Contents
Preface xi
Introduction xiii
Chapter 1. Operators on Graphs. Quantum graphs 1
1.1. Main graph notions and notation 2
1.2. Difference operators. Discrete Laplace operators 5
1.3. Metric graphs 7
1.4. Differential operators on metric graphs. Quantum graphs 12
1.4.1. Vertex conditions. Finite graphs. 15
1.4.2. Scale invariance 22
1.4.3. Quadratic form 22
1.4.4. Examples of vertex conditions 24
1.4.5. Infinite graphs 27
1.4.6. Non-local vertex conditions 32
1.5. Further remarks and references 33
Chapter 2. Quantum Graph Operators. Special Topics 37
2.1. Quantum graphs and scattering matrices 37
2.1.1. Scattering on vertices 37
2.1.2. Bond scattering matrix and the secular equation 41
2.2. First order operators and scattering matrices 44
2.3. Factorization of quantum graph Hamiltonians 51
2.4. Index of quantum graph operators 52
2.5. Dependence on vertex conditions 54
2.5.1. Variations in the edge lengths 58
2.6. Magnetic Schr¨ odinger operator 59
2.7. Further remarks and references 62
Chapter 3. Spectra of Quantum Graphs 65
3.1. Basic spectral properties of compact quantum graphs 66
3.1.1. Discreteness of the spectrum 66
3.1.2. Dependence on the vertex conditions 67
3.1.3. Eigenfunction dependence 68
3.1.4. An Hadamard-type formula 68
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