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Contents

4.2. Classical oscillation theory 78

4.3. Renormalized oscillation theory 81

Chapter 5. Random Jacobi operators 87

5.1. Random Jacobi operators 87

5.2. The Lyapunov exponent and the density of states 91

5.3. Almost periodic Jacobi operators 100

Chapter 6. Trace formulas 105

6.1. Asymptotic expansions 105

6.2. General trace formulas and xi functions 109

Chapter 7. Jacobi operators with periodic coefficients 115

7.1. Floquet theory 115

7.2. Connections with the spectra of finite Jacobi operators 119

7.3. Polynomial identities 123

7.4. Two examples: period one and two 124

7.5. Perturbations of periodic operators 126

Chapter 8. Reflectionless Jacobi operators 133

8.1. Spectral analysis and trace formulas 133

8.2. Isospectral operators 140

8.3. The finite-gap case 142

8.4. Further spectral interpretation 150

Chapter 9. Quasi-periodic Jacobi operators and Riemann theta functions 153

9.1. Riemann surfaces 153

9.2. Solutions in terms of theta functions 155

9.3. The elliptic case, genus one 162

9.4. Some illustrations of the Riemann-Roch theorem 165

Chapter 10. Scattering theory 167

10.1. Transformation operators 167

10.2. The scattering matrix 171

10.3. The GePfand-Levitan-Marchenko equations 175

10.4. Inverse scattering theory 180

Chapter 11. Spectral deformations - Commutation methods 185

11.1. Commuting first order difference expressions 185

11.2. The single commutation method 187

11.3. Iteration of the single commutation method 191

11.4. Application of the single commutation method 194

11.5. A formal second commutation 196