zbMATH — the first resource for mathematics

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.

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