Gauge Equivalence and the Inverse Spectral Problem for the Magnetic Schr\"odinger Operator on the Torus

We study the inverse spectral problem for the Schr\"odinger operator $H$ on the two-dimensional torus with even magnetic field $B(x)$ and even electric potential $V(x)$. V.Guillemin [11] proved that the spectrum of $H$ determines $B(x)$ and $V(x)$. A simple proof of Guillemin's results was given by the authors in [3]. In the present paper we consider gauge equivalent classes of magnetic potentials and give conditions which imply that the gauge equivalence class and the spectrum of $H$ determine the magnetic field and the electric potential. We also show that generically the spectrum and the magnetic field determine the "extended" gauge equivalence class of the magnetic potential. The proof is a modification of the proof in [3] with some corrections and clarifications.