Positivity, essential self-adjointness, and spectral properties of a class of Schrödinger operators with multipolar inverse-square potentials are discussed. In particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established.

Asymptotics of solutions to Schrödinger equations with singular dipole-type potentials is investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular multiples of radial inverse-square functions. Both the linear and the semilinear (critical and subcritical) cases are considered.Dedicated to Prof. Norman Dancer on the occasion of his 60th birthday.

We study positivity, localization of binding and essential self-adjointness properties of a class of Schrödinger operators with many anisotropic inverse square singularities, including the case of multiple dipole potentials.

Once well-posedness is attained, the study of the asymptotic properties of the solution semigroup becomes a meaningful and interesting question, that will be possibly addressed in forthcoming papers.

In this paper we consider a state constrained differential inclusion ẋ ∈ Ax+ F(t; x), with A generator of a strongly continuous semigroup in an infinite dimensional separable Banach space. Under an “inward pointing condition” we prove a relaxation result stating that the set of trajectories lying in the interior of the constraint is dense in the set of constrained trajectories of the convexified inclusion (formula presented). Some applications to control problems involving PDEs are given