We study a magnetic Schr{o}dinger Hamiltonian, with axisymmetric potential in any dimension. The associated magnetic field is unitary and non constant. The problem reduces to a 1D family of singular Sturm-Liouville operators on the half-line indexed by a quantum number. We study the associated band functions. They have finite limits that are the Landau levels. These limits play the role of thresholds in the spectrum of the Hamiltonian. We provide an asymptotic expansion of the band functions at infinity. Each Landau level concerns an infinity of band functions and each energy level is intersected by an infinity of band functions. We show that among the band functions that intersect a fixed energy level, the derivative can be arbitrary small. We apply this result to prove that even if they are localized in energy away from the thresholds, quantum states possess a bulk component. A similar result is also true in classical mechanics.

中文翻译：

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具有轴对称势的幺正磁场中的量子哈密顿量

我们研究磁性 Schr{o}dinger Hamiltonian，在任何维度上都具有轴对称势。相关磁场是单一的且非恒定的。该问题简化为由量子数索引的半线上的奇异 Sturm-Liouville 算子的一维族。我们研究相关的带函数。它们具有有限的限制，即朗道水平。这些限制在哈密顿量的谱中起到阈值的作用。我们提供了带函数在无穷远处的渐近扩展。每个朗道能级涉及无穷个能带函数，每个能级都与无穷个能带函数相交。我们表明，在与固定能级相交的带函数中，导数可以任意小。我们应用这个结果来证明即使它们在能量上远离阈值，量子态具有体成分。类似的结果在经典力学中也成立。