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.
Highlights
The spectral analysis of a non-selfadjoint di¤erential operators with continuous and discrete spectrum was investigated by Naimark [1]
We have proved that spectral expansion of L in terms of the principal functions under the condition p q 2 AC(R+); lim q(x) = 0; x!1 sup [e" xjq0(x)j] < 1; x2R+
We have proved the convergence of the spectral expansion
Summary
The spectral analysis of a non-selfadjoint di¤erential operators with continuous and discrete spectrum was investigated by Naimark [1]. The Laurent expansion of the resolvents of non-selfadjoint operators in neigbourhood of spectral singularities was investigated by Gasymov-Maksudov [3] and Maksudov-Allakhverdiev [4]. They studied the e¤ect of spectral singularities in the spectral analysis of these operators. Using the boundary uniqueness theorems of analytic functions, the structure of the eigenvalues and the spectral singularities of a quadratic pencil of Schrödinger, Klein-Gordon, discrete Dirac and discrete Schrödinger operators was investigated in [5]-[10].
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