| Preface | p. xi |
| Historical survey | p. 1 |
| Development from 1817 to 1926 | p. 1 |
| Carlini's pioneering work | p. 1 |
| The work by Liouville and Green | p. 3 |
| Jacobi's contribution towards making Carlini's work known | p. 4 |
| Scheibner's alternative to Carlini's treatment of planetary motion | p. 4 |
| Publications 1895-1912 | p. 5 |
| First traces of a connection formula | p. 5 |
| Publications 1915-1921 | p. 6 |
| Both connection formulas are derived in explicit form | p. 7 |
| The method is rediscovered in quantum mechanics | p. 7 |
| Development after 1926 | p. 8 |
| Description of the phase-integral method | p. 12 |
| Form of the wave function and the q-equation | p. 12 |
| Phase-integral approximation generated from an unspecified base function | p. 13 |
| F-matrix method | p. 21 |
| Exact solution expressed in terms of the F-matrix | p. 22 |
| General relations satisfied by the F-matrix | p. 25 |
| F-matrix corresponding to the encircling of a simple zero of Q[superscript 2](z) | p. 26 |
| Basic estimates | p. 26 |
| Stokes and anti-Stokes lines | p. 28 |
| Symbols facilitating the tracing of a wave function in the complex z-plane | p. 29 |
| Removal of a boundary condition from the real z-axis to an anti-Stokes line | p. 30 |
| Dependence of the F-matrix on the lower limit of integration in the phase integral | p. 32 |
| F-matrix expressed in terms of two linearly independent solutions of the differential equation | p. 33 |
| F-matrix connecting points on opposite sides of a well-isolated turning point, and expressions for the wave function in these regions | p. 35 |
| Symmetry relations and estimates of the F-matrix elements | p. 36 |
| Parameterization of the matrix F(x[subscript 1], x[subscript 2]) | p. 38 |
| Changes of [alpha], [beta] and [gamma] when x[subscript 1] moves in the classically forbidden region | p. 40 |
| Changes of [alpha], [beta] and [gamma] when x[subscript 2] moves in the classically allowed region | p. 41 |
| Limiting values of [alpha], [beta] and [gamma] | p. 42 |
| Wave function on opposite sides of a well-isolated turning point | p. 43 |
| Power and limitation of the parameterization method | p. 45 |
| Phase-integral connection formulas for a real, smooth, single-hump potential barrier | p. 46 |
| Exact expressions for the wave function on both sides of the barrier | p. 48 |
| Phase-integral connection formulas for a real barrier | p. 50 |
| Wave function given as an outgoing wave to the left of the barrier | p. 53 |
| Wave function given as a standing wave to the left of the barrier | p. 54 |
| Problems with solutions | p. 59 |
| Base function for the radial Schrodinger equation when the physical potential has at the most a Coulomb singularity at the origin | p. 59 |
| Base function and wave function close to the origin when the physical potential is repulsive and strongly singular at the origin | p. 61 |
| Reflectionless potential | p. 62 |
| Stokes and anti-Stokes lines | p. 63 |
| Properties of the phase-integral approximation along an anti-Stokes line | p. 66 |
| Properties of the phase-integral approximation along a path on which the absolute value of exp[iw(z)] is monotonic in the strict sense, in particular along a Stokes line | p. 66 |
| Determination of the Stokes constants associated with the three anti-Stokes lines that emerge from a well isolated, simple transition zero | p. 69 |
| Connection formula for tracing a phase-integral wave function from a Stokes line emerging from a simple transition zero t to the anti-Stokes line emerging from t in the opposite direction | p. 72 |
| Connection formula for tracing a phase-integral wave function from an anti-Stokes line emerging from a simple transition zero t to the Stokes line emerging from t in the opposite direction | p. 73 |
| Connection formula for tracing a phase-integral wave function from a classically forbidden to a classically allowed region | p. 74 |
| One-directional nature of the connection formula for tracing a phase-integral wave function from a classically forbidden to a classically allowed region | p. 77 |
| Connection formulas for tracing a phase-integral wave function from a classically allowed to a classically forbidden region | p. 79 |
| One-directional nature of the connection formulas for tracing a phase-integral wave function from a classically allowed to a classically forbidden region | p. 81 |
| Value at the turning point of the wave function associated with the connection formula for tracing a phase-integral wave function from the classically forbidden to the classically allowed region | p. 83 |
| Value at the turning point of the wave function associated with a connection formula for tracing the phase-integral wave function from the classically allowed to the classically forbidden region | p. 87 |
| Illustration of the accuracy of the approximate formulas for the value of the wave function at a turning point | p. 88 |
| Expressions for the a-coefficients associated with the Airy functions | p. 91 |
| Expressions for the parameters [alpha], [beta] and [gamma] when Q[superscript 2](z) = R(z) = -z | p. 96 |
