The eikonal approximation for the Schrödinger equation (Q908427)

The Schrödinger equation \([\Delta -gq(x)+k^ 2]\psi =0\) is considered under the following assumptions: 1) q(x) satisfies the inequality \(q(x)\leq const(1+| x|)^{- \alpha}\) 2) \(N=g(2k)^{-1}\) is an arbitrary constant, where g is the coupling constant, k is the wave vector. It is known that the time independent perturbation theory can be used only for \(N\to 0\). The asymptotic behaviour both of wave function (\(\psi)\) and its derivative with respect to k are found in the eikonal approximation. This allows one to determine both the amplitude of forward scattering and the total cross section. The formulae of Born approximation for the wave function, the amplitude and for the cross section follow from the obtained results under \(k\to \infty\), \({\mathcal N}\to 0\). The method of Green functions and the theory of Hermitian linear operators in Hilbert spaces are used in the work. The main results have been published by the author [Ann. Inst. Henri Poncaré, Phys. Theor. 44, 397-425 (1986; Zbl 0608.35054)].