Schroedinger and elliptic operators with distributional coefficients on a bounded domain

We study the operator $L=-\mathrm{\Delta }+q$ on a bounded domain $\Omega\subset\mathbb R^n$,where$q(x)$ is a distributional potential. We find sufficient conditions for $q(x)$ which guarantee that $L$ is well–defined with Dirichlet and generalized Neumann boundary conditions. The asymptotics of the eigenvalues and the basis properties of the eigen- and associated functions of such operators are studied. The results are generalized for strongly elliptic operator of order $2m$ with Dirichlet boundary conditions.