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

Esra Kir, Gülen Bascanbaz-Tunca, Canan Yanik

Resumen


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).


Palabras clave


Spectrum ; Spectral Singularities ; Non-Selfadjoint System of Differential Equations.

Texto completo:

PDF

Referencias


O. Akın and E. Bairamov, On the structure of discrete spectrum of the non-selfadjoint system of differential equations in the first order, J. Korean Math. Soc. No 3, 32, pp. 401-413, (1995).

Yu. M. Berezanski, Expansion in Eigenfunctions of Selfadjoint Operators, Amer. Math. Soc., Providence R. I. (1968).

E. P. Dolzhenko, Boundary Value Uniqueness Theorems for Analytic Functions, Math. Notes 25 No 6, pp. 437-442, (1979).

N. B. Kerimov, A Boundary Value Problem for the Dirac System with a Spectral Parameter in the Boundary Conditions, Differential Equations, Vol. 38, No 2, pp. 164-174, (2002).

E. Kır, Spectral Properties of Non-Selfadjoint System of Differential Equations, Commun. Fac. Sci. Univ. Ank. Series A1, Vol.49 (2000),111- 116.

V.E. Lyance, A differential Operator with Spectral Singularities, I,II, Amer. Math. Soc.Trans. Ser. 2, Vol. 60, pp. 185-225, pp. 227-283, (1967).

M.A.Naimark, Investigation of the Spectrum and the Expansion in Eigenfunctions of a Non-selfadjoint Operator of Second Order on a Semi-axis, Amer. Math. Soc. Trans. Ser. 2, Vol 16, pp.103-193, Amer. Math. Soc., Providence, (1960).

J.T.Schwartz, Some non-selfadjoint operators, Comm. Pure and Appl.Math. 13, pp. 609-639, (1960).




DOI: http://dx.doi.org/10.4067/10.4067/S0716-09172005000100005

Enlaces refback

  • No hay ningún enlace refback.