Quasi-Exactly Solvable Models in Quantum Mechanics: 1st Edition (Hardback) book cover

Quasi-Exactly Solvable Models in Quantum Mechanics

1st Edition

By A.G Ushveridze

CRC Press

480 pages

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Hardback: 9780750302661
pub: 1994-01-01
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Description

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 Schrödinger 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.

Table of Contents

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

Subject Categories

BISAC Subject Codes/Headings:
SCI040000
SCIENCE / Mathematical Physics
SCI057000
SCIENCE / Quantum Theory