Exponential Estimates of Solutions of Pseudodifferential Equations with Operator-valued Symbols: Applications to Schrodinger Operators with Operator-valued Potentials

We consider a class of pseudodifferential operators with operator-valued symbols a(x, xi) under the assumption that a(x, xi) can be analytically extended with respect to xi onto a tube domain R-n + iB where B is a convex bounded domain in R-n containing the origin. The main result of the paper is exponential estimates at infinity of solutions of pseudodifferential equations Op(a)u = f. We apply this result to Schrodinger operators with operator-valued potentials and give applications to spectral properties of quantum waveguides. Our approach is based on the construction of the local inverse operator at infinity and on formulas for commutators of pseudodifferential operators with exponential weights.