Abstract Title

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.

Comments

This is spectral theory of differential operators with harmonic analysis. Very suitable for graduate students

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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.