Second Order Perturbations of Divergence Type Operators with a Spectral Gap

While it follows from Floquet theory that the spectrum of periodic elliptic operators $$ A = - \sum {{\partial _j}{a_{ij}}{\partial _i},} $$ acting in the Hilbert space L 2 (R), has band structure, we construct operators of this type with a spectral gap. We also report on some recent results of Alama et al. (1992) concerning spectral properties of divergence form operators $$ A + \lambda B $$ where $$ {\text{B = - }}\sum {{\partial _j}{b_{ij}}{\partial _i}} $$ is a non-negative operator whose coefficients tend to zero at ∞. Here we ask for eigenvalues of A + λB, λ > 0, in a spectral gap of A.