2. Spectral problems

Problem 2.1.
[S. Fu] Let $L=\partial_b + f\bar\partial_b$ on the standard CR threesphere. Does $L$ have closed range if and only if the Paneitz operator is nonnegative?
Remark. If the CR Yamabe constant is positive, it is known that $L$ has closed range if the Paneitz operator is nonnegative.

Remark. More ambitiously, one would like to know if the spectrum of the Kohn Laplacian determines the CR mass.


Problem 2.2.
[J. Dâ€™Angelo and S. Fu] Consider the domains
\[ \Omega_k = \left\{ (z,w) \in \mathbb{C}^{n+1} \colon \mathrm{Im}(w) < \lvert z\rvert^{2k} \right\} \]
and
\[ U_k = \left\{ (z,w) \in \mathbb{C}^{n+1} \colon \mathrm{Im}(w) < \sum_{j=1}^n \lvert z\rvert^{2k} \right\} . \]
(a) Can one compute the spectrum of the Kohn Laplacian on $(0,1)$forms in either case?
(b) Can one distinguish $\Omega_k$ from $U_k$ using the spectrum?
Cite this as: AimPL: Analysis and geometry on pseudohermitian manifolds, available at http://aimpl.org/pshermitian.