Spectral properties of a Schrödinger equation with a class of complex potentials and a general boundary condition

In this paper we investigate the spectrum and the spectral singularities of an operator L generalized in L2(R+) by the differential expression l(y) = y − n−1 k=0 λkqk(x)y, x ∈ R+ = [0,∞), and the boundary condition ∞ 0 K(x)f (x)dx + αf (0) −βf (0) = 0, where λ is a complex parameter, qk , k = 0, 1, . . . ,n − 1, are complex valued functions, q0,q1, . . . , qn−1 are differentiable on (0,∞), K ∈ L2(R+), and α,β ∈ C with |α| + |β| = 0. Discussing the spectrum we obtain that L has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions lim x→∞ qk(x) = 0, sup x∈R+ eε√x n−1 k=0 q k (x) + K(x) <∞ hold, where k = 0, 1, . . . ,n− 1 and ε >0.  2003 Elsevier Inc. All rights reserved.