Index Abel transform, 5 Abel's map, 281 Abel's theorem, 282 Abelian differentials, 275 Additive process, 91 Almost Mathieu operator, 101 Almost periodic sequence, 100 Anderson model, 88 Antisymmetric part, 15 Asymptotic expansion, 106 Backlund transformation, 259 Baker-Akhiezer function, 148 Banach space, 3 Bessel function, 25 Birkhoff's ergodic theorem, 91 Birman-Schwinger type bound, 70 Borel measure, 297 set, 297 transform, 298 Boundary condition, 16, 18 Branch number, 274 Burchnall-Chaundy polynomial, 234 Canonical divisor, see divisor Canonical form, 40 Canonical homology basis, 274 Casoratian, 189 Cauchy transform, 298 Characteristic polynomial, 12 Closure essential, 301 operator, 47 Commutation method, 185 Dirichlet deformation method, 207 double, 198 single, 187 Commutator, 224 Conservation law, 252 Continued fraction, 10 Crystal exponential interaction, 221 linear interaction (harmonic), 22 C* -algebra, 227 Cyclic vector, 40 Deficiency indices, 48 Degree, 274 Degree of a divisor, see divisor Density of states, 95 Deterministic sequences, 100 Difference expression, 4 commuting, 231 matrix representation, 4 order, 4 second order, symmetric, 5 skew-symmetric, 4 symmetric, 4 triangular parts, 5 Dirac operator, 262 Dirichlet boundary condition, 16 Dirichlet eigenvalues, 134 Discriminant, 116 Dispersion, 25 relation, 25 Divisor, 277 associated with eigenvalues, 154 canonical, 278 degree, 277 index of specialty, 279 nonspecial, 279 positive, 279 principal, 278 relatively prime, 279 Divisor class group, 278 Divisor of poles, 279 Divisor of zeros, 279 Double commutation method, 198 Dynamical system, 87 ergodic, 88 Eigenfunction expansion, 40, 45 Elliptic functions, 162 Elliptic integral, 163 Ergodic, 88 Essential spectrum, 60

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