. Kac, Probability, Number theory, and Statistical Physics, K. Baclawski and M.D. Donsker (eds.), Mathematicians of Our Time 14 (Cambridge, Mass.-London, 1979). 274 SEUNG JUN CHANG, JAE GIL CHOI, AND HYUN SOO CHUNG
  15. M. Kac, Integration in Function Spaces and Some of its Applications, Lezione Fermiane, Scuola Normale Superiore, Pisa, 1980.
  16. H.-H. Kuo, Gaussian Measure in Banach Space, Lecture Notes in Math. 463, Springer, Berlin, 1975.
  17. E. Merzbacher, Quantum Mechanics 3rd ed., Wiley, NJ (1998), Chap. 5.
  18. C. S. Park, M. G. Jeong, S. K. Yoo, and D. K. Park, Double-well potential: The WKB approximation with phase loss and anharmonicity effect, Phys. Rev. A 58 (1998), 3443-3447. https://doi.org/10.1103/PhysRevA.58.3443
  19. B. Simon, Functioanl Integration and Quantum Physis, Academic Press, New York, 1979.
  20. V. K. Tuan, Paley-Wiener type theorems, Frac. Calc. Appl. Anal. 2 (1999), no. 2, 135-143.
  21. T. Zastawniak, The equaivalence of two approaches to the Feynman integral for the anharmonic oscillator, Univ. Iagel. Acta Math. 28 (1991), 187-199.

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