Eigenfunctions of Quadratic Pencil Operators

Institution

University of West Georgia

Abstract

We give a new and direct proof of the celebrated Gasymov and Guseinov representation of the eigenfunctions of quadratic pencil operators defined by
-y''(x,r)+2r p(x)y(x,r)+q(x)y(x,r)= r2y(x,r)where 01(0,1) and q is L2(0,1). By using Paley-Wiener spaces, we show that y(x,r)-cos(r(x-a(x)) belongs to PWx where a is a certain antiderivative of p. Applications of the Gasymov-Guseinov formula to the spectral theory of the quadratic pencil and the stability of evolution equations generated by the quadratic pencil equation will be discussed.

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Eigenfunctions of Quadratic Pencil Operators

We give a new and direct proof of the celebrated Gasymov and Guseinov representation of the eigenfunctions of quadratic pencil operators defined by
-y''(x,r)+2r p(x)y(x,r)+q(x)y(x,r)= r2y(x,r)where 01(0,1) and q is L2(0,1). By using Paley-Wiener spaces, we show that y(x,r)-cos(r(x-a(x)) belongs to PWx where a is a certain antiderivative of p. Applications of the Gasymov-Guseinov formula to the spectral theory of the quadratic pencil and the stability of evolution equations generated by the quadratic pencil equation will be discussed.