Direct and inverse scattering on the line:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
1988
|
| Series: | Mathematical surveys and monographs
28 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 209 S. |
| ISBN: | 082181530X |
Staff View
MARC
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| 100 | 1 | |a Beals, Richard |d 1938- |e Verfasser |0 (DE-588)10795320X |4 aut | |
| 245 | 1 | 0 | |a Direct and inverse scattering on the line |c Richard Beals ; Percy Deift ; Carlos Tomei |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 1988 | |
| 300 | |a XIII, 209 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Mathematical surveys and monographs |v 28 | |
| 650 | 4 | |a Dispersion (Mathématiques) | |
| 650 | 7 | |a Dispersion (mathématiques) |2 ram | |
| 650 | 7 | |a Operadores (Analise Funcional) |2 larpcal | |
| 650 | 4 | |a Problème inverse de diffusion | |
| 650 | 7 | |a Scattering inverse |2 Jussieu | |
| 650 | 7 | |a Scattering |2 Jussieu | |
| 650 | 4 | |a Inverse scattering transform | |
| 650 | 4 | |a Scattering (Mathematics) | |
| 650 | 0 | 7 | |a Dimension |0 (DE-588)4149932-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Inverse Transformation |0 (DE-588)4162229-7 |2 gnd |9 rswk-swf |
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Record in the Search Index
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| adam_text | Contents
Preface xiii
Introduction 1
Part I. The Forward Problem
1. Distinguished Solutions 7
A strategy to obtain the matrix fundamental solution ip
2. Fundamental Matrices 9
The set £ and the associated ordering in a sector
The fundamental matrices m and rh: definition and uniqueness
2 bis. The Second Order Case 13
Jost functions and scattering data
Relation to the fundamental matrices of §2
3. Fundamental Tensors 17
The fundamental tensors fk{x, z) = mi A m^ A • • • A mj and
gk{x, z)=mkA mk+i A ¦ ¦ • A mn
Volterra equations, existence and uniqueness in an open sector fi;
smoothness and extendability to fi 0
4. Behavior of Fundamental Tensors as x — oo; the Functions A* 21
Asymptotics of fk as x —» +oo and of gk as x —* — oo
The functions Ajt: existence, analyticity in £7, and extendability
to fi; integral representations
vii
Vlii CONTENTS
5. Behavior of Fundamental Tensors as z — oo 22
Asymptotics of fk and g^ as z — oo
Asymptotics of A^ as z —+ oo
6. Behavior of Fundamental Tensors as z — 0 24
Asymptotics of fk and (? as z —» 0
Asymptotics of Afc as^ »0
7. Construction of Fundamental Matrices 28
Decomposition of fk{9k) into m/s (m/s)
The relation between the poles of rrik and the zeros of Afc_i
The residue of m at a pole
8. Global Properties of Fundamental Matrices; the Transition Matrix 6 33
Relation between m and m: the matrix 6
Asymptotics of m as z — oo and as z — 0
Asymptotics of ^jip{x,0) as x — ±oo
9. Symmetries of Fundamental Matrices 40
The a symmetry of m, m, and Afc
The relation between mL (x,z) and rriL{x,z)
10. The Green s Function for L 42
The Green s kernel expressed in terms of the normalized
eigenfunctions of L and L*
11. Generic Operators and Scattering Data 44
Definition of genericity
Definition of scattering data
Injectivity of the map between generic operators and scattering
data
12. Algebraic Properties of Scattering Data 46
The a symmetry of v
The block structure of v on £
Diagonal entries of v in terms of 6
Integral representation of the entries of v
13. Analytic Properties of Scattering Data 52
CONTENTS IX
Smoothness and decay properties of v on S
Properties of the asymptotic expansion of v at z = 0
14. Scattering Data for m; Determination of v from v 54
Definition of v and its relation to v
Obtaining 8 from v as the solution of a factorization problem
15. Scattering Data for L* 57
Relations between vL and vL~ and between A*, and Al
16. Generic Selfadjoint Operators and Scattering Data 58
Definition of generic selfadjoint operator
