Abstract:

This paper investigates the Dirichlet eigenvalue problem for the 2D Grushin operator $\triangle_X=\partial_{x_{1}}^2+x_{1}^2\partial_{x_{2}}^2$ on a bounded open set $\Omega$ in $\mathbb{R}^2$. The Grushin operator is an important class of Hörmander operators in the non-equiregular case, where the 2D Lebesgue measure of its singular degenerate set $H$ is zero ($|H|=0$), making Métivier's asymptotic formula no longer applicable. By utilizing the explicit expression of the global heat kernel and refined estimates for the error term of the Dirichlet heat kernel, we establish a Weyl asymptotic law with a logarithmic term: $\lambda_k \sim \frac{4\pi}{s_{\Omega}(0)} \frac{k}{\ln k}$. Furthermore, we show that $s_{\Omega}(0)$, the 1D Lebesgue measure of the projection of the singular degenerate set $H$ onto the $x_2$-axis, is a geometric spectral invariant characterizing the asymptotics of the Dirichlet eigenvalues for this operator.

Key words: Grushin operator, Dirichlet eigenvalue, Weyl's Law