Abstract
In this talk we will investigate the inverse problem of recovering an unbounded
potential for the Schrödinger equation in two dimensions. In contrary with its analogy
in higher dimensions, we need to construct semiclassical Carleman estimates for the
operator
with holomorphic weight and a better order of decay. This will be done by exploiting the
factorisation
and we explain how a classical result of Gunning-Narasimhan that every Riemann
surface admits a holomorphic function with non-vanishing gradient provides the
natural weight. Finally, we will discuss some limitations of the existing method.