On the negative spectrum of an elliptic operator

New estimates are given for the number of points in the negative spectrum for an elliptic operator or arbitrary order. These estimates generalize and refine the well-known results of Rozenblyum, Lieb, Cwikel, the authors, and others. The proofs have a simple geometric character, and are based on uncomplicated dimensionless imbedding theorems. Also given are results for degenerate elliptic operators, for operators in a domain that contracts or expands in a definite way at infinity, and so on. Theorem 10 gives conditions under which the essential spectrum of an operator contains infinitely many points.