Spectrum of the Kohn Laplacian on the Rossi sphere (Q1639626)

Let \(S^3\) denote the unit sphere in \(\mathbb{C}^2\) and \[ \square_b^t=-L_t \frac{1+|t|^2}{(1-|t|^2)^2}\overline{L}_t \] be the Kohn Laplacian on the Rossi example \((S^3,L_t)\) where \[ L_t= \overline{z}_1\frac{\partial}{\partial z_2}-\overline{z}_2\frac{\partial}{\partial z_1} +\overline{t}\left( z_1\frac{\partial}{\partial \overline{z}_2} -z_2\frac{\partial}{\partial \overline{z}_1} \right) \] and \(|t|<1\). \textit{H. Rossi} [in: Proc. Conf. Complex Analysis, Minneapolis 1964, 242--256 (1965; Zbl 0143.30301)] proved that the CR-manifold \((S^3,L_t)\) is not CR-embeddable in \(\mathbb{C}^n\). \textit{J. J. Kohn} [Proc. Symp. Pure Math. 43, 207--217 (1985; Zbl 0571.58027)] showed that a CR-manifold is CR-embeddable in \(\mathbb{C}^n\) if and only if the tangential Cauchy-Riemann operator \(\overline{\partial}_{b,t}\) has closed range. In this paper the authors show that zero is in the essential spectrum of \(\square_b^t\) on the Rossi example. Their analysis involves spherical harmonics to construct finite-dimensional subspaces of \(L^2(\mathbb{S}^3)\) on which \(\square_b^t\) has a tridiagonal matrix representation. Then they find upper bounds for the small eigenvalues of the matrices and show that the bounds converge to 0. The authors combine their result with Kohn's theorem to give another proof of Rossi's result: \((S^3,L_t)\) is not CR-embeddable in \(\mathbb{C}^n\).