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Quasi-exactly solvable models in quantum mechanics. (English) Zbl 0834.58042
Bristol: Institute of Physics Publishing. xiv, 465 p. (1994).
The dynamics of a quantum system is determined by its Hamiltonian which is a selfadjoint operator on a complex Hilbert space. The author uses the following terminology: the model is exactly solvable if the spectrum of the Hamiltonian is known and it is quasi-exactly solvable if there is a known subspace (usually finite-dimensional) which is invariant under the Hamiltonian and the spectrum of the restriction to this subspace of the Hamiltonian is known. It is rarely the case that an actual quantum system is either exactly or quasi-exactly solvable – more often such a system can be approximated by exactly or quasi-exactly solvable models. The author’s contribution to the literature of quasi-exactