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]<LaTeX>\mbox{\:}</LaTeX>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^{ε√x}|q′(x)|]<∞,  ε>0

taking into account the spectral singularities. We have also proved the convergence of the spectral expansion.

Keywords

• Eigenvalues,spectral singularities,principal functions,resolvent,spectral expansion.

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