We consider the asymptotic behavior of the spectrum of the Landau Hamiltonian plus a short-range continuous potential. The spectrum of the operator forms eigenvalue clusters. We obtain a Szegő limit theorem for the eigenvalues in the clusters as the cluster index and the field strength B tend to infinity with a fixed ratio \({\mathcal E}\). The answer involves the averages of the potential over circles of radius \(\sqrt{{\mathcal E}/2}\) (classical orbits). After rescaling, this becomes a semiclassical problem where the role of Planck’s constant is played by 2/B. We also discuss a related inverse spectral result.

中文翻译：

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Landau Hamiltonian 的扰动：特征值簇的渐近

我们考虑朗道哈密顿量谱的渐近行为加上短程连续势。算子的频谱形成特征值簇。我们获得了簇中特征值的 Szegő 极限定理，因为簇索引和场强B趋于无穷大，具有固定的比率\({\mathcal E}\)。答案涉及半径\(\sqrt{{\mathcal E}/2}\)（经典轨道）圆上的势能平均值。重新标度后，这变成了一个半经典问题，其中普朗克常数的作用由 2/ B扮演。我们还讨论了相关的逆谱结果。