Quasi-Exactly Solvable Models in Quantum Mechanics

Collecting the results of QES models in a unified and accessible form, this book 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 Schrdinger 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.

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.

QUASI-EXACT SOLVABILITY-WHAT DOES THAT MEAN?IntroductionCompletely algebraizable spectral problemsThe quartic oscillatorThe sextic oscillatorNon-perturbative effects in an explicit form and convergent perturbation theoryPartial algebraization of the spectral problemThe two-dimensional harmonic oscillatorCompletely integrable quantum systemsDeformation of completely integrable modelsQuasi-exact solvability and the Gaudin modelThe classical multi-particle Coulomb problemClassical formulation of quantal problemsThe Infeld-Hull factorization method and quasi-exact solvabilityThe Gelfand-Levitan equationSummaryHistorical commentsSIMPLEST ANALYTIC METHODS FOR CONSTRUCTING QUASI-EXACTLY SOLVABLE MODELS The Lanczos tridiagonalization procedureThe sextic oscillator with a centrifugal barrierThe electrostatic analogue-the quartic oscillatorHigher oscillators with centrifugal barriersThe electrostatic analogue-the general caseThe inverse method of separation of variablesThe Schrödinger equations with separable variablesMulti-dimensional modelsThe "field-theoretical" caseOther quasi-exactly solvable models THE INVERSE METHOD OF SEPARATION OF VARIABLESMulti-parameter spectral equationsThe method-general formulationThe case of differential equationsAlgebraically solvable multi-parameter spectral equationsAn analytic methodReduction to exactly solvable modelsThe one-dimensional case-classificationElementary exactly solvable modelsThe multi-dimensional case-classificationCLASSIFICATION OF QUASI-EXACTLY SOLVABLE MODELS WITH SEPARATE VARIABLEPreliminary commentsThe one-dimensional non-degenerate caseThe non-degenerate case-the first typeThe non-degenerate case-the second typeThe non-degenerate case-the third typeThe one-dimensional simplest degenerate caseThe simplest degenerate case-the first typeThe simplest degenerate case-the second typeThe simplest degenerate case-the third typeThe one-dimensional twice-degenerate caseThe twice-degenerate-the first typeThe twice-degenerate case-the second typeThe one-dimensional most degenerate caseThe multi-dimensional caseCOMPLETELY INTEGRABLE GAUDIN MODELS AND QUASI-EXACT SOLVABILITYHidden symmetriesPartial separation of variablesSome properties of simple Lie algebrasSpecial decomposition in simple Lie algebrasThe generalized Gaudin model and its solutionsQuasi-exactly solvable equationsReduction to the Schrödinger formConclusionsAppendices A: The Inverse Schrödinger Problem and Its Solution for Several Given StatesAppendices B: The Generalized Quantum Tops and Exact SolvabilityAppendices C: The Method of Raising and Lowering OperatorsAppendices D: Lie Algebraic Hamiltonians and Quasi-Exact SolvabilityReferencesIndex

Ushveridze\, A.G