We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schrödinger operators on Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For non-zero potentials, we obtain new geometric invariants determined by the spectrum. In particular, for constant potentials, which give rise to the parameter-dependent 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.

中文翻译：

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曲面上 Dirichlet-to-Neumann 算子的谱不变量

我们获得了与带边界的黎曼曲面上的薛定谔算子相关联的狄利克雷到诺依曼映射的特征值的完整渐近展开。对于零电位，我们恢复了 Steklov 问题的众所周知的谱渐近线。对于非零电位，我们获得了由频谱确定的新几何不变量。特别是，对于引起参数相关 Steklov 问题的恒定电位，边界的每个连接分量上的总测地曲率是谱不变量。在恒定曲率假设下，这允许我们从这些边界算子的频谱中获得一些内部信息。