# 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)= r^{2}y(x,r)where 01(0,1) and q is L^{2}(0,1). By using Paley-Wiener spaces, we show that y(x,r)-cos(r(x-a(x)) belongs to PW_{x} 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.

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)= r^{2}y(x,r)where 01(0,1) and q is L^{2}(0,1). By using Paley-Wiener spaces, we show that y(x,r)-cos(r(x-a(x)) belongs to PW_{x} 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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