Abstract
We show how the Wentzel-Kramers-Brillouin (WKB) approximation works for potentials with sharp corners.
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K. W. Ford, D. L. Hill, M. Wakano, J. A. Wheeler: Ann. Phys. (NY)7, 239 (1959) K. W. Ford, J. A. Wheeler: Ann. Phys. (NY)7, 259 (1959); Ann. Phys. (NY)7, 287 (1959)
For a review on the WKB method, see for example J. Heading:An Introduction to Phase Integral Methods (Methuen, London 1962)
N. Fröman, P. O. Früman:JWKB Approximation: Contributions to the Theory (North-Holland, Amsterdam 1965)
M. V. Berry, K. E. Mount: Rep. Prog. Phys.35, 315 (1972)
J. A. Wheeler: InStudies in Mathematical Physics, ed. by E. H. Lieb, B. Simon, A. S. Wightman (Princeton Univ. Press, Princeton 1976)
For a review on scattering of light off rough surfaces we refer to P. Beckmann, Spizzichino:The Scattering of Electro-magnetic Waves from Rough Surfaces (Pergamon, New York 1963) J. A. DeSanto, G. S. Brown:Analytical Techniques for Multiple Scattering from Rough Surfaces, Progr. Opt. Vol. XXIII (North-Holland, Amsterdam 1986) p. 3
For the application of a refined WKB method to symmetric planar wave guides with truncated-index and graded-index profiles, that is, wave guides where the index of refraction involves corners, see Jingyi Wang, Li Qiao: IEEE J. QE-27, 878 (1991) and references therein
For a discussion of the importance of reflected waves to detect airplanes, and, in particular, for methods to suppress such waves, see the stealth bomber analyzed in the articles inThe New York Times by R. Holloran, pages A1 and B7, May 16 (1988) and by R. W. Stevenson pages A1 and A15, July 18 (1989) J. A. Adam: IEEE Spectrum25 (4), 26 (1988) G. Sweetman:Stealth Bomber, Invisible Warplane, Black Budget (Motorbooks, Osceola 1989)
Here, we neglect the exponentially small contribution of a reflected wave appearing ordinarily at a smooth potential discussed for the first time by V. L. Pokrovskii, S. K. Savvinykh, F. R. Ulinich: J. Exp. Theoret. Phys.34, 1272 (1958); Sov. Phys.-JETP34, 879 (1958) V. L. Pokrovskii, F. R. Ulinich, S. K. Savvinykh: Sov. Phys.-JETP34, 1119 (1958); Sov. Phys.-JETP34, 1629 (1958) V. L. Pokrovskii, I. M. Khalatnikov: Sov. Phys.-JETP13, 1207 (1961) L. D. Landau, E. M. Lifschitz:Quantum Mechanics (Non-relativistic Theory) (Pergamon, Oxford 1965) p. 78 For a beautiful summary of this problem we refer to Sect. II of [4]. The birth of exponentially small reflected waves as a result of the crossing of a Stokes line from the nearest complex turning point with the ϰ-axis is an immediate application of M. V. Berry's work on uniform asymptotic smoothing of Stokes' discontinuities, see for example, M. V. Berry: Proc. R. Soc. London A422, 7 (1989) For more on the treatment of exponentially small terms coined hyperasymptotics see M. V. Berry, C. J. Howls: Proc. R. Soc. London A430, 653 (1990) and references therein
L. S. Schulman:Techniques and Applications of Path Integration (Wiley, New York 1982)
A. Anderson: Am. J. Phys.57, 230 (1989)
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Dedicated to H. Walther on the of occasion his 60th birthday
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Bestle, J., Schleich, W.P. & Wheeler, J.A. Anti-stealth: WKB grapples with a corner. Appl. Phys. B 60, 289–299 (1995). https://doi.org/10.1007/BF01135876
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DOI: https://doi.org/10.1007/BF01135876