Spectral analysis of family of singular non-self-adjoint differential operators of even order

This work examines the spectrum of a family of certain non-self-adjoint singular differential operators of even order on a whole axis. The coefficients of such operators depend on a complex spectral parameter in a polynomial manner. The scope of our work is also engaged in the construction of the resolvent and a multiple spectral expansion which is corresponding to such operators. This process is performed under the hypothesis that the coefficients of the differential expression are not infinitely small. The similar problems on a semi-axis and a whole axis were investigated in earlier papers [F.G. Maksudov, E.E. Pashayeva, About multiple expansion in terms of eigenfunctions for one-dimensional non-self-adjoint differential operator of even order on a semi-axis, in: Spectral Theory of Operators and Its Applications, vol. 3, Elm Press, Baku, 1980, pp. 34–101 (in Russian)] and [E.E. Pashayeva, About one multiple expansion in terms of solutions of differential equation on the whole axis, in: Spectral Theory of Operators and Its Applications, vol. 5, Elm Press, Baku, 1984, pp. 145–151 (in Russian)], respectively. However, in those papers, the coefficients of the differential expression were decreasing rapidly enough as x was approaching to infinity. © 2007 Elsevier Inc. All rights reserved.