CR Paneitz operator on non-embeddable CR manifolds

TL;DR

This paper studies the properties of the CR Paneitz operator on non-embeddable CR manifolds, revealing its spectral characteristics and differences from the embeddable case, especially on the Rossi sphere.

Contribution

It demonstrates that the CR Paneitz operator on non-embeddable CR manifolds is self-adjoint with discrete spectrum and exhibits infinitely many negative eigenvalues on the Rossi sphere.

Findings

Operator is essentially self-adjoint with discrete spectrum.

Eigenfunctions form finite-dimensional subspaces.

Rossi sphere has infinitely many negative eigenvalues.

Abstract

The CR Paneitz operator is closely related to some important problems in CR geometry. In this paper, we consider this operator on a non-embeddable CR manifold. This operator is essentially self-adjoint and its spectrum is discrete except zero. Moreover, the eigenspace corresponding to each non-zero eigenvalue is a finite dimensional subspace of the space of smooth functions. Furthermore, we show that the CR Paneitz operator on the Rossi sphere, an example of non-embeddable CR manifolds, has infinitely many negative eigenvalues, which is significantly different from the embeddable case.