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Fredholm property and essential spectrum of pseudodifferential operators with operator-valued symbols

Commun. Math. Anal. Conf.
Conference 3 (2011), 50 - 65

Fredholm property and essential spectrum of pseudodifferential operators with operator-valued symbols

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Abstract

We consider the Fredholm property and essential spectrum of pseudodifferential operators with symbols having values in the space of bounded operators acting from a separable Hilbert space H1 into a separable Hilbert space H2 and satisfying additional estimates. There are many problems in mathematical physics which are reduced to the study of associated pseudodifferential operators with operator-valued symbols. In particular, this happens for problems of wave propagation in acoustic, electromagnetic and quantum waveguides. We consider so-called locally Fredholm pseudodifferential operators with operator-valued symbols for which the study of their Fredholmness is reduced to the problem of local invertibility at infinity. For investigation of the local invertibility at infinity we use the method of limit operators. We give applications of general results to the investigation of the essential spectra of Schr¨odinger operators with the Robin condition at the boundary in a cylindrical domain D = Rn £W; where W is a bounded domain in Rm with a smooth boundary ( quantum waveguides). We obtain the exact descriptions of the essential spectra of such operators in terms of the spectra of limit operators. In the case of slowly oscillating at infinity electric potentials the description of the essential spectra has an explicit form.