发表机构
Nantes Université; American University of Beirut（南特大学; 贝鲁特美国大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带Aharonov-Bohm通量的奇异谐振子的谱渐近，通过半经典方法与合流超几何函数均匀渐近理论，刻画了隧穿、高激发及过渡区域的谱行为，并给出Kummer函数零点的显式渐近公式。
AI 中文摘要
受R. Vanlaere（2026）近期文章的启发，我们研究了奇异谐振子的谱渐近，该谐振子源于单位圆盘上存在恒定磁场和Aharonov-Bohm通量时磁Schrödinger算子的径向约化。相应的特征值问题等价于研究Kummer合流超几何函数关于其第一个参数的零点。将标准的半经典方法与Dunster（1989）、Gabutti-Gatteschi（2001）以及最近由Dunster（2026）改进的Whittaker和合流超几何函数的均匀渐近理论相结合，我们获得了不同谱区域的全面描述。当边界位于经典禁区时，我们推导出由隧穿效应引起的特征值的指数小渐近修正。对于高于隧穿区域的能级，我们在固定磁场下推导出高激发态的移位Bohr-Sommerfeld渐近。最后，对于能级触及边界的过渡区域，我们推导出特征值的三项渐近展开。在附录中，我们将这些谱结果转化为Kummer函数零点的显式渐近公式。
英文摘要
Motivated by a recent article of R. Vanlaere (2026), we investigate the spectral asymptotics of a singular harmonic oscillator arising from the radial reduction of the magnetic Schrödinger operator on the unit disk in the presence of a constant magnetic field and an Aharonov-Bohm flux. The corresponding eigenvalue problem is equivalent to the study of the zeros of the Kummer confluent hypergeometric function with respect to its first parameter. Combining now standard semiclassical methods with the uniform asymptotic theory of Whittaker and confluent hypergeometric functions developed by Dunster (1989), Gabutti-Gatteschi (2001) and improved quite recently by Dunster (2026), we obtain a comprehensive description of the different spectral regimes. When the boundary lies in the classically forbidden region, we derive the exponentially small asymptotic correction to the eigenvalue caused by the tunneling effect. For energy levels above the tunneling regime, we derive a shifted Bohr-Sommerfeld asymptotics for highly excited states at a fixed magnetic field. Finally, for the transition regime where the energy level touches the boundary, we derive a three-term asymptotic expansion of the eigenvalue. In an appendix, we translate these spectral results into explicit asymptotic formulas for the zeros of the Kummer function.
Comments51 pages, 1 figure
