Discrete spectrum and principal functions of non-selfadjoint differential operator

Abstract

In this article, we consider the operator L defined by the differential expression \(l(y) = - y'' + q(x)y,{\text{ }} - \infty < x < \infty \) in L 2(−∞, ∞), 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 \(\mathop {{\text{sup}}}\limits_{ - \infty < x < \infty } \;\left\{ {\exp \left( {\varepsilon \sqrt {\left| x \right|} } \right)\left| {q(x)} \right|} \right\} < \infty ,\;\;\;\;\varepsilon > 0\) holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.