The Cauchy problem of the one dimensional Schrödinger equation with non-local potentials

Abstract

For a large class of operators A, not necessarily local, it is proved that the Cauchy problem of the Schrödinger equation: −d2f(z)dz2+Af(z)=s2f(z), f(0)=0, f′(0)=1 possesses a unique solution in the Hilbert (H2(Δ)) and Banach (H1(Δ)) spaces of analytic functions in the unit disc Δ={z:|z|<1}.