发表机构
Departamento de Física, Universidad de Santiago de Chile（智利圣地亚哥大学物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
通过莫尔斯哈密顿量连接平面开普勒-库仑问题与双曲平面上的朗道问题，揭示两者在谱、散射和对称性上的对应关系。
AI 中文摘要
我们展示了两个典范系统——平面开普勒-库仑问题和双曲平面$H^2$上的朗道问题——通过一个共同的一维中介者连接：莫尔斯哈密顿量。在开普勒侧，刘维尔变换和耦合常数变形将径向动力学转化为莫尔斯问题，其中开普勒极角成为莫尔斯演化参数。在朗道侧，双曲磁动力学的水平圆约化给出相同的莫尔斯哈密顿量，并带有量子半密度修正。因此，径向开普勒问题和双曲朗道问题的固定水平圆动量扇区被映射到同一个莫尔斯谱问题，关联它们的束缚谱、连续阈值、共振和散射数据。我们进一步证明朗道时间演化具有开普勒圆锥形式，并约化为莫尔斯系统的束缚、阈值和散射轨迹。由此得到的词典将开普勒圆锥与磁圆、水平圆和超圆连接起来，并将朗道问题的磁$SL(2,\mathbb R)$对称性转化为莫尔斯系统的谱生成代数结构。
英文摘要
A common Morse Hamiltonian bridges the planar Kepler--Coulomb and hyperbolic Landau systems, linking flat electric dynamics to curved magnetic dynamics. A radial Liouville transformation with coupling-constant metamorphosis maps the separated Kepler problem to the Morse system, while fixed-horocyclic-momentum reduction maps the hyperbolic Landau problem to the same system. In a common normalization, the Coulomb coupling is the signed product of the magnetic field and conserved horocyclic momentum, while the quantum Landau reduction has the universal half-density shift $1/4$. Beyond this shared reduction, we construct a direct classical orbit map on the regular nonradial sectors for all signs of the Kepler energy. It maps the Landau height to a scaled inverse Kepler radius, Kepler ellipses, parabolas and hyperbolas to magnetic circles, horocycles and hypercycles, and Hamilton's hodograph affinely to the Landau carrier circle. The reduced Kepler symplectic form and the Binet-selected complex structure determine a Poincaré Kähler metric. Its nondegenerate Binet orbits have constant geodesic curvature of magnitude equal to the inverse eccentricity, while zero coupling gives geodesics. The orbit map identifies this metric isometrically with the Poincaré configuration-space metric of the hyperbolic Landau problem. Composing the quantum Liouville maps eliminates the Morse wavefunction and yields an exact weighted intertwining identity between the radial Kepler and horocyclic Landau operator pencils. At distinguished half-integer magnetic fields, the angular-momentum grading of a fixed Kepler shell is reorganized into the finite reduced Landau tower and its threshold endpoint. These exact correspondences hold in the stated reduced sectors despite the unitary inequivalence of the complete parent systems.
Comments25 pages, 2 figures, 1 table. Substantially revised and extended version, with new direct classical and reduced quantum Kepler--Landau correspondences. Accepted for publication in JHEP
