Bitangential Direct and Inverse Problems for Systems of Integral and Differential Equations (Encyclopedia of Mathematics and its Applications, Series Number 145)

"This largely self-contained treatment surveys, unites and extends some 20 years of research on direct and inverse problems for canonical systems of integral and differential equations and related systems. Five basic inverse problems are studied in which the main part of the given data is either a monodromy matrix; an input scattering matrix; an input impedance matrix; a matrix valued spectral function; or an asymptotic scattering matrix. The corresponding direct problems are also treated. The book incorporates introductions to the theory of matrix valued entire functions, reproducing kernel Hilbert spaces of vector valued entire functions (with special attention to two important spaces introduced by L. de Branges), the theory of J-inner matrix valued functions and their application to bitangential interpolation and extension problems, which can be used independently for courses and seminars in analysis or for self-study. A number of examples are presented to illustrate the theory"-- "This largely self-contained treatment surveys, unites and extends some 20 years of research on direct and inverse problems for canonical systems of integral and differential equations and related systems. Five basic inverse problems are studied in which the main part of the given data is either a monodromy matrix, an input scattering matrix, an input impedance matrix, a matrix-valued spectral function, or an asymptotic scattering matrix. The corresponding direct problems are also treated. The book incorporates introductions to the theory of matrix-valued entire functions, reproducing kernel Hilbert spaces of vector-valued entire functions (with special attention to two important spaces introduced by L. de Branges), the theory of J-inner matrix-valued functions and their application to bitangential interpolation and extension problems, which can be used independently for courses and seminars in analysis or for self-study. A number of examples are presented to illustrate the theory"-- Cover 1 BITANGENTIAL DIRECT AND INVERSE PROBLEMS FOR SYSTEMS OF INTEGRAL AND DIFFERENTIAL EQUATIONS 3 Series 4 Title 5 Copyright 6 Dedication 7 CONTENTS 9 PREFACE 15 1 Introduction 17 1.1 The matrizant as a chain of entire J-inner mvf's 18 1.2 Monodromy matrices of regular systems 20 1.3 Canonical integral systems 21 1.4 Singular, right regular and right strongly regular matrizants 22 1.5 Input scattering matrices 24 1.6 Chains of associated pairs of the first kind 25 1.7 The bitangential direct input scattering problem 27 1.8 Bitangential inverse monodromy and inverse scattering problems 28 1.9 The generalized Schur interpolation problem 29 1.10 Identifying matrizants as resolvent matrices when J=jpq 30 1.11 Input impedance matrices and spectral functions 31 1.12 de Branges spaces 33 1.13 Bitangential direct and inverse input impedance and spectral problems 35 1.14 Krein extension problems and Dirac systems 37 1.15 Direct and inverse problems for Dirac–Krein systems 39 1.16 Supplementary notes 41 2 Canonical systems and related differential equations 43 2.1 Canonical integral systems 43 2.2 Connections with canonical differential systems 46 2.3 The matrizant and its properties 49 2.4 Regular case: Monodromy matrix 52 2.5 Multiplicative integral formulas for matrizants and monodromy matrices; Potapov's theorems 53 2.6 The Feller–Krein string equation 59 2.7 Differential systems with potential 63 2.8 Dirac–Krein systems 65 2.9 The Schrödinger equation 67 2.10 Supplementary notes 69 3 Matrix-valued functions in the Nevanlinna class 72 Basic classes of functions 72 3.1 Preliminaries on the Nevanlinna class Npq 74 The Smirnov maximum principle 75 Inner–outer factorization 77 The Beurling–Lax theorem 78 3.2 Linear fractional transformations and Redheffer transformations 81 Matrix balls 83 3.3 The Riesz–Herglotz–Nevanlinna representation 85 Some proper subclasses of the Carathéodory class 87 3.4 The class ENpq of entire mvf's in Npq 90 3.5 The class pq of mvf's in Npq with pseudocontinuations 93 3.6 Fourier transforms and Paley–Wiener theorems 95 3.7 Entire inner mvf's 97 3.8 J contractive, J-inner and entire J-inner mvf's 100 The Potapov--Ginzburg transform of P(J) into Smm and U(J) into Sinmm 100 Connections with the classes Cmm and Csingmm 106 3.9 Associated pairs of the first kind 107 3.10 Singular and right (and left) regular J-inner mvf's 109 3.11 Linear fractional transformations of Spq into itself 112 3.12 Linear fractional transformations in Cpp and from Spp into Cpp 114 Affine generalizations of Cpp 118 3.13 Associated pairs of the second kind 118 3.14 Supplementary notes 121 4 Interpolation problems, resolvent matrices and de Branges spaces 123 4.1 The Nehari problem 124 4.2 The generalized Schur interpolation problem 128 4.3 Right and left strongly regular J-inner mvf's 134 4.4 The generalized Carathéodory interpolation problem 135 4.5 Detour on scalar determinate interpolation problems 140 4.6 The reproducing kernel Hilbert space H(U) 143 An isometry from H(S) onto H(U) when S=PG(U) 151 4.7 de Branges' inclusion theorems 151 4.8 A description of H(W)Lm2 153 4.9 The classes UAR(J) and UBR(J) of A-regular and B-regular J-inner mvf"s 156 4.10 de Branges matrices and de Branges spaces B 159 Regular de Branges matrices and spaces B 75 Connections between mvf's AEU(Jp) and entire de Branges matrices 77 4.11 A coisometry from H(A) onto B 164 4.12 Formulas for resolvent matrices W E UrsR(jpq) 165 4.13 Formulas for resolvent matrices AEUrsR(Jp) 167 4.14 Supplementary notes 171 5 Chains that