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Applicable Analysis
An International Journal
Volume 93, 2014 - Issue 3
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Articles

Uniformization of WKB functions by Wigner transform

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Pages 624-645 | Received 12 Mar 2013, Accepted 12 Mar 2013, Published online: 08 Jul 2013

References

  • Bensoussan A, Lions J-L, Papanicolaou G. Asymptotic analysis for periodic structures. Amsterdam: North-Holland; 1987.
  • Babich VB, Buldyrev VS. Short-wavelength diffraction theory. Asymptotic methods. Berlin: Springer-Verlag; 1991.
  • Tolstoy I, Clay CS. Ocean acoustics. Theory and experiment in underwater sound. New York: American Institute of Physics; 1966.
  • Cervenỳ V. Seismic ray theory. New York: Cambridge University Press; 2001.
  • Babich VM, Kirpichnikova NY. The boundary-layer method in diffraction problems. Berlin: Springer-Verlag; 1979.
  • Kravtsov YuA. Two new asymptotic methods in the theory of wave propagation in inhomogeneous media(review). Soviet Physics: Acoustics. 1968;14(1):1–17.
  • Kravtsov YuA, Orlov YuI. Caustics, catastrophes and wave fields. Vol. 15, Springer Series on Wave Phenomena. Berlin: Springer-Verlag; 1999.
  • Ludwig D. Uniform asymptotic expansions at a caustic. Communications on Pure and Applied Mathematics. 1966;XIX:215–250.
  • Maslov VP, Fedoryuk VM. Semi-classical approximations in quantum mechanics. Dordrecht: D. Reidel; 1981.
  • Mishchenko A, Shatalov V, Sternin B. Lagrangian manifolds and the Maslov operator. Berlin: Springer-Verlag; 1990.
  • Gerard P, Markowich PA, Mauser NJ, Poupaud F. Homogenization limits and Wigner transforms. Communications on Pure and Applied Mathematics. 1997;50:323–380.
  • Markowich P. On the equivalence of the Schrödinger and the quantum Liouville equations. Mathematical Methods in the Applied Sciences. 1999;11:4106–4118.
  • Lions PL, Paul T. Sur les measures de Wigner. Revista Matemática Iberoamericana. 1993;9:563–618.
  • Filippas S, Makrakis GN. Semiclassical Wigner function and geometrical optics. Multiscale Modeling and Simulation. 2003;1(4):674–710.
  • Filippas S, Makrakis GN. On the evolution of the semi-classical Wigner function in higher dimensions. European Journal of Applied Mathematics. 2003;17:33–62.
  • Sparber C, Markowich PA, Mauser NJ. Wigner functions versus WKB-methods in multivalued geometrical optics. Asymptote Analysis. 2003;33(2):153–187.
  • Berry MV. Semi-classical mechanics in phase space: a study of Wigner’s function. Philosophical Transactions of the Royal Society of London. 1977;287(1343):237–273.
  • Nazaikinskii VE, Schulze B-W, Sternin BYu. Quantization methods in differential equations. London: Taylor & Francis; 2002.
  • Vainberg BR. Quasiclassical approximation in stationary scattering problems. Functional Analysis and its Applications. 1977;11:247–257.
  • Kucherenko VV. Quasiclassical asymptotics of a point-source function for the stationary Schrödinger equation. Theoretical and Mathematical Physics (English Translation). 1969;1(3):294–310.
  • Rauch J. Hyperbolic partial differential equations and geometric optics. Providence (RI): American Mathematical Society; 2012.
  • Evans LC. Partial differential equations. Providence (RI): American Mathematical Society; 1998.
  • Littlejohn RG. The Van Vleck formula, Maslov theory and phase space geometry. Journal of Statistical Physics. 1992;68(1/2):7–50.
  • Courant R, Hilbert D. Methods of mathematical physics. Vol. II. New York: Wiley; 1962.
  • McDonald SW. Phase-space representations of wave equations with applications to the eikonal approximation for short-wavelength waves. Physics Reports. 1988;158(6):337–416.
  • Avila G, Keller J. The high-frequency asymptotic field of a point source in an inhomogeneous medium. Communications on Pure and Applied Mathematics. 1963;XVI:363–381.
  • Katsaounis T, Kossioris GT, Makrakis GN. Computation of high frequency fields near caustics. Mathematical Methods in the Applied Sciences. 2001;11(2):1–30.
  • Papanikolaou G, Ryzhik L. Waves and transport, hyperbolic equations and frequency interactions. In: Caffarelli L, Weinan E, editor. IAS/Park City Mathematical Series, Vol. 5, AMS; 1998, 305–382.
  • Chester C, Friedman B, Ursell F. An extension of the method of steepest descent. Proceedings of the Cambridge Philosophical Society. 1957;53:599–611.
  • Borovikov VA. Uniform stationary phase method. London: The Institution of Electrical Engineers; 1994.
  • Vallee O, Soares M. Airy functions and applications to physics. London: Imperail College Press; 2004.
  • Ozorio de Almeida AM. The Wigner function for two dimensional tori: uniform approximation and projections. Annals of Physics. 1983;145:100–115.
  • Chapman SJ, Lawry JMH, Ockendon JR, Tew RH. On the theory of complex rays. SIAM Review. 1999;41(3):417–509.
  • Seckler BD, Keller JB. Geometrical theory of diffraction in inhomogeneous media. Journal of the Acoustical Society of America. 1959;31:192–205.
  • Bleistein N, Handelsman R. Asymptotic expansions of integrals. New York: Dover; 1986.

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