The div A grad Operator with Indefinite Coefficient Matrix A

这个方案的主要目的是研究n维空间中系数矩阵A不定的有界域上的div A梯度形式的微分算子，这种算子出现在具有负折射率的光学超材料的建模中。在某些假设下，与微分表达式div A grad相关的二次型可以用一个纯离散谱的非半有界自伴算子来表示。最近，我们成功地证明了不定二次型的一些新的表示定理，这些定理允许研究当自伴算子不仅具有离散谱而且具有本质谱的情形。

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.