Gaussian decay of the magnetic eigenfunctions

We investigate whether the eigenfunctions of the two-dimensional magnetic Schrödinger operator have a Gaussian decay of type exp(-Cx2) at infinity (the magnetic field is rotationally symmetric). We establish this decay if the energy (E) of the eigenfunction is below the bottom of the essential spectrum (B), and if the angular Fourier components of the external potential decay exponentially (real analyticity in the angle variable). We also demonstrate that almost the same decay is necessary. The behavior of C in the strong field limit and in the small (B - E) limit is also studied.