| Solutions of the Airy differential equation that at a fixed point on one side of the turning point are represented by a single, pure phase-integral function, and their representation on the other side of the turning point | p. 98 |
| Connection formulas and their one-directional nature demonstrated for the Airy differential equation | p. 102 |
| Dependence of the phase of the wave function in a classically allowed region on the value of the logarithmic derivative of the wave function at a fixed point x[subscript 1] in an adjacent classically forbidden region | p. 105 |
| Phase of the wave function in the classically allowed regions adjacent to a real, symmetric potential barrier, when the logarithmic derivative of the wave function is given at the centre of the barrier | p. 107 |
| Eigenvalue problem for a quantal particle in a broad, symmetric potential well between two symmetric potential barriers of equal shape, with boundary conditions imposed in the middle of each barrier | p. 115 |
| Dependence of the phase of the wave function in a classically allowed region on the position of the point x[subscript 1] in an adjacent classically forbidden region where the boundary condition [psi](x[subscript 1]) = 0 is imposed | p. 117 |
| Phase-shift formula | p. 121 |
| Distance between near-lying energy levels in different types of physical systems, expressed either in terms of the frequency of classical oscillations in a potential well or in terms of the derivative of the energy with respect to a quantum number | p. 123 |
| Arbitrary-order quantization condition for a particle in a single-well potential, derived on the assumption that the classically allowed region is broad enough to allow the use of a connection formula | p. 125 |
| Arbitrary-order quantization condition for a particle in a single-well potential, derived without the assumption that the classically allowed region is broad | p. 127 |
| Displacement of the energy levels due to compression of an atom (simple treatment) | p. 130 |
| Displacement of the energy levels due to compression of an atom (alternative treatment) | p. 133 |
| Quantization condition for a particle in a smooth potential well, limited on one side by an impenetrable wall and on the other side by a smooth, infinitely thick potential barrier, and in particular for a particle in a uniform gravitational field limited from below by an impenetrable plane surface | p. 137 |
| Energy spectrum of a non-relativistic particle in a potential proportional to cot[superscript 2](x/a[subscript 0]), where 0 < x/a[subscript 0] < [pi] and a[subscript 0] is a quantity with the dimension of length, e.g. the Bohr radius | p. 140 |
| Determination of a one-dimensional, smooth, single-well potential from the energy spectrum of the bound states | p. 142 |
| Determination of a radial, smooth, single-well potential from the energy spectrum of the bound states | p. 144 |
| Determination of the radial, single-well potential, when the energy eigenvalues are -mZ[superscript 2]e[superscript 4 2h[superscript 2](l + s + 1)[superscript 2], where l is the angular momentum quantum number, and s is the radial quantum number | p. 147 |
| Exact formula for the normalization integral for the wave function pertaining to a bound state of a particle in a radial potential | p. 150 |
| Phase-integral formula for the normalized radial wave function pertaining to a bound state of a particle in a radial single-well potential | p. 152 |
| Radial wave function [psi](z) for an s-electron in a classically allowed region containing the origin, when the potential near the origin is dominated by a strong, attractive Coulomb singularity, and the normalization factor is chosen such that, when the radial variable z is dimensionless, [psi](z)/z tends to unity as z tends to zero | p. 155 |
| Quantization condition, and value of the normalized wave function at the origin expressed in terms of the level density, for an s-electron in a single-well potential with a strong attractive Coulomb singularity at the origin | p. 160 |
| Expectation value of an unspecified function f(z) for a non-relativistic particle in a bound state | p. 163 |
| Some cases in which the phase-integral expectation value formula yields the expectation value exactly in the first-order approximation | p. 166 |
| Expectation value of the kinetic energy of a non-relativistic particle in a bound state. Verification of the virial theorem | p. 167 |
| Phase-integral calculation of quantal matrix elements | p. 169 |
| Connection formula for a complex potential barrier | p. 171 |
| Connection formula for a real, single-hump potential barrier | p. 181 |
| Energy levels of a particle in a smooth double-well potential, when no symmetry requirement is imposed | p. 186 |
| Energy levels of a particle in a smooth, symmetric, double-well potential | p. 190 |
| Determination of the quasi-stationary energy levels of a particle in a radial potential with a thick single-hump barrier | p. 192 |
| Transmission coefficient for a particle penetrating a real single-hump potential barrier | p. 197 |
| Transmission coefficient for a particle penetrating a real, symmetric, superdense double-hump potential barrier | p. 200 |
| References | p. 205 |
| Author index | p. 209 |
| Subject index | p. 211 |
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