Quasi-Exactly Solvable Models in Quantum Mechanics

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Exactly solvable models (that is, models with explicitly and completely diagnolizable hamiltonians) are too few in number, and sufficiently 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. This book collects them together, and expounds them in a unified and accessible form, providing an invaluable resource for any physicists using quantum mechanics and applied mathematicians dealing with linear differential equations. By generalizing from one-dimensional QES models. Alex Ushveridze 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 ansatz solution of the Gaudin model, discuss hidden symmetry properties of QES hamiltonians, and explains various Lie-algebraic and analytic approaches to the problem of quasi-exact solubility in quantum mechanics. The applications of QES models are very wide - for example for investigating non-perturbative phenomena, or as a good approximation to exactly non-solvable problems. Researchers in any field involving quantum mechanics cannot afford to be unaware of the possibilities of QES models.

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