Two classes of special functions for the phase-integral approximation

1979 ◽  
Vol 12 (8) ◽  
pp. 1149-1154 ◽  
Author(s):  
J A Campbell
Keyword(s):  
Special Functions ◽  
Integral Approximation ◽  
Phase Integral

scholarly journals Comments on ’’Higher order modified potentials for the effective phase integral approximation’’

1981 ◽  
Vol 22 (6) ◽  
pp. 1190-1191 ◽  
Author(s):  
Nanny Fröman ◽  
Per Olof Fröman
Keyword(s):  
Higher Order ◽  
Integral Approximation ◽  
Phase Integral ◽  
Effective Phase

Phase‐Integral Approximation in Momentum Space and the Bound States of an Atom. II

1969 ◽  
Vol 10 (6) ◽  
pp. 1004-1020 ◽  
Author(s):  
Martin C. Gutzwiller
Keyword(s):  
Momentum Space ◽  
Bound States ◽  
Integral Approximation ◽  
Phase Integral

Transmission through cutoffs and resonances in the double phase‐integral approximation

1984 ◽  
Vol 25 (9) ◽  
pp. 2655-2661 ◽  
Author(s):  
Andrzej A. Skorupski
Keyword(s):  
Integral Approximation ◽  
Double Phase ◽  
Phase Integral

A large-N phase integral approximation of Coulomb-type systems using SO(2,1) coherent states

1986 ◽  
Vol 19 (18) ◽  
pp. 3797-3806 ◽  
Author(s):  
C C Gerry ◽  
J B Togeas
Keyword(s):  
Coherent States ◽  
Type Systems ◽  
Integral Approximation ◽  
Large N ◽  
Coulomb Type ◽  
Phase Integral

Phase‐Integral Approximation in Momentum Space and the Bound States of an Atom

1967 ◽  
Vol 8 (10) ◽  
pp. 1979-2000 ◽  
Author(s):  
Martin C. Gutzwiller
Keyword(s):  
Momentum Space ◽  
Bound States ◽  
Integral Approximation ◽  
Phase Integral

Waves near Stokes lines

1990 ◽  
Vol 427 (1873) ◽  
pp. 265-280 ◽  
Keyword(s):  
Refractive Index ◽  
Error Function ◽  
Divergent Series ◽  
Stokes Line ◽  
Stokes Phenomenon ◽  
Total Change ◽  
Integral Approximation ◽  
Reflected Waves ◽  
Stokes Lines ◽  
Phase Integral

The large- k asymptotics of d 2 u ( z )/d z 2 = k 2 R 2 ( z ) u ( z ) are studied near a Stokes line ( ω ≡ ∫ z z 0 R d z real, where z 0 is a zero of R 2 ( z ), of any order), on which there is greatest disparity between the dominant and subdominant exponential waves in the phase-integral (WKB) approximations. The aim is to establish precisely how the multiplier b _ of the subdominant wave varies across the Stokes line. Although b _ always has a total change proportional to i times the multiplier of the dominant wave (the Stokes phenomenon), the form of the change depends on the convention used to define the two waves. The optimal convention, for which the variation is maximally compact and smooth, is to define them by the phase-integral approximation truncated at its least term, whose order is proportional to k and therefore large (‘asymptotics of asymptotics’). Then the variation of b _ is proportional to the error function of the natural Stokes-crossing variable Im ω √( k /Re ω ). This result is obtained without resumming divergent series (thereby avoiding ‘asymptotics of asymptotics of asymptotics’). An application is given, to the birth of exponentially weak reflected waves in media with smoothly varying refractive index.

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