The div A grad Operator with Indefinite Coefficient Matrix A

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The main objective of this proposal is the study of differential operators of the form div A grad on bounded domains in the n-dimesional space with the indefinite coefficient matrix A. Such operators occur in the modeling of optical metamaterials with the negative refractive index. Under some assumptions the quadratic form associated with the differential expression div A grad can be respresented by a non-semibounded self-adjoint operator with purely discrete spectrum. Recently we have succeeded to prove some new representation theorems for indefinite quadratic forms, which allows to study cases, when the self-adjoint operator possesses not only the discrete but also an essential spectrum.