Inverse boundary value problem for Schrödinger equation in two dimensions (Q2902721)

The paper under review considers the problem of reconstructing the potential \(q\) of the two-dimensional Schrödinger equation NEWLINE\[NEWLINE (\Delta + q) u = 0 \text{ in } \Omega, NEWLINE\]NEWLINE where \(\Omega\in \mathbb R^2\) is a bounded smooth domain whose boundary consists of a number of smooth contours, from Cauchy data given on the whole boundary or a part \(\tilde\Gamma\) of it.NEWLINENEWLINEThe authors prove that the potential \(q\) can be uniquely determined if \(q\in C^\alpha(\Omega)\), for some \(\alpha\in (0,1) \), in the case of full Cauchy data, or \(q\in W^1_p(\Omega)\) with some \(p > 2\) in the case of partial Cauchy data.NEWLINENEWLINEThe proof makes use of the complex geometrical optics solution and stationary phase method.NEWLINENEWLINEThis result improves previous ones concerning the regularity of the potential.