Periodic quantum graphs and open waveguides

The project deals with spectral properties of periodic quantum graphs and their perturbations. The name ``quantum graph`` is used for a pair (G,H), where G is a metric graph, i.e. the set of points (vertices) and a set of segments (edges) connecting some of the vertices, moreover to each edge a positive length is assigned, H is a second order self-adjoint differential operator on G (Hamiltonian), which is determined by differential operations on the edges and interface conditions at the vertices. They serve as natural models of wave propagation in systems looking like a thin neighbourhood of a graph. Periodic quantum graphs attracts a lot of attention in recent years, largely due to numerous applications - graphene and carbon nano-structures, photonic crystals etc. This project is aimed to make new steps for a better understanding of spectral properties of periodic quantum graphs and graph-like structures, and also to investigate how their spectrum changes in a presence of unbounded defects. The project consists of two parts. The first part is devoted to a problem falling within one of the traditional mathematical-physics categories, asking about construction of differential operators with prescribed spectral properties. Our goal is to construct a periodic quantum graph with prescribed spectrum. It is assumed, that the combinatorial structure of the graph is prescribed, and thus the required structure for the spectrum must be achieved by a suitable choice of coupling conditions at the graph vertices. As we noted quantum graphs are used to model real graph-like structures with small transverse size. In this connection we are going to address similar problem for Laplace operators posed on the domains with graph- like geometry. In the second part of the project we investigate how the spectral properties of periodic quantum graphs change if perturb it by inserting some defect (e.g., by changing the geometry of the underlying metric graph). So far this situation has been considered episodically and mostly for localized defects. In contrast we are going to investigate the case of non-local defects supported by an infinite chain of vertices or/and edges. The aim is to detect and describe an additional spectrum, which eventually may appear in the gaps of the unperturbed problem. To achieved the pursued goals we are going to combine rather standard methods, which are used in similar situations (Floque-Bloch theory, tools from asymptotic analysis, Birman-Schwinger principle, relations between the spectra of quantum graphs and certain discrete graphs) and more abstract methods from extension and spectral theory of symmetric and selfadjoint operators (e.g., boundary triple techniques and abstract Titchmarsh-Weyl m-functions). Combining several approaches we expect to obtain a complete description the spectral problems under investigation.