Consider the eigenvalue problem

$LaTeX: Hu=\lambda u,$

on a (weighted) graph consisting of LaTeX: n vertices, where LaTeX: H is a symmetric matrix

$LaTeX: H_{mn}=\begin{cases} v_n, \quad & m=n \\ h_{mn}\neq 0,\quad & n\in \mathcal{N}(m), \end{cases}$

where $LaTeX: \mathcal{N}(n)$ denotes a set of neighboring sites connected to site LaTeX: n . We then construct a lattice by copying the above graph, which is referred to as the unit cell lattice, and coupling them through one common vertex. This gives us a periodic lattice, and we expect that the spectrum of the full lattice Hamiltonian consists of only essential spectrum, i.e., no eigenvalues of finite multiplicity. The question is now if we can, by locally perturbing the couplings between the unit cell lattices, construct embedded eigenvalues (eigenvalues of finite multiplicity inside the continuous spectrum) of the full lattice Hamiltonian. Should the answer be yes, how can we continue to perturb the coupling without the embedded eigenvalues disappearing?

The project is intended as part of a proposed research project together with the Quantum Center at University of Tennessee at Chattanooga, USA and relates to my own research on perturbation theory for periodic Schrödinger operators.

Main Supervisor: Wilhelm Treschow, Department of Mathematics, Lund University

Contact: wilhelm.treschow@math.lth.se