Microlocal properties of sheaves and complex WKB:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Paris
Soc. Math. de France
2013
|
| Series: | Astérisque
356 |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | IX, 110 S. graph. Darst. |
| ISBN: | 9782856297728 |
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MARC
| LEADER | 00000nam a2200000 cb4500 | ||
|---|---|---|---|
| 001 | BV041724425 | ||
| 003 | DE-604 | ||
| 005 | 20140505 | ||
| 007 | t | ||
| 008 | 140307s2013 d||| |||| 00||| eng d | ||
| 020 | |a 9782856297728 |9 978-2-85629-772-8 | ||
| 035 | |a (OCoLC)872513929 | ||
| 035 | |a (DE-599)HBZHT018191182 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-83 |a DE-19 |a DE-824 |a DE-384 |a DE-355 |a DE-29T | ||
| 084 | |a SI 832 |0 (DE-625)143196: |2 rvk | ||
| 084 | |a 35J10 |2 msc | ||
| 100 | 1 | |a Getmanenko, Alexander |e Verfasser |0 (DE-588)1048627454 |4 aut | |
| 245 | 1 | 0 | |a Microlocal properties of sheaves and complex WKB |c Alexander Getmanenko & Dmitry Tamarkin |
| 264 | 1 | |a Paris |b Soc. Math. de France |c 2013 | |
| 300 | |a IX, 110 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Astérisque |v 356 | |
| 700 | 1 | |a Tamarkin, Dmitry |e Verfasser |0 (DE-588)1048246183 |4 aut | |
| 830 | 0 | |a Astérisque |v 356 |w (DE-604)BV002579439 |9 356 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027171319&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027171319&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-027171319 | ||
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| _version_ | 1804152004217405441 |
|---|---|
| adam_text | CONTENTS
1.
Introduction
........................................................... 1
1.1.
Cauchy problem
..................................................... 1
1.1.1.
Initial data
..................................................... 1
1.2.
Multi-valued solution to a multi-valued Cauchy problem
............ 2
1.3.
Formulation of the result
............................................ 2
1.4.
Introducing sheaves
................................................. 3
1.4.1.
A covering space X
............................................. 3
1.4.2.
Solution sheaf and its singular support
......................... 3
1.4.3.
Initial value problem in sheaf-theoretical terms
................. 4
1.4.4.
Semi-orthogonal decomposition of RgtZs,, [~2]
................. 4
1.4.5.
Étalé
space of
Фп
and solving the initial data problem
......... 5
2.
Conventions and Notations
.......................................... 7
2.1.
Various subsets of
С
................................................ 7
2.2.
Sector Sa
........................................................... 7
2.3.
Potential r{x). Stokes curves. Assumptions
........................ 7
2.3.1.
Stokes curves and further assumptions
......................... 7
2.3.2.
Further assumptions
............................................ 8
2.4.
Universal cover X
................................................... 8
2.5.
Initial point xo
...................................................... 8
2.6.
Action function on X
............................................... 8
2.7.
Subdivision of X into
α
-strips ......................................
8
2.7.1.
Weakest Possible Assumptions on V{x)
........................ 9
2.7.2.
Boundary rays
.................................................. 9
2.7.3.
Strips form a tree
.............................................. 9
2.S. (-a)-Strips
......................................................... 9
2.9.
Interaction of
α
and
-α
-strips ......................................
10
2.1Ü.
Categories
......................................................... 10
2.10.1.
Sub-categories
ď
: -
Є* ......................................
10
2.11.
Sheaves
............................................................ 11
3.
Statement of the problem and Main results
..................... 13
3.1.
Transfer of the equation
-Ф„
+
ľ(x)*„ =
0
to
X x C
............ 13
3.2.
Singular support of the solution sheaf Sol
........................... 13
CONTEXTS
3.3.
Initial conditions
.................................................... 15
3.3.1.
Definition of a solution
......................................... 16
3.3.2.
Equivalent formulation
......................................... 16
3.3.3.
Formulation of the analytic continuation problem
.............. 17
3.4.
Semi-orthogonal decomposition of Rg(Lsa
[—2] ...................... 17
3.4.1.
Factorization of the initial condition
............................ 17
3.4.2.
Truncation
..................................................... 18
3.5.
Étalé
space of
Фо
................................................... 19
3.5.1.
Choice of a covering space
Σ
................................... 19
3.5.2.
Solving the initial value problem
............................... 19
3.5.3.
Solving the analytic continuation problem
...................... 19
3.6.
Structure of the object
Φ
............................................ 19
3.6.1.
Decomposition of
чсѕЈ^зл
Є
D(C)
............................. 20
3.6.2.
Semi-orthogonal decomposition for ZXo
χ-,
ZXüX^ ,ZXl)xr±fV .....
21
3.6.3.
Фс
.............................................................. 22
3.7.
Notation: convolution functor D(X
x C) x D(C)
->
Ό
(X
x
С)
..... 22
3.8.
Construction of
Фк
................................................. 22
3.8.1.
Subdivision into
α
-strips
....................................... 22
3.8.2.
Words
.......................................................... 23
3.8.3.
Sheaves Sf,Sw oi C
............................................ 23
3.8.4.
Definition of
ф£
................................................ 24
3.8.5.
Construction of the identification
Г
J1/2
........................ 24
3.8.6.
Description of the map
¿$к
:
ZX(,xa [—
2] —»
ФА
................ 26
3.9.
Alternative construction of
Фк
via -a-strips
........................ 27
3.9.1.
Notation for -a-strips
........................................... 27
3.9.2.
Sheaves
Ф£
.................................................... 28
3.9.3.
Gluing maps
.................................................... 28
3.10.
The map
/ФФ
...................................................... 29
3.10.1.
Decomposing
ѓцр
into components
............................ 30
3.10.2.
Identification W~a
->·
W
.................................... 30
3.10.3.
Formulation of the result
...................................... 31
3.11.
Description of
Φ1
................................................. 31
3.12.
Description of
Фг
................................................ 31
3.13.
Constructing the map
(30) ......................................... 32
3.13.1.
The map
а^Гу
................................................. 32
3.13.2.
Map qKr^a
:
ФА
-+
ΦΓ~ή .....................................
33
3.13.3.
Мар№га :ФА
->
Фг-
........................................ 33
3.13.4.
Restriction of Q to a parallelogram
............................ 33
3.13.5.
The map </;>tv revisited
........................................ 34
3.13.6.
The map QKr lt
............. ................................. 34
3.13.7.
The map qKFft
................................................. 34
3.14.
Σ
and
φ
are Hausdorff
............................................ 34
ASTÉRISQUE
356
CONTENTS
3.14.1.
Generalities
on étalé
spaces
................................... 35
3.14.2.
Reduction to rigidity on
Π Π Ρ ................................
35
3.14.3.
Filtration on
ФојппРхС .......................................
36
3.14.4.
Sheaf
F¡t
D
F„ ............................................... 36
3.14.5.
Further nitrations on
íf £n
,
F*n
.............................. 36
3.14.6.
Finishing the proof
............................................ 36
3.15.
Surjectivity of the projection
p,.ţ
:
φ
—»
Χ
.......................... 37
3.15.1.
Constructing
%ί
................................................ 37
3.15.2.
Verifying
1) ................................................... 38
3.15.3.
Verifying
2) ................................................... 38
3.15.4.
Reformulation of
3) ........................................... 38
3.15.5.
Subset
W
С
Sa
............................................... 39
3.15.6.
Finishing
the proof
............................................ 40
3.16.
Infinite continuation in the direction of
К
......................... 41
3.16.1.
Parallelogram
U
.............................................. 41
3.16.2.
Small sets
..................................................... 41
3.16.3.
Theorem
...................................................... 42
3.16.4.
Reformulation in terms of sheaves
............................. 42
3.16.5.
Writing ftf in terms of its components
......................... 43
3.16.6.
Restriction to a sub-parallelogram V
.......................... 44
3.16.7.
Proof of a weaker version of the Theorem
..................... 44
3.16.8.
Proof of the theorem for
U
.................................... 46
4.
Orthogonality criterion
—
a simplified version
..................... 47
4.1.
Formulation
of the Theorem
........................................ 47
4.2.
Fourier-Sato Kernel
................................................. 48
4.2.1.
