Non-compactness of the Neumann operator for the Kohn Laplacian on the Heisenberg ball

TL;DR

This paper demonstrates that the Neumann operator for the Kohn Laplacian on certain forms is not compact on the Heisenberg ball, providing explicit examples for specific form degrees.

Contribution

It offers explicit examples showing non-compactness of the Neumann operator for the Kohn Laplacian on the Heisenberg ball for certain form degrees.

Findings

Neumann operator is non-compact for specific form degrees

Explicit examples are constructed in the Heisenberg space

Results contribute to understanding boundary behavior of the Kohn Laplacian

Abstract

For $1\leq q \leq n-2$, we provide explicit examples to demonstrate non-compactness of the Neumann operator for the Kohn Laplacian acting on $L^2$ $(0,q)$-forms on the unit ball in $(2n+1)$-dimensional Heisenberg space.