Inverse spectral results for the magnetic Schrödinger operator in low dimensions

We consider the semiclassical magnetic Schrödinger operator in dimensions one and two. The first result is that an even one dimensional semiclassical potential is spectrally determined by the Schrödinger operator. The second result states that a two dimensional radial magnetic field is spectrally determined by a purely magnetic Schrödinger operator. This extends the earlier work of Guillemin-Wang and answers a conjecture of theirs therein.