Properties of the modified
Föurier-Sato
transform
.............. 48
4.2.2.
Singular support estimation
.................................... 49
4.2.3.................................................................. 51
4.2.4.
Representation of
G
............................................ 51
5.
Orthogonality criterion for a generalized strip
.................... 53
5.1.
Conventions and notations
.......................................... 53
5.1.1.
Convolution
.................................................... 53
5.1.2.
The category
ďs
................................................ 54
5.1.3.
Rays L· and L·
................................................. 54
5.1.4.
Projectors P-t
.................................................. 54
5.2.
Formulation of the criterion
......................................... 54
5.3.
Fourier-Sato decomposition
......................................... 54
5.4.
Transfer of the conditions RP^iF
— 0
to IFF
........................ 55
5.5.
Fourier-Sato decomposition for sheaves satisfying
(103) ............. 56
5.5.1.
Computing ZLz
* ->ƒ._ ............................................ 57
5.5.2.
Further reformulation
.......................................... 59
SOCIÉTÉ
11АЇНЕМЛТІ^).К
DE FiìAXCl·! 2«Ki
CONTENTS
5.5.3.
Rewriting the map
(123) ....................................... 59
5.5.4.
Transferring Claim
4
to
Ф_р
.................................... 60
56.
Rewriting the condition of orthogonality to
β
..................... 61
5.7.
Subdivision into three cases
......................................... 62
5.7.1.
Subdivision of
M x S x
Я .......................................
63
5.7.2.
Subdivision of
ФР ...............................................
63
5.7.3.
Subdivision of
M
................................................. 63
5.7.4.
Subdivision of Claim
5 ......................................... 64
5.7.5.
Further reduction
............................................... 64
5.7.6.................................................................. 65
5.8.
The case U^
==
I<>
χ
(—сю, со)
x R
................................. 65
5-9.
Proof of Claim
9
for U<>
== !<>
x
(0,
со)
x M
......................... 66
5.9.1.
Representation of
С
............................................ 66
5.10.
Proof of Claim
10 .................................................. 67
5.10.1.
Functors
r
ι
and r2 and their properties
....................... 67
5.10.2.
Construction of the object
Η
and proof of the Claim
10 1) .... 68
5.10.3.
Reduction of part
2)
of the Claim
10 .......................... 69
5.10.4.
Subdivision into three cases
................................... 69
5.10.5.
Proof of the 1-st and the 2-nd vanishing
....................... 69
5.11.
Finishing proof of Claim
9 ......................................... 72
6.
Proof of Theorem
3.4 ................................................ 75
6.1.
Proof of
Φκ
Є
if
.................................................... 75
6.2.
Proof of orthogonality
............................................... 76
6.2.1.
Regular sequences
.............................................. 76
6.2.2.
Admissible rays
................................................. 76
6.2.3.
Subset PXiW
.................................................... 77
6.2.4.
Subsheaves
Af f
,„................................................. 77
6.2.5.
Subsheaves $J X
с ф£
......................................... 77
6.2.6.
Sheaves
Фр
match on the intersections
....................... 77
6.2.7.
Definition of a filtration on
Фк
................................. 78
6.2.8.
Computing Fl&K
......................................,......... 79
6.2.9.
The map iy factorizes through FX^K
.......................... 79
6.2.10.
Computing successive quotients of the filtration
............... 80
6.2.11.
Description of gn
............................................. 83
6.2.12.
Reduction of the orthogonality property
....................... 85
6.2.13.
Conventions
................................................... 85
6.2.14.
Orthogonality of Aw
.......................................... 85
6.2.15.
Orthogonality of Bw
.......................................... 87
6.2.16.
Orthogonality of Cone(Z{xo}XjRr[-2]
—►
ΡλΦκ) ................
89
7.
Identification of
Φ*
and
Фх
......................................... 91
7.1.
Endomorphisms of AK+
*
S+
θ
Л*^
♦
5L |(Pnn)xC
.................. 91
ASTÉRISQUE
356
CONTENTS
7.1.1.
Filtration on
Ноту
xC(8Wl, Sw2)
.............................. 93
7.1.2.
Lemma on composition
......................................... 94
7.1-3.
Lemma on extension
........................................... 95
7.1.4.
Decomposition Lemma
......................................... 96
7.2.
Restriction
Ф^јп
.................................................... 96
7.2.1.
Notation
........................................................ 96
7.2.2.
Prescription of
0¿|(nnPj.)xC
.................................... 97
7.2.3.
Extension of
0¿
to
ПхС
....................................... 97
7.2.4.
Estimate
....................................................... 98
7.2.5.
Construction of
фп
............................................. 98
7.2.6.
The map
фц
is an isomorphism
................................. 98
7.3.
The maps
(/»Пи
Фп2
f°r a Pair neighboring strips Hj and LI2
........ 99
7.3.1.
Identifications
.................................................. 102
7.4.
The isomorphism
/ФФ
:
Фк
-+
Фк
.................................. 103
7.4.1.
Estimate
....................................................... 106
7.5.
Inductive construction of the maps
Un
.............................. 107
7.5.1.
Rewriting the gluing condition
................................. 107
7.5.2.
Constructing
Č7#
............................................... 108
7.5.3.
Estimate
....................................................... 108
7.5.4.
Proof of Proposition
(3.10.1) ................................... 109
Acknowledgements
.................................................... 110
Bibliography
..............................................................
Ill
société mathématique
db
france
2013
1
Г
Kashiwara-Schapira
style sheaf theory is used to justify ana¬
lytic continuability of solutions of the Laplace transformed
Schrödinger
equation with a small parameter. This partially
proves the description of the Stokes phenomenon for WKB
asymptotics predicted by Voros in
1983,
|
| any_adam_object | 1 |
| author | Getmanenko, Alexander Tamarkin, Dmitry |
| author_GND | (DE-588)1048627454 (DE-588)1048246183 |
| author_facet | Getmanenko, Alexander Tamarkin, Dmitry |
| author_role | aut aut |
| author_sort | Getmanenko, Alexander |
| author_variant | a g ag d t dt |
| building | Verbundindex |
| bvnumber | BV041724425 |
| classification_rvk | SI 832 |
| ctrlnum | (OCoLC)872513929 (DE-599)HBZHT018191182 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV041724425 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:03:49Z |
| institution | BVB |
| isbn | 9782856297728 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027171319 |
| oclc_num | 872513929 |
| open_access_boolean | |
| owner | DE-83 DE-19 DE-BY-UBM DE-824 DE-384 DE-355 DE-BY-UBR DE-29T |
| owner_facet | DE-83 DE-19 DE-BY-UBM DE-824 DE-384 DE-355 DE-BY-UBR DE-29T |
| physical | IX, 110 S. graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Soc. Math. de France |
| record_format | marc |
| series | Astérisque |
| series2 | Astérisque |
| spelling | Getmanenko, Alexander Verfasser (DE-588)1048627454 aut Microlocal properties of sheaves and complex WKB Alexander Getmanenko & Dmitry Tamarkin Paris Soc. Math. de France 2013 IX, 110 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Astérisque 356 Tamarkin, Dmitry Verfasser (DE-588)1048246183 aut Astérisque 356 (DE-604)BV002579439 356 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027171319&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027171319&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Getmanenko, Alexander Tamarkin, Dmitry Microlocal properties of sheaves and complex WKB Astérisque |
| title | Microlocal properties of sheaves and complex WKB |
| title_auth | Microlocal properties of sheaves and complex WKB |
| title_exact_search | Microlocal properties of sheaves and complex WKB |
| title_full | Microlocal properties of sheaves and complex WKB Alexander Getmanenko & Dmitry Tamarkin |
| title_fullStr | Microlocal properties of sheaves and complex WKB Alexander Getmanenko & Dmitry Tamarkin |
| title_full_unstemmed | Microlocal properties of sheaves and complex WKB Alexander Getmanenko & Dmitry Tamarkin |
| title_short | Microlocal properties of sheaves and complex WKB |
| title_sort | microlocal properties of sheaves and complex wkb |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027171319&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027171319&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002579439 |
| work_keys_str_mv | AT getmanenkoalexander microlocalpropertiesofsheavesandcomplexwkb AT tamarkindmitry microlocalpropertiesofsheavesandcomplexwkb |