A/t and v arising from selfadjoint operators
Forbidden regions
17. The Green s Function Revisited 65
The Green s function in the selfadjoint case
Derivation of the analyticity properties and the jump relations
across cr(L) from the algebraic properties of v
18. Genericity at z = 0 67
Obtaining generic behavior of Afc s at z — 0 by perturbation
19. Genericity at z £ 0 70
Ensuring by perturbation that A^ has only simple zeros in
a sector U and no roots on dfi
Ensuring by perturbation in the selfadjoint case that only
allowable zeros occur on dQ
20. Summary of Properties of Scattering Data 77
Definition of generic scattering data, and some immediate
consequences
Part II. The Inverse Problem
21. Normalized Eigenfunctions for Odd Order Inverse Data 84
Strategy for solving the inverse problem
Definition of the normalized eigenfunctions i
Statement of basic inverse theorem
X CONTENTS
22. The Vanishing Lemma 86
Null vectors and their triviality
Uniqueness of normalized eigenfunctions
23. The Cauchy Operator 87
Basic properties of the Cauchy transform in Hl (£)
24. Equations for the Inverse Problem 91
The extension of fi to (E 0) U Z
The integral equation for fi
Uniqueness of the solution of the integral equation
25. Factorization of v near z = 0 and Property (20.6) 98
26. Reduction to a Fredholm Equation 104
Construction of a parametrix for the integral equation for
n(x, •) using the factorization in §25
The change of variables /x t i#
27. Existence of h* 114
Solution of the integral equation for h* n# 1 for negative x
28. Properties of h* 117
Continuous extension of /i(x, •) to the boundary of a sector
Asymptotics of fx(x, ¦) as z —? 0
Smoothness with respect to x
29. Properties of fi*(x, z) and //(x, z) as z — oo and as x — —oo 123
Asymptotics of n{x, z) and n#{x, z) as z —? oo
Decay of the coefficients in the expansions as x —* — oo
Decay of the a; derivatives of /z and /i#asi » oo
30. Proof of the Basic Inverse Theorem 127
Existence of an operator L for which i solves the generalized
eigenvalue problem
31. The Scalar Factorization Problem for 6 130
Existence of 6: explicit formula in terms of v
32. The Inverse Problem at x = +oo and the bijectivity of the
map L^S{L) = (Z{L),v{L)) 134
CONTENTS xi
Existence of the normalized eigenfunctions fi at x = +00
Relation of /i to /i
Selfadjointness of L
Bijectivity of the scattering map (odd order case)
33. The Even Order Case 137
Modification of the inverse procedure to include negative
spectrum
34. The Second Order Problem 143
Reduction of the inverse procedure to classical Faddeev
Marchenko theory; nongeneric data
Part III. Applications
35. Flows 149
The Cauchy problem for the Gelfand Dikii evolutions
Well posed and ill posed problems; the geometry of phase space
36. Eigenfunction Expansions and Classical Scattering Theory 162
Spectral decomposition of L and the quantum mechanical
wave operators in terms of bounded eigenfunctions
The scattering operator in terms of v(L)
37. Inserting and Removing Poles 170
Algebraic procedure to insert and remove poles of A* while
leaving the rest of the scattering data unchanged
Backlund transformations for the Gelfand Dikii evolutions
38. Matrix Factorization and First Order Systems 181
Scattering data for ordinary differential operators viewed
as scattering data for first order systems
Algebraic nature of the correspondence between operators
and their associated first order systems
Nonuniqueness of the inverse problem for operators with
coefficients in L1 (R)
The matrix factorization problem
Flows on systems induced by the Gelfand Dikii hierarchy:
modified KdV and others
xii CONTENTS
Appendix A. Rational approximation 197
Appendix B. Some formulas 201
References 203
Notation Index 207
Index 209
|
| any_adam_object | 1 |
| author | Beals, Richard 1938- Deift, Percy 1945- Tomei, Carlos |
