Quantum Integrability and Complete Separation of Variables for Projectively Equivalent Metrics on the Torus

Abstract

Let two Riemannian metrics $g$ and $\bar{g}$ on the torus $T^n$ have the same geodesics (considered as unparameterized curves). Then we can construct invariantly $n$ commuting differential operators of second order. The Laplacian $\Delta_g$ of the metric $g$ is one of these operators. For any $x \in T^n$, consider the linear transformation $G$ of $T_xT^n$ given by the tensor $g^{i\alpha}{\bar{g}}_{{\alpha}j}$. If all eigenvalues of $G$ are different at one point of the torus then they are different at every point; the operators are linearly independent and we can globally separate the variables in the equation $\Delta_gf = {\mu}f$ on this torus.