Abstract:
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.
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