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 Schrodinger operators and a (not
quite)Lieb-Thirring bound for perturbations of algebro-geometric almost
periodic Jacobi matrices