Quasi-Exactly Solvable Models in Quantum Mechanics

ISBN-13: 9780750302661 / Angielski / Twarda / 1994 / 480 str.

Alexander G. Ushveridze

Opis

Exactly solvable models, that is, models with explicitly and completely diagonalizable Hamiltonians are too few in number and insufficiently diverse to meet the requirements of modern quantum physics. Quasi-exactly solvable (QES) models (whose Hamiltonians admit an explicit diagonalization only for some limited segments of the spectrum) provide a practical way forward.
Although QES models are a recent discovery, the results are already numerous. Collecting the results of QES models in a unified and accessible form, Quasi-Exactly Solvable Models in Quantum Mechanics provides an invaluable resource for physicists using quantum mechanics and applied mathematicians dealing with linear differential equations. By generalizing from one-dimensional QES models, the expert author constructs the general theory of QES problems in quantum mechanics. He describes the connections between QES models and completely integrable theories of magnetic chains, determines the spectra of QES Schrodinger equations using the Bethe-Iansatz solution of the Gaudin model, discusses hidden symmetry properties of QES Hamiltonians, and explains various Lie algebraic and analytic approaches to the problem of quasi-exact solubility in quantum mechanics.
Because the applications of QES models are very wide, such as, for investigating non-perturbative phenomena or as a good approximation to exactly non-solvable problems, researchers in quantum mechanics-related fields cannot afford to be unaware of the possibilities of QES models."

Spis treści

QUASI-EXACT SOLVABILITY-WHAT DOES THAT MEAN?
Introduction
Completely algebraizable spectral problems
The quartic oscillator
The sextic oscillator
Non-perturbative effects in an explicit form and convergent perturbation theory
Partial algebraization of the spectral problem
The two-dimensional harmonic oscillator
Completely integrable quantum systems
Deformation of completely integrable models
Quasi-exact solvability and the Gaudin model
The classical multi-particle Coulomb problem
Classical formulation of quantal problems
The Infeld-Hull factorization method and quasi-exact solvability
The Gelfand-Levitan equation
Summary
Historical comments

SIMPLEST ANALYTIC METHODS FOR CONSTRUCTING QUASI-EXACTLY SOLVABLE MODELS
The Lanczos tridiagonalization procedure
The sextic oscillator with a centrifugal barrier
The electrostatic analogue-the quartic oscillator
Higher oscillators with centrifugal barriers
The electrostatic analogue-the general case
The inverse method of separation of variables
The Schrödinger equations with separable variables
Multi-dimensional models
The "field-theoretical" case
Other quasi-exactly solvable models

THE INVERSE METHOD OF SEPARATION OF VARIABLES
Multi-parameter spectral equations
The method-general formulation
The case of differential equations
Algebraically solvable multi-parameter spectral equations
An analytic method
Reduction to exactly solvable models
The one-dimensional case-classification
Elementary exactly solvable models
The multi-dimensional case-classification

CLASSIFICATION OF QUASI-EXACTLY SOLVABLE MODELS WITH SEPARATE VARIABLE
Preliminary comments
The one-dimensional non-degenerate case
The non-degenerate case-the first type
The non-degenerate case-the second type
The non-degenerate case-the third type
The one-dimensional simplest degenerate case
The simplest degenerate case-the first type
The simplest degenerate case-the second type
The simplest degenerate case-the third type
The one-dimensional twice-degenerate case
The twice-degenerate-the first type
The twice-degenerate case-the second type
The one-dimensional most degenerate case
The multi-dimensional case

COMPLETELY INTEGRABLE GAUDIN MODELS AND QUASI-EXACT SOLVABILITY
Hidden symmetries
Partial separation of variables
Some properties of simple Lie algebras
Special decomposition in simple Lie algebras
The generalized Gaudin model and its solutions
Quasi-exactly solvable equations
Reduction to the Schrödinger form
Conclusions

Appendices A: The Inverse Schrödinger Problem and Its Solution for Several Given States
Appendices B: The Generalized Quantum Tops and Exact Solvability
Appendices C: The Method of Raising and Lowering Operators
Appendices D: Lie Algebraic Hamiltonians and Quasi-Exact Solvability

References

Index

O autorze

Ushveridze\, A.G