论文标题
次生laplacians的光谱渐近学
Spectral asymptotics for sub-Riemannian Laplacians
论文作者
论文摘要
我们研究少年laplacians的光谱特性，即低纤维化算子。主要目的是获得量子性成果结果，这是我们在3D接触案例中所取得的成就。在一般情况下，我们研究了伊曼尼亚次热核的小渐近造物。我们证明它们是由Nilpotentized Heat内核给出的。在等效的情况下，我们推断局部和微局部Weyl定律，将Weyl Measuy置于SR几何形状中。该措施与POPP度量相吻合，但在低维度中与其一般不同。我们证明，光谱浓度发生在长度R-1的Lie括号中，其中R是非遗传学的程度。在奇异的情况下，例如马丁内特或格鲁什（Martinet）或格鲁什（Grushin），情况更多，但在某些情况下，我们获得了热核和Weyl Law的小渐近扩张。最后，我们在一般单数案例中给出了Weyl定律，假设奇异集是分层的。
We study spectral properties of sub-Riemannian Laplacians, which are hypoelliptic operators. The main objective is to obtain quantum ergodicity results, what we have achieved in the 3D contact case. In the general case we study the small-time asymptotics of sub-Riemannian heat kernels. We prove that they are given by the nilpotentized heat kernel. In the equiregular case, we infer the local and microlocal Weyl law, putting in light the Weyl measure in sR geometry. This measure coincides with the Popp measure in low dimension but differs from it in general. We prove that spectral concentration occurs on the shief generated by Lie brackets of length r-1, where r is the degree of nonholonomy. In the singular case, like Martinet or Grushin, the situation is more involved but we obtain small-time asymptotic expansions of the heat kernel and the Weyl law in some cases. Finally, we give the Weyl law in the general singular case, under the assumption that the singular set is stratifiable.
