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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