Discrete spectrum and principal functions of non-selfadjoint differential operator

In this article, we consider the operator L defined by the differential expression \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$l(y) = - y'' + q(x)y,{\text{ }} - \infty < x < \infty $$ \end{document} in L2(−∞, ∞), 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathop {{\text{sup}}}\limits_{ - \infty < x < \infty } \;\left\{ {\exp \left( {\varepsilon \sqrt {\left| x \right|} } \right)\left| {q(x)} \right|} \right\} < \infty ,\;\;\;\;\varepsilon > 0$$ \end{document} holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.