Inverse problems for the Schrodinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect

Weak dispersive estimates for fractional Aharonov-Bohm-Schrödinger groups

We prove local smoothing, local energy decay and weighted Strichartz inequalities for fractional Schrödinger equations with a Aharonov-Bohm magnetic field in 2D. Explicit representations of the flows in terms of spherical expansions of the Hamiltonians are involved in the study. An improvement of the free estimate is proved, when the total flux of the magnetic field through the unit sphere is not an integer.

Gauge Equivalence and Inverse Scattering for Aharonov-Bohm Effect

We consider the Aharonov-Bohm effect for the Schroedinger operator and the related inverse problem in an exterior domain the 2-dim. Euclidean space with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain and the Hamiltonian up to Gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle is convex.

量子隧穿与Aharonov-Bohm效应

Quantum Tunneling and the Aharonov-Bohm effect

研究含径向势阱和双极点Aharonov-Bohm矢量势的哈密顿量，推导基态与第二态能量分裂的半经典渐近，发现通量效应为低阶。

On Aharonov-Bohm operators with two colliding poles

We consider Aharonov-Bohm operators with two poles and prove sharp asymptotics for simple eigenvalues as the poles collapse at an interior point out of nodal lines of the limit eigenfunction.

Dispersive estimates for two-particles Schrödinger and wave equations in the Aharonov-Bohm field

We study the dispersive behaviors of two-particles Schrödinger and wave equations in the Aharonov-Bohm field. In particular, we prove the Strichartz estimates for Schrödinger and wave equations in this setting. The key point is to construct spectral measure of Schrödinger operator with an Aharonov-Bohm type potential in $\R^4$. As applications, we finally prove a scattering theory for the nonlinear defocusing subcritical two-particles Schrödinger equation with Aharonov-Bohm potential.

Multiple Aharonov--Bohm eigenvalues: the case of the first eigenvalue on the disk

It is known that the first eigenvalue for Aharonov--Bohm operators with half-integer circulation in the unit disk is double if the potential's pole is located at the origin. We prove that in fact it is simple as the pole $a\neq 0$.

Rate of convergence for eigenfunctions of Aharonov-Bohm operators with a moving pole

We study the behavior of eigenfunctions for magnetic Aharonov-Bohm operators with half-integer circulation and Dirichlet boundary conditions in a planar domain. We prove a sharp estimate for the rate of convergence of eigenfunctions as the pole moves in the interior of the domain.

On the leading term of the eigenvalue variation for Aharonov-Bohm operators with a moving pole

We study the behavior of eigenvalues for magnetic Aharonov-Bohm operators with half-integer circulation and Dirichlet boundary conditions in a planar domain. We analyse the leading term in the Taylor expansion of the eigenvalue function as the pole moves in the interior of the domain, proving that it is a harmonic homogeneous polynomial and detecting its exact coefficients.

Generalized Strichartz estimates for wave and Dirac equations in Aharonov-Bohm magnetic fields

We prove generalized Strichartz estimates for wave and massless Dirac equations in Aharonov-Bohm magnetic fields. Following a well established strategy to deal with scaling critical perturbations of dispersive PDEs, we make use of Hankel transform and rely on some precise estimates on Bessel functions. As a complementary result, we prove a local smoothing estimate for the Klein-Gordon equation in the same magnetic field.