Discrete spectrum and principal functions of non-selfadjoint differential operator

In this article, we consider the operator $L$ defined by the differential expression \[ \ell (y)=-y^{\prime \prime }+q(x) y ,\quad - \infty < x < \infty \] 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 \lbrace \exp \big…