The integrated density of states for periodic elliptic pseudo-differential operators in dimension one

• Main Author: Hughes, David J. J.
• Published: University of Sussex 2005
• Subjects: 515.7242
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.410366

In this thesis, we study an elliptic, one dimensional, pseudo-differential operator, with homogeneous symbol, which is perturbed by a lower order, symmetric, pseudo-differential operator, whose symbol is periodic, magnified by a real coupling constant. The main goal is to prove that such an operator generates a complete asymptotic expansion of the integrated density of states for large energies and an arbitrary large coupling constant. The Floquet theory, which may be viewed as the foundation of the study of periodic operators, is rigorously developed for pseudo-differential operators in arbitrary dimension. We prove the existence, through the use of a developed calculus, of a "Gauge transformation" which is a unitary operator, transforming the original operator into an operator whose symbol, up to some controllable perturbation and remainder, has constant coefficients. This operator with constant coefficients is shown to admit a complete asymptotic expansion. Finally, we show, by the use of a suitable perturbation argument, that these asymptotics coincide with that of the original operator