| author_GND | (DE-588)10795320X (DE-588)140992995 |
| author_facet | Beals, Richard 1938- Deift, Percy 1945- Tomei, Carlos |
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| author_sort | Beals, Richard 1938- |
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| dewey-full | 515.7/24 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.7/24 |
| dewey-search | 515.7/24 |
| dewey-sort | 3515.7 224 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV001791734 |
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| indexdate | 2024-07-09T15:36:05Z |
| institution | BVB |
| isbn | 082181530X |
| language | English |
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| physical | XIII, 209 S. |
| publishDate | 1988 |
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| series | Mathematical surveys and monographs |
| series2 | Mathematical surveys and monographs |
| spelling | Beals, Richard 1938- Verfasser (DE-588)10795320X aut Direct and inverse scattering on the line Richard Beals ; Percy Deift ; Carlos Tomei Providence, RI American Math. Soc. 1988 XIII, 209 S. txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs 28 Dispersion (Mathématiques) Dispersion (mathématiques) ram Operadores (Analise Funcional) larpcal Problème inverse de diffusion Scattering inverse Jussieu Scattering Jussieu Inverse scattering transform Scattering (Mathematics) Dimension (DE-588)4149932-3 gnd rswk-swf Inverse Transformation (DE-588)4162229-7 gnd rswk-swf Streutheorie (DE-588)4183697-2 gnd rswk-swf Streutheorie (DE-588)4183697-2 s Dimension (DE-588)4149932-3 s DE-604 Inverse Transformation (DE-588)4162229-7 s Deift, Percy 1945- Verfasser (DE-588)140992995 aut Tomei, Carlos Verfasser aut Erscheint auch als Online-Ausgabe 978-1-4704-1255-5 Mathematical surveys and monographs 28 (DE-604)BV000018014 28 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001207367&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Beals, Richard 1938- Deift, Percy 1945- Tomei, Carlos Direct and inverse scattering on the line Mathematical surveys and monographs Dispersion (Mathématiques) Dispersion (mathématiques) ram Operadores (Analise Funcional) larpcal Problème inverse de diffusion Scattering inverse Jussieu Scattering Jussieu Inverse scattering transform Scattering (Mathematics) Dimension (DE-588)4149932-3 gnd Inverse Transformation (DE-588)4162229-7 gnd Streutheorie (DE-588)4183697-2 gnd |
| subject_GND | (DE-588)4149932-3 (DE-588)4162229-7 (DE-588)4183697-2 |
| title | Direct and inverse scattering on the line |
| title_auth | Direct and inverse scattering on the line |
| title_exact_search | Direct and inverse scattering on the line |
| title_full | Direct and inverse scattering on the line Richard Beals ; Percy Deift ; Carlos Tomei |
| title_fullStr | Direct and inverse scattering on the line Richard Beals ; Percy Deift ; Carlos Tomei |
| title_full_unstemmed | Direct and inverse scattering on the line Richard Beals ; Percy Deift ; Carlos Tomei |
| title_short | Direct and inverse scattering on the line |
| title_sort | direct and inverse scattering on the line |
| topic | Dispersion (Mathématiques) Dispersion (mathématiques) ram Operadores (Analise Funcional) larpcal Problème inverse de diffusion Scattering inverse Jussieu Scattering Jussieu Inverse scattering transform Scattering (Mathematics) Dimension (DE-588)4149932-3 gnd Inverse Transformation (DE-588)4162229-7 gnd Streutheorie (DE-588)4183697-2 gnd |
| topic_facet | Dispersion (Mathématiques) Dispersion (mathématiques) Operadores (Analise Funcional) Problème inverse de diffusion Scattering inverse Scattering Inverse scattering transform Scattering (Mathematics) Dimension Inverse Transformation Streutheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001207367&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018014 |
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