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

Abstract

In this paper we investigate the spectrum and the spectral singularities of an operator L generalized in L 2( R +) by the differential expression l(y)=y″− ∑ k=0 n−1 λ kq k(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, q k , k=0,1,…, n−1, are complex valued functions, q 0, q 1,…, q n−1 are differentiable on (0,∞), K∈L 2( 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→∞ q k(x)=0, sup x∈ R + e ε x ∑ k=0 n−1 q′ k(x) + K(x) <∞ hold, where k=0,1,…, n−1 and ε>0.