Spectral Expansion of a Non-Self-Adjoint Differential Operator on the Whole Axis

In this article, we consider an operator L defined by the differential expression l y y Ž . qŽx.y, x R Ž , . in L2ŽR., where q is a complex-valued function. Under the condition sup expŽ ʹ x .qŽx.4 , 0, x we have proved a spectral expansion of L in terms of the principal functions, taking into account the spectral singularities. We have also investigated the convergence of the spectral expansion of L.