Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditions

Abstract

In this paper, we consider the operator L generated in L₂(R₊) by the differential expression l(y)=-y′′+q(x)y,x∈R₊:=[0,∞) and the boundary condition ((y′(0))/(y(0)))=α₀+α₁λ+α₂λ², where q is a complex valued function and α_{i}∈C,[mbox] \mbox{\:} i=0,1,2α₂. We have proved that spectral expansion of L in terms of the principal functions under the condition q∈AC(R₊), lim_{x→∞}q(x)=0, sup[e^{e√x}|q′(x)|] 0 taking into account the spectral singularities. We have also proved the convergence of the spectral expansion.