论文标题
Dirichlet到Neumann操作员在表面上的光谱不变
Spectral invariants of Dirichlet-to-Neumann operators on surfaces
论文作者
论文摘要
我们为与Schrödinger操作员在紧凑的Riemannian表面上的Dirichlet到Neumann地图的特征值获得了完整的渐近扩展。对于零电位，我们为Steklov问题恢复了众所周知的光谱渐进性。对于非零势，我们获得了由我们所谓的参数steklov问题的频谱确定的新几何不变。特别是，对于恒定电势参数steklov问题，边界的每个连接组件上的总测量曲率是光谱不变的。在恒定的曲率假设下，这使我们能够从这些边界运算符的光谱中获取一些内部信息。
We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schrödinger operators on compact Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For nonzero potentials, we obtain new geometric invariants determined by the spectrum of what we call the parametric Steklov problem. In particular, for constant potentials parametric Steklov problem, the total geodesic curvature on each connected component of the boundary is a spectral invariant. Under the constant curvature assumption, this allows us to obtain some interior information from the spectrum of these boundary operators.
