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.