Strongly singular couplings supported by manifolds of codimension one and two
Inverse Problems and Applications
29 April 14:00 - 15:00
In this talk I am going to discuss Schrödinger operators with an attractive singular potential supported by a manifold of a lower dimensionality, more specifically, of the type of a delta function supported by a curve or surface. After illustrating that the geometry of the interaction support can give rise to an effective interaction, we concentrate on the strong-coupling asymptotic behaviour. We show that after the natural energy renormalization, the leading term is given by an operator on the curve or surface with a geometrically induced potential. We discuss the several situations with different dimensions and codimensions situations including periodic manifolds and magnetic Schrödinger operators. In the simplest case of a planar arc we also prove a conjecture about the asymptotic behaviour for manifolds with a boundary.