Asymptotic formulae with remainder estimates for eigenvalue branches of the Schrödinger operator $H-\lambda W$ in a gap of $H$

The Floquet theory provides a decomposition of a periodic
Schrödinger operator into a direct integral, over a torus, of operators on a basic period cell. In this paper, it is proved that the same transform establishes a unitary equivalence between a multiplier by a decaying potential and a pseudo-differential operator on the torus, with an operator-valued symbol. A formula for the symbol is given. As applications, precise remainder estimates and two-term asymptotic formulas for spectral problems for a perturbed periodic Schrödinger operator are obtained.