有限面积非紧双曲曲面上常磁场作用下Laplacian算子的特征值
Eigenvalues of Laplacian with constant magnetic field on non-compact hyperbolic surfaces with finite area
摘要：
考虑有限面积非紧双曲曲面$\mM $上的磁Laplacian $-Δ_A=(id+A)^\star (id+A)$. $A$是一个实的形式，磁场$dA$在每个尖点是恒定的。当$A$的谐波分量满足一定的量化条件时，$-Δ_A$的频谱是离散的。在这种情况下，我们证明了$-Δ_{A}$的本征值的计数函数满足经典的Weyl公式，即使当$dA=0. $
英文摘要：
We consider a magnetic Laplacian $-Δ_A=(id+A)^\star (id+A)$ on a noncompact hyperbolic surface $\mM $ with finite area. $A$ is a real one-form and the magnetic field $dA$ is constant in each cusp. When the harmonic component of $A$ satifies some quantified condition, the spectrum of $-Δ_A$ is discrete. In this case we prove that the counting function of the eigenvalues of $-Δ_{A}$ satisfies the classical Weyl formula, even when $dA=0. $
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