This dissertation provides an introduction to the diffraction and scattering theory for the Aharonov--Bohm Hamiltonian with one or multiple poles on $\mathbf{R}^2$. It shows the propagation of diffractive singularities of the wave equation with the magnetic Hamiltonians with singular vector potential, which is related to the so-called Aharonov--Bohm effect. Based on the diffractive theorem, a Duistermaat--Guillemin type trace formula is given and singularities of the wave trace are computed. Then a Vainberg parametrix method is used to prove a resolvent estimate for the Aharonov--Bohm Hamiltonian with multiple poles. Finally, some applications of these results are given; in particular, scattering resonances generated by diffractions between the multiple singular vector potentials are discussed.

Creator

• Yang, Mengxuan

DOI

• https://doi.org/10.21985/n2-hzr2-n635

Subject

• Mathematics

Language

• en

Alternate Identifier

• http://dissertations.umi.com/northwestern:16059
• etdadmin_upload_901993

Keyword

• Scattering
• Wave Trace
• Resonance
• Diffraction
• Aharonov-Bohm
• Resolvent

Date created

• 2022-01-01

Resource type

• Dissertation

Rights statement

• In Copyright