Czechoslovak Mathematical Journal, Vol. 49, No. 4, pp. 689-700, 1999

# Discrete spectrum and principal functions of non-selfadjoint differential operator

## Gulen Bascanbaz Tunca, Elgiz Bairamov

Ankara University, Department of Mathematics, Faculty of Science, 06100 Tandogan-Ankara, Turkey, e-mail: tunca@science.ankara.edu.tr

Abstract: In this article, we consider the operator $L$ defined by the differential expression
\ell(y)=-y"+q(x) y ,\quad- \infty< x < \infty
in $L_2(-\infty,\infty)$, where $q$ is a complex valued function. Discussing the spectrum, we prove that $L$ has a finite number of eigenvalues and spectral singularities, if the condition
\sup_{-\infty< x < \infty} \Big\{ \exp\bigl(\epsilon\sqrt{|x|}\bigr) |q(x)|\Big\} < \infty, \quad\epsilon> 0
holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.

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