A SPECTRAL EXPANSION FOR SCHRÖDINGER OPERATOR

In this paper we consider the Schrödinger operator L generated in L2 (R+) by y00 + q (x) y = µy, x ∈ R+: = [0,∞) subject to the boundary condition y0 (0) − hy (0) = 0, where,q is a complex valued function summable in [0, ∞ and h 6 = 0 is a complex constant, µ is a complex parameter. We have assumed that supx∈R+ {exp (ε x) |q (x)|} <∞, ε> 0, holds which is the minimal condition that the eigenvalues and the spec-tral singularities of the operator L are finite with finite multiplicities. Under this condition we have given the spectral expansion formula for the operator L using an integral representation for the Weyl func-tion of L. Moreover we also have investigated the convergence of the spectral expansion