Asymptotic solutions of the one-dimensional Schrödinger equation:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
1996
|
| Series: | Translations of mathematical monographs
151 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus dem Russ. übers. |
| Physical Description: | XVI, 190 S. |
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MARC
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| 100 | 1 | |a Slavjanov, Sergej Ju. |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Asimptotika rešenij odnomernogo uravnenija Šredingera |
| 245 | 1 | 0 | |a Asymptotic solutions of the one-dimensional Schrödinger equation |c S. Yu. Slavyanov |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 1996 | |
| 300 | |a XVI, 190 S. | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface to the English Edition xi
Preface xiii
Chapter I. Comparison Functions 1
1.1. Basic concepts of asymptotic methods 1
1. Symbols O, o, and ~ 1
2. Asymptotic series and their properties. Asymptotic ansatz 3
3. Properties of asymptotic expansions 5
1.2. The Airy functions and their asymptotics 7
1. The Airy equation. Standard solutions. Relations between so¬
lutions 7
2. Formal solutions of the Airy equation at infinity 10
3. Derivation of asymptotic expansion for the Airy function from
integral representation at | argz 2tt/3 — e 11
4. The Stokes phenomenon for the Airy equation 14
5. Justifying formal asymptotic solutions of the Airy equation by
using integral equations 17
1.3. Parabolic cylinder functions and their asymptotics 21
1. The Weber equation. Standard solutions and relations between
solutions 21
2. Asymptotics of the parabolic cylinder functions for large argu¬
ments 25
3. Modified parabolic cylinder functions and their asymptotics 30
1.4. The Bessel functions and their asymptotics 31
1. The Bessel equation. Standard solutions and relations between
solutions 31
2. Asymptotics of cylinder functions for large arguments 32
3. Asymptotic solutions of the equation
y (z) + [1/z + (1 - m2)/(4z2)]y(z) = 0 35
4. The equation w {z) —azmw(z) = 0: Solutions and their asymp¬
totics 37
1.5. Confluent hypergeometric function and its asymptotics 40
1. Confluent hypergeometric equation. The functions $ (a, c, z) and
P(a, c, z) and relations between them 40
2. Asymptotics of the functions $(a, c, z) and $(a, c, z) 43
3. The Whittaker functions and their asymptotics 45
Comments 47
vii
viii CONTENTS
Chapter II. Derivation of Asymptotics 49
II. 1. General theory 49
1. Reduction of second order equations to the canonical form 49
2. Formal theory for equations without transition points 50
3. The Liouville-Green transformation 56
11.2. Asymptotic solutions on the complex plane 59
1. Turning points, Stokes lines, canonical domains 59
2. Primary fundamental system of solutions in a canonical domain 62
3. Relation matrices 65
11.3. Method of comparison equations for equations with one transition
point 69
1. Formal procedure of the method of comparison equations 69
2. Method of comparison equations for equations with one simple
turning point 71
3. Asymptotics far from a turning point 73
4. Local asymptotic expansions near a turning point 76
5. Turning point of multiplicity m 77
6. Equations with one simple pole 80
11.4. Method of comparison equations for equations with two transition
points 81
1. Formal analysis of equations with two simple turning points 81
2. Regularization of phase integrals 85
3. Formal analysis of equations with one simple turning point and
one simple pole 90
11.5. Method of comparison equations for equations with close transition
points 92
1. Scaling transformations 92
2. Two close turning points 94
3. Close pole and turning point 97
Comments 99
Chapter III. Physical Problems 101
111.1. The WKB method for bound states in quantum mechanics 101
1. Anharmonic oscillator. Highly excited states 101
2. Anharmonic oscillator. Small perturbations 107
3. Quantization for potentials Coulomb-type singularity 111
111.2. Normal modes in ocean waveguide 118
1. Formulation of the problem 118
2. Asymptotic formulas for normal modes and phase velocities 120
111.3. Exponential spectrum splitting 123
1. Two symmetric potential wells 123
2. Symmetric two-center problem 127
111.4. Quasistationary states 131
1. Stark effect in hydrogen 131
2. Ionization in electric field 134
111.5. One-dimensional scattering problem 136
1. Semiclassical asymptotics of the Jost functions and scattering
phases for potentials with Coulomb singularity 136
2. Wave transition through a potential barrier 139
CONTENTS ix
3. Overbarrier reflection 143
III.6. Band spectrum 147
1. Equations with periodic potential 147
2. Asymptotic formulas for bandwidths 149
3. The Mathieu equation 152
Comments 152
Chapter IV. Supplements 155
IV. 1. Numerical realization of asymptotic methods 155
1. Approximation of potential and evaluation of phase integrals 155
2. Approximation of the derivatives of a potential at a point 158
IV.2. The Priifer transformation and iterative modification of the WKB
method 160
1. The Priifer transformation 160
2. Iterative procedure for solving equations for amplitude and
phase and its connection with asymptotic expansions 162
IV.3. Solutions of z2w - (z3 + a2z2 + axz + ao)w = 0 164
1. Standard solutions 164
2. Representation of solutions in terms of the Mellin-Barnes inte¬
grals 167
3. Connection between the solutions il (2) and Ka (2) 174
4. Difference equation for connection factors 180
Comments 183
References 185
|
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| indexdate | 2024-07-09T18:03:10Z |
| institution | BVB |
| language | English |
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| physical | XVI, 190 S. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spelling | Slavjanov, Sergej Ju. Verfasser aut Asimptotika rešenij odnomernogo uravnenija Šredingera Asymptotic solutions of the one-dimensional Schrödinger equation S. Yu. Slavyanov Providence, RI American Math. Soc. 1996 XVI, 190 S. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 151 Aus dem Russ. übers. Schrödinger-Gleichung - Asymptotische Methode Asymptotisches Lösungsverhalten (DE-588)4134367-0 gnd rswk-swf Schrödinger-Gleichung (DE-588)4053332-3 gnd rswk-swf Schrödinger-Gleichung (DE-588)4053332-3 s Asymptotisches Lösungsverhalten (DE-588)4134367-0 s DE-604 Translations of mathematical monographs 151 (DE-604)BV000002394 151 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007400308&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Slavjanov, Sergej Ju Asymptotic solutions of the one-dimensional Schrödinger equation Translations of mathematical monographs Schrödinger-Gleichung - Asymptotische Methode Asymptotisches Lösungsverhalten (DE-588)4134367-0 gnd Schrödinger-Gleichung (DE-588)4053332-3 gnd |
| subject_GND | (DE-588)4134367-0 (DE-588)4053332-3 |
| title | Asymptotic solutions of the one-dimensional Schrödinger equation |
| title_alt | Asimptotika rešenij odnomernogo uravnenija Šredingera |
| title_auth | Asymptotic solutions of the one-dimensional Schrödinger equation |
| title_exact_search | Asymptotic solutions of the one-dimensional Schrödinger equation |
| title_full | Asymptotic solutions of the one-dimensional Schrödinger equation S. Yu. Slavyanov |
| title_fullStr | Asymptotic solutions of the one-dimensional Schrödinger equation S. Yu. Slavyanov |
| title_full_unstemmed | Asymptotic solutions of the one-dimensional Schrödinger equation S. Yu. Slavyanov |
| title_short | Asymptotic solutions of the one-dimensional Schrödinger equation |
| title_sort | asymptotic solutions of the one dimensional schrodinger equation |
| topic | Schrödinger-Gleichung - Asymptotische Methode Asymptotisches Lösungsverhalten (DE-588)4134367-0 gnd Schrödinger-Gleichung (DE-588)4053332-3 gnd |
| topic_facet | Schrödinger-Gleichung - Asymptotische Methode Asymptotisches Lösungsverhalten Schrödinger-Gleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007400308&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000002394 |
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