Preface
Introduction
Chapter 1 Operators on Graphs. Quantum graphs
1.1 Main graph notions and notation
1.2 Difference operators. Discrete Laplace operators
1.3 Metric graphs
1.4 Differential operators on metric graphs. Quantum graphs
1.4.1 Vertex conditions. Finite graphs
1.4.2 Scale invarianee
1.4.3 Quadratic form
1.4.4 Examples of vertex conditions
1.4.5 Infinite graphs
1.4.6 Non-local vertex conditions
1.5 Further remarks and references
Chapter 2 Quantum Graph Operators. Special Topics
2.1 Quantum graphs and scattering matrices
2.1.1 Scattering on vertices
2.1.2 Bond scattering matrix and the secular equation
2.2 First order operators and scattering matrices
2.3 Factorization of quantum graph Hamiltonians
2.4 Index of quantum graph operators
2.5 Dependence on vertex conditions
2.5.1 Variations in the edge lengths
2.6 Magnetic SchrSdinger operator
2.7 Further remarks and references
Chapter 3 Spectra of Quantum Graphs
3.1 Basic spectral properties of compact quantum graphs
3.1.1 Discreteness of the spectrum
3.1.2 Dependence on the vertex conditions
3.1.3 Eigenfunction dependence
3.1.4 An Hadamard-type formula
3.1.5 Generic simplicity of the spectrum
3.1.6 Eigenvalue bracketing
3.1.7 Dependence on the coupling constant at a vertex
3.2 The Shnol' theorem
3.3 Generalized eigenfunctions
3.4 Failure of the unique continuation property. Scars
3.5 The ubiquitous Dirichlet-to-Neumann map
3.5.1 DtN map for a single edge
3.5.2 DtN map for a compact graph with a "boundary
3.5.3 DtN map for a single vertex boundary
3.5.4 DtN map and the secular equation
3.5.5 DtN map and number of negative eigenvalues
3.6 Relations between quantum and discrete graph spectra
3.7 Trace formulas
3.7.1 Secular equation
3.7.2 Weyl's law
3.7.3 Derivation of the trace formula
3.7.4 Expansion in terms of periodic orbits
3.7.5 Other formulations of the trace formula
