Weyl's spectral asymptotic formula for Dirichlet Kohn-Laplacian (Q2735873)

The authors consider the Kohn-Laplace operator \(\Delta_H= \sum_{j=0}^n (X^j)^2+ (Y^j)^2\), \(X^j= \frac{\partial} {\partial x^j}+ 2y^j \frac{\partial}{\partial z}\), \(Y^j= \frac{\partial} {\partial y^j}+ 2x^j \frac{\partial} {\partial z}\), which is well-known as an invariant operator of the Heisenberg group of noncommutative translations in \(\mathbb{R}^{2n+1}\) along \(0z\), in an arbitrary bounded domain \(\Omega\) of an odd-dimensional space \(\mathbb{R}^{2n+1}\) with variables \(x^1,\dots, x^n\), \(y^1,\dots, y^n\), \(z\). An asymptotic formula of Weyl type \(N(\lambda,\Omega)\sim C_n|\Omega|\lambda^{n+1}\) for the quantity \(N(\lambda, \omega)\) of eigenvalues less or equal than \(\lambda\) of the operator \(-\Delta_H u= u|_{\partial\Omega}= 0\) is obtained. An explicit expression for \(C_n\) is established.