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}.