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The quasi-classical limit of scattering amplitude - finite range potentials. (English) Zbl 0591.35079
Schrödinger operators, Lect. 2nd 1984 Sess. C.I.M.E., Como/Italy, Lect. Notes Math. 1159, 242-263 (1985).
Summary: [For the entire collection see Zbl 0565.00010.]
In these lectures the author considers the asymptotic behavior as Planck’s constant \(\hslash \to 0\) of the scattering operator \(S^ h\) associated with the pair of time dependent Schrödinger equations \[ (1)\quad i\hslash (\partial u/\partial t)=-(\hslash^ 2/2m)\Delta u+V(x)u\equiv H^ hu, \] \[ (2)\quad i\hslash (\partial u/\partial t)=- (\hslash^ 2/2m)\Delta u\equiv H^ h_ 0u. \] He obtains the asymptotic expansion of the scattering amplitude \(T^ h(p,q)\) (in an average sense) and proves the well-known conjecture that the limit as \(h\to 0\) of the quantum mechanical total cross section is twice the one of classical mechanics.

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35P25 Scattering theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
35C20 Asymptotic expansions of solutions to PDEs