Preface to the English Edition ................................. xi
Preface ...................................................... xiii
Chapter I Comparison Functions ................................. 1
1.1 Basic concepts of asymptotic methods ...................... 1
1 Symbols О, 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 solutions ...................................... 7
2 Formal solutions of the Airy equation at infinity ..... 10
3 Derivation of asymptotic expansion for the Airy
function from integral representation at |arg z| ≤
2π/3 - ε .............................................. 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 arguments ....................................... 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
asymptotics ........................................... 37
1.5 Confluent hypergeometric function and its asymptotics .... 40
1 Confluent hypergeometric equation. The functions
Φ(а, с, z) and ψ(а, с, z) and relations between them .. 40
2 Asymptotics of the functions Φ(а, с, z) and
ψ(а, с, z) ............................................ 43
3 The Whittaker functions and their asymptotics ......... 45
Comments ................................................. 47
Chapter II Derivation of Asymptotics ........................... 49
II.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
II.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
II.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
II.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
II.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
III.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
III.2 Normal modes in ocean waveguide ......................... 118
1 Formulation of the problem ........................... 118
2 Asymptotic formulas for normal modes and phase
velocities ........................................... 120
III.3 Exponential spectrum splitting .......................... 123
1 Two symmetric potential wells ........................ 123
2 Symmetric two-center problem ......................... 127
III.4 Quasistationary states .................................. 131
1 Stark effect in hydrogen ............................. 131
III.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
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 Prtifer transformation and iterative modification
of the WKB method ....................................... 160
1 The Prüfer 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 + a1z + a0)w = 0 ........ 164
1 Standard solutions ................................... 164
2 Representation of solutions in terms of the Mellin-
Barnes integrals ..................................... 167
3 Connection between the solutions Iα(j)(z) and
Kα(k,r)(z) ........................................... 174
4 Difference equation for connection factors ........... 180
Comments ...................................................... 183
References .................................................... 185
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