Geodesic
Equations of the quantum inverse scattering method in the semiclassical limit
Teoretičeskaâ i matematičeskaâ fizika, Tome 63 (1985) no. 1, pp. 32-49 Cet article a éte moissonné depuis la source Math-Net.Ru

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The transition to the semiclassicat limit in Marchenko's method for the inverse scattering problem for fixed angular momentum in the $s$-wave case is investigated. It is shown that the kernel $K(r, r')$ of the transformation operator is determined by the classically forbidden region and is exponentially large. Therefore, the linear integral equation for $K(r, r')$ cannot be reduced to a relationship between semiclassical physical quantities. Instead, one uses the equivalent nonlinear equation for the kernel $L(r, r')$ of the inverse operator of the transformation, continued with respect to the first argument to the complete axis. Under semiclassical conditions, the kernel $L(r, r')$ is a rapidly oscillating function having a simple physical meaning, and the nonlinear equation for $L(r, r')$ goes over into the well-known semiclassical relation between the phase shift and the potential. As an example, $s$-wave scattering by an exponential potential is considered. 
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     author = {D. I. Abramov},
     title = {Equations of the quantum inverse scattering method in the semiclassical limit},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {32--49},
     year = {1985},
     volume = {63},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1985_63_1_a2/}
}
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D. I. Abramov. Equations of the quantum inverse scattering method in the semiclassical limit. Teoretičeskaâ i matematičeskaâ fizika, Tome 63 (1985) no. 1, pp. 32-49. http://geodesic.mathdoc.fr/item/TMF_1985_63_1_a2/

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