Spectral analysis of magnetic tunneling on the circle

Specialeforsvar: Libei Wang

Titel: Spectral analysis of magnetic tunneling on the circle

Abstract: This thesis is aimed to analyze the double-well electromagnetic tunneling effect on the circle. More precisely, we study the Laplacian
                 (hDs + a(s))2 + V (s),
where V (s) induces two electric wells with a(s) induces a magnetic field, and provide an estimate of the gap between the first two eigenvalues. We shall demonstrate that this gap can be controlled pointwise by an approximation to the eigenfunctions. This is achieved by analyzing a 2×2 Hermitian matrix constructed step by step. The thesis is organized into three parts. Chapter 2 forms the first part and discusses general spectral theory. Note that the basic knowledge of bounded operator and distributions is assumed. The second and third parts, which are primarily based on [1], form the core discussion of the thesis. The second part consists of Chapter 3. In this part, the problem is reduced into a single-well electric case. Some of the conclusions in this chapter are quite classical, such as the WKB approximation and Agmon estimate. The third part includes Chapter 4 to Chapter 6. The interaction matrix is constructed step by step. And finally, in Chapter 6, we compute the interaction term using the WKB approximation and obtain the final result.

Vejleder: Søren Fournais
Censor:    Horia Cornean, Aalborg Universitet