are matrizants and chains of associated pairs 174 5.1 Continuous chains of entire J-inner mvf's 174 5.2 Chains that are matrizants 178 5.3 Continuity of chains of associated pairs 183 5.4 Type functions for chains 186 5.5 Supplementary notes 193 6 The bitangential direct input scattering problem 194 6.1 The set Sd scat(dM) of input scattering matrices 194 Input scattering for arbitrary J#Im 196 6.2 Parametrization of Sd scatd(dM) in terms of Redheffer transforms 197 6.3 Regular canonical integral systems 199 6.4 Limit balls for input scattering matrices 200 6.5 The full rank case 205 6.6 Rank formulas 208 6.7 Regular systems (= full rank) case 209 6.8 The limit point case 209 6.9 The diagonal case 211 6.10 A Weyl–Titchmarsh like characterization for input scattering matrices 211 A property of the semiradii of the limit ball 212 The Weyl property of the input scattering matrices 214 6.11 Supplementary notes 216 7 Bitangential direct input impedance and spectral problems 218 7.1 Input impedance matrices 218 7.2 Limit balls for input impedance matrices 222 7.3 Formulas for the ranks of semiradii of the limit ball 225 7.4 Bounded mass functions and full rank end points 227 7.5 The limit point case 229 7.6 The Weyl–Titchmarsh characterization of the input impedance 231 Initial data that generate square summable solutions 72 The limit point case again 234 7.7 Spectral functions for canonical systems 235 Regular canonical integral systems 237 Spectral functions for the spaces H(A) and B 239 7.8 Parametrization of the set (H(A))psf 241 7.9 Parametrization of psfd(dM) for regular canonical integral systems 246 7.10 Pseudospectral and spectral functions for singular systems 248 7.11 Supplementary notes 253 8 Inverse monodromy problems 257 8.1 Some simple illustrative examples 260 8.2 Extremal solutions when J=Im 263 8.3 Solutions for UUAR(J) when J=Im 269 8.4 Connections with the Livsic model of a Volterra node 274 Products and projections of LB J-nodes 277 8.5 Conditions for the uniqueness of normalized Hamiltonians 285 8.6 Solutions with symplectic and/or real matrizants 293 Symplectic and/or real matrizants for J=jp 75 8.7 Entire homogeneous resolvent matrices 297 8.8 Solutions with homogeneous matrizants 301 8.9 Extremal solutions for J=Im 307 8.10 The unicellular case for J=Im 310 8.11 Solutions with symmetric type 311 8.12 The inverse monodromy problem for 2*2 differential systems 315 8.13 Examples of 2*2 Hamiltonians with constant determinant 320 8.14 Supplementary notes 324 9 Bitangential Krein extension problems 326 9.1 Helical extension problems 327 9.2 Bitangential helical extension problems 333 9.3 The Krein accelerant extension problem 336 9.4 Continuous analogs of the Schur extension problem 344 9.5 A bitangential generalization of the Schur extension problem 350 9.6 The Nehari extension problem for mvf's in Wiener class 354 9.7 Continuous Analogs of the Schur extension problem in the Wiener class 363 9.8 Bitangential Schur extension problems in the Wiener class 366 9.9 Supplementary notes 370 10 Bitangential inverse input scattering problems 371 10.1 Existence and uniqueness of solutions 372 10.2 Formulas for the solution of the inverse input scattering problem 373 10.3 Input scattering matrices in the Wiener class 377 10.4 Examples with diagonal mvf's b1t and b2t 378 10.5 Supplementary notes 387 11 Bitangential inverse input impedance and spectral problems 388 11.1 Existence and uniqueness of solutions 389 11.2 Formulas for the solutions 391 11.3 Input impedance matrices in the Wiener class 394 Two basic theorems 72 Reduction to the class of impedances in +pp(Ip) 397 11.4 Examples with diagonal mvf's b3t and b4t=Ip 399 A generalized Krein system 408 11.5 The bitangential inverse spectral problem 409 11.6 An example 412 Connections with strings 421 11.7 Supplementary notes 422 12 Direct and inverse problems for Dirac–Krein systems 425 12.1 Factoring Hamiltonians corresponding to DK-systems 427 12.2 Matrizants of canonical differential systems corresponding to DK-systems 432 12.3 Direct and inverse monodromy problems for DK-systems 439 12.4 Direct and inverse input scattering problems for DK-systems 440 12.5 Direct and inverse input impedance problems for DK-systems 443 12.6 Direct and inverse spectral problems for DK-systems 447 12.7 The Krein algorithms for the inverse input scattering and impedance problems 449 12.8 The left transform 451 12.9 Asymptotic equivalence matrices 454 12.10 Asymptotic scattering matrices (S-matrices) 454 12.11 The inverse asymptotic scattering problem 459 12.12 More on spectral functions of DK-systems 461 12.13 Supplementary notes 463 REFERENCES 466 SYMBOL INDEX 481 INDEX 485 "This largely self-contained treatment surveys, unites and extends some 20 years of research on direct and inverse problems for canonical systems of integral and differential equations and related systems. Five basic inverse problems are studied in which the main part of the given data is either a monodromy matrix; an input scattering matrix; an input impedance matrix; a matrix valued spectral function; or an asymptotic scattering matrix. The corresponding direct problems are also treated. The book incorporates introductions to the theory of matrix valued entire functions, reproducing kernel Hilbert spaces of vector valued entire functions (with special attention to two important spaces introduced by L. de Branges), the theory of J-inner matrix valued functions and their application to bitangential interpolation and extension problems, which can be used independently for courses and seminars in analysis or for self-study. A number of examples are presented to illustrate the theory"-- Résumé de l'éditeur