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.