An Exact Version of the Egorov Theorem for Schrödinger Operators in $L^2 (Bbb T)$

We provide an exact version of the Egorov Theorem for a class of Schrödinger oper- ators in $L^2(Bbb T)$, where $Bbb T = Bbb R / 2 pi Bbb Z$ is the one-dimensional torus. We show that the classical Hamiltonian, after the symplectic transformation to action coordinates, can be composed with a toroidal semiclassical ψdo in order to recover the Schrödinger operator. This result turns out to be strictly related to the Bohr-Sommerfeld quantiza- tion rules as well as to the inverse spectral problem and the periodic homogenization of Hamilton-Jacobi equations.