圆锥层上Aharonov-Bohm Laplacian算子的谱跃迁
Spectral transitions for Aharonov-Bohm Laplacians on conical layers
摘要：
我们考虑拉普拉斯算子在一个管状邻域内的一个圆锥表面的旋转，受Aharonov-Bohm磁场支持的对称轴和Dirichlet边界条件的边界上的域。我们表明，存在一个临界总磁通量取决于孔径的圆锥表面，该系统经历了突然的光谱过渡，从无限多的本征值以下的本质谱的空离散谱频谱。对于临界通量，我们建立了一个Hardy型不等式。在具有无限离散谱的情况下，我们得到了清晰的谱渐近性和对余项的精确估计，并研究了特征值对表面孔径和磁场通量的依赖性。
英文摘要：
We consider the Laplace operator in a tubular neighbourhood of a conical surface of revolution, subject to an Aharonov-Bohm magnetic field supported on the axis of symmetry and Dirichlet boundary conditions on the boundary of the domain. We show that there exists a critical total magnetic flux depending on the aperture of the conical surface for which the system undergoes an abrupt spectral transition from infinitely many eigenvalues below the essential spectrum to an empty discrete spectrum. For the critical flux we establish a Hardy-type inequality. In the regime with infinite discrete spectrum we obtain sharp spectral asymptotics with refined estimate of the remainder and investigate the dependence of the eigenvalues on the aperture of the surface and the flux of the magnetic field.
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