Resumen de Spectral properties of a non selfadjoint system of differential equations with a spectral parameter in the boundary condition

Esra Kir Arpat, Gülen Bascanbaz Tunca, Canan Yanik

• In this paper we investigated the spectrum of the operator L(λ) generated in Hilbert Space of vector-valued functions L2 (R+, C2) by the system iy0 1 + q1 (x) y2 = λy1, −iy0 2 + q2 (x) y1 = λy2 (0.1) , x ∈R+ := [0,∞), and the spectral parameter- dependent boundary condition (a1λ + b1) y2 (0, λ) − (a2λ + b2) y1 (0, λ)=0, where λ is a complex parameter, qi, i = 1, 2 are complex-valued functions ai 6= 0, bi 6= 0, i = 1, 2 are complex constants. Under the condition sup x∈R+ {exp εx |qi (x)|} < ∞, i = 1, 2,ε> 0, we proved that L(λ) has a finite number of eigenvalues and spectral singularities with finite multiplicities. Furthermore we show that the principal functions corresponding to eigenvalues of L(λ) belong to the space L2 (R+, {C2) and the principal functions corresponding to spectral singularities belong to a Hilbert space containing L2 (R+, C2).