Abstract. We study the forward problem of the magnetic Schrödinger operator with potentials that have a strong singularity at the origin. We obtain new resolvent estimates and give some applications on the spectral measure and on the solutions of the associated evolution problem.

We study the following Helmholtz equationin R d with magnetic and electric potentials that are singular at the origin and decay at infinity. We prove the existence of a unique solution satisfying a suitable Sommerfeld radiation condition, together with some a priori estimates. We use the limiting absorption method and a multiplier technique of Morawetz type.

We study the electromagnetic Helmholtz equationwith the magnetic vector potential b(x) and n(x) a variable index of refraction that does not necessarily converge to a constant at infinity, but can have an angular dependence like n(x) → n ∞ ( x |x| ) as |x| → ∞. We prove an explicit Sommerfeld radiation conditionfor solutions obtained from the limiting absorption principle and we also give a new energy estimatewhich explains the main physical effect of the angular dependence of n at infinity and deduces that the energy concentrates in the directions given by the critical points of the potential.

Abstract. We study the inverse Robin problem for the Schrödinger equation in a half-space. The potential is assumed to be compactly supported. We first solve the direct problem for dimensions two and three. We then show that the Robin-to-Robin map uniquely determines the potential q.

Abstract. In this paper we study unique continuation theorems for magnetic Schrödinger equation via Carleman estimates. We use integration by parts techniques in order to show these estimates. We consider electric and magnetic potentials with strong singularities at the origin and some decay at infinity.