The Fredholm property and the essential spectrum of pseudo-differential operators (PDO) with operator-valued symbols is studied. The method of limit operators is developed and the essential spectrum of operators of quantum waveguides is considered. Two types of electric potentials are analyzed, slowly oscillating at infinity and perturbations of periodic potentials by slowly oscillating ones. The boundary value problem is found to be associated with an unbounded self-adjoint operator with the domain and the operator describes the coupled states of the quantum system on the configuration space with a potential equal to infinity outside. For the unbounded operator of boundary value problem, the union is taken over the spectra of all the limit operators of the sequences for which limits exists.
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