The discrete spectrum in the gaps of a second-order perturbed periodic operator
The author studies perturbations of the Schrödinger operator by a potential which leads to a discrete spectrum in ``continuous gaps''. There are investigated the properties of the spectrum of the operator on \(L_ 2({\mathbb{R}}^ d)\), \(d\geq 3\), generated by \(-div(a(x)\text{grad} u)+f(x)u,\) where a(x) is \(C^ 1\) and the potential \(f\in L_{\infty}\) is real and periodic. In the case of regular perturbations there are described integral representations for some functionals defined in terms of eigenvalues of perturbation operators and gaps in the spectrum. There is also discussed the case of irregular perturbations.
scientific article; zbMATH DE number 4212632