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. |
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
a₁ absolutely continuous algorithm analysis analytic boundary data boundary values bounded Bruckstein Cauchy sequence causal solution Chapter compute consider constant construct continuous function converges corresponding D(iA+,t defined derive discontinuity discrete eigenvalues everywhere F(wn fi(z finite fn(x function f(x Gel'fand-Levitan generalised function given gives Goupillaud medium Green's function grid h₁ impulse response inequality integral equation interval inverse problem inverse scattering problem ISBN Kailath Lebesgue Lebesgue integral Lemma linear Marchenko Marchenko equation matrix metric space non-causal non-zero notation obtain orthogonal plane polynomials quiescent real axis real numbers recurrence relation regular solution Riemann Riemann integral satisfies Schrödinger equation shown in Fig Suppose t₁ tends to zero Theorem Tn+1 Toeplitz variable vector wave write Wronskian z-transform Z₁ θε др მა მთ
Popular passages
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