Universal inequalities for the eigenvalues of a power of the Laplace operator

TL;DR

This paper introduces a new abstract formula connecting eigenvalues of self-adjoint operators with symmetric and skew-symmetric operators, enabling simpler proofs of known results and deriving new bounds for eigenvalues of powers of the Laplace operator.

Contribution

It generalizes existing eigenvalue inequalities by providing a unified abstract framework applicable to various operators, including the Kohn Laplacian.

Findings

Unified abstract formula for eigenvalues of self-adjoint operators.

Simplified proofs for known eigenvalue inequalities.

New bounds for eigenvalues of powers of the Laplace operator.

Abstract

In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use this abstract formulation both for giving dierent and simpler proofs for all the known results obtained for the eigenvalues of a power of the Laplace operator (i.e. the Dirichlet Laplacian, the clamped plate problem for the bilaplacian and more generally for the polyharmonic problem on a bounded Euclidean domain) and to obtain new ones. In a last paragraph, we derive new bounds for eigenvalues of any power of the Kohn Laplacian on the Heisenberg group.