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

Authors : Nihal YOKUŞ, Esra KIR ARPAT

Pages : 1316-1334

Doi:10.31801/cfsuasmas.526270

View : 20 | Download : 11

Publication Date : 2019-08-01

Article Type : Research Paper

Abstract :In this paper, we consider the operator L generated in L₂insert ignore into journalissuearticles values(R₊); by the differential expression linsert ignore into journalissuearticles values(y);=-y′′+qinsert ignore into journalissuearticles values(x);y,x∈R₊:=[0,∞);  and the boundary condition insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(y′insert ignore into journalissuearticles values(0););/insert ignore into journalissuearticles values(yinsert ignore into journalissuearticles values(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∈ACinsert ignore into journalissuearticles values(R₊);,  lim_{x→∞}qinsert ignore into journalissuearticles values(x);=0,  sup[e^{ε√x}|q′insert ignore into journalissuearticles values(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