COUNTING SPECTRUM VIA THE MASLOV INDEX FOR ONE DIMENSIONAL θ–PERIODIC SCHRÖDINGER OPERATORS

We study the spectrum of the Schrödinger operators with n × n matrix valued potentials on a finite interval subject to θ–periodic boundary conditions. For two such operators, corresponding to different values of θ, we compute the difference of their eigenvalue counting functions via the Maslov index of a path of Lagrangian planes. In addition we derive a formula for the derivatives of the eigenvalues with respect to θ in terms of the Maslov crossing form. Finally, we give a new shorter proof of a recent result relating the Morse and Maslov indices of the Schrödinger operator for a fixed θ.