Asymptotic formulae for eigenvalues and eigenfunctions of Sturm--Liouville operators with potentials---distributions. Dirichlet--Neumann boundary conditions

We deal with the Sturm–Liouville operator $Ly=l(y)=-{\displaystyle \frac{d^{2}y}{d{x}^2}}+q(x)y,$ with Dirichlet–Neumann boundary conditions $y(0)=y^{\mathrm{\prime }}(\pi )=0$ in the space $L_2[0,\pi ]$. We assume that the potential $q$ is complex-valued and has the form $q(x)=u^{\mathrm{\prime }}(x)$, where $u\in L_2[0,\pi]$. Here the derivative is treated in the distributional sense. Our aim is to obtain the detailed asymptotic formulae for eigenvalues and eigen- and associated functions of the operator $L$.