## Inverse Problems in Scattering: An IntroductionInverse Problems in Scattering exposes some of the mathematics which has been developed in attempts to solve the one-dimensional inverse scattering problem. Layered media are treated in Chapters 1--6 and quantum mechanical models in Chapters 7--10. Thus, Chapters 2 and 6 show the connections between matrix theory, Schur's lemma in complex analysis, the Levinson--Durbin algorithm, filter theory, moment problems and orthogonal polynomials. The chapters devoted to the simplest inverse scattering problems in quantum mechanics show how the Gel'fand--Levitan and Marchenko equations arose. The introduction to this problem is an excursion through the inverse problem related to a finite difference version of Schrödinger's equation. One of the basic problems in inverse quantum scattering is to determine what conditions must be imposed on the scattering data to ensure that they correspond to a regular potential, which involves Lebesque integrable functions, which are introduced in Chapter 9. |

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### Contents

II | 1 |

IV | 10 |

V | 15 |

VI | 16 |

VII | 18 |

VIII | 23 |

IX | 27 |

X | 33 |

LVII | 181 |

LVIII | 182 |

LIX | 189 |

LXI | 190 |

LXII | 191 |

LXIII | 195 |

LXIV | 201 |

LXV | 211 |

XI | 39 |

XII | 40 |

XIII | 46 |

XIV | 48 |

XV | 55 |

XVII | 59 |

XVIII | 60 |

XIX | 63 |

XX | 67 |

XXII | 71 |

XXIII | 74 |

XXIV | 80 |

XXV | 86 |

XXVI | 91 |

XXVIII | 92 |

XXIX | 97 |

XXX | 101 |

XXXI | 103 |

XXXII | 106 |

XXXIII | 111 |

XXXIV | 115 |

XXXV | 119 |

XXXVI | 123 |

XXXIX | 128 |

XL | 131 |

XLI | 133 |

XLII | 135 |

XLIII | 138 |

XLIV | 147 |

XLVI | 149 |

XLVII | 150 |

XLVIII | 152 |

XLIX | 157 |

LII | 161 |

LIII | 163 |

LIV | 169 |

LV | 172 |

LVI | 178 |

LXVI | 220 |

LXVII | 232 |

LXVIII | 233 |

LXIX | 237 |

LXX | 241 |

LXXI | 242 |

LXXII | 246 |

LXXIII | 251 |

LXXVI | 257 |

LXXVII | 261 |

LXXVIII | 268 |

LXXIX | 270 |

LXXX | 273 |

LXXXI | 276 |

LXXXII | 280 |

LXXXIII | 283 |

LXXXIV | 289 |

LXXXVI | 290 |

LXXXVII | 292 |

LXXXVIII | 295 |

LXXXIX | 300 |

XC | 303 |

XCI | 306 |

XCII | 310 |

XCIII | 312 |

XCIV | 316 |

XCV | 319 |

XCVI | 321 |

XCVII | 326 |

XCVIII | 328 |

XCIX | 331 |

C | 334 |

CI | 337 |

CII | 345 |

355 | |

359 | |

### Common terms and phrases

algorithm analysis analytic apply argument assume axis boundary bounded called causal Chapter circle combination complex compute consider constant construct continuous converges corresponding defined definition depend derive described difference differential discontinuity discrete eigenvalues equation everywhere Exercises expresses finite follows function given gives governing grid hand holds implies initial integral equation interval introduce inverse problem inverse scattering Kailath known limit linear matrix means measure medium methods multiple namely obtain operator orthogonal particular plane polynomials positive Proof prove quantities recurrence reflection regular relation response result Riemann integral satisfies sequence shown in Fig side simple solution solve space step Suppose Theorem theory values variable vector wave write written zero

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