An eigenfunction expansion of the non-selfadjoint Sturm–Liouville operator with a singular potential

Abstract

In this paper, we consider the operator $L$ generated in $L^{2}\left( \mathbb{R }_{+}\right)$ by the differential expression

$$\begin{aligned} l\left( y\right) =-y^{\prime \prime }+\left[ \frac{\nu ^{2}-\frac{1}{4}}{x^{2}}+q\left( x\right) \right] y,\,\,x\in \mathbb{R }_{+}:=\left( 0,\infty \right) \end{aligned}$$

and the boundary condition

$$\begin{aligned} \underset{x\rightarrow 0}{\lim }x^{-\nu -\frac{1}{2}}y\left( x\right) =1, \end{aligned}$$

where $q$ is a complex valued function and $\nu$ is a complex number with $Re\nu >0$. We have proved a spectral expansion of L in terms of the principal functions under the condition

$$\begin{aligned} \underset{x\in \mathbb{R }_{+}}{Sup}\left\{ e^{\epsilon \sqrt{x}}\left| q(x)\right| \right\} <\infty , \epsilon >0 \end{aligned}$$

taking into account the spectral singularities. We have also investigated the convergence of the spectral expansion.