Eigenvalue bounds in the gaps of Schrödinger operators and Jacobi matrices

We consider C = A+B where A is selfadjoint with a gap (a, b) in its spectrum and B is (relatively) compact.We prove a general result allowing B of indefinite sign and apply it to obtain a (δV )d/2 bound for perturbations of suitable periodic Schrödinger operators and a (not quite) Lieb–Thirring bound for perturbations of algebro-geometric almost periodic Jacobi matrices. © 2007 Elsevier Inc. All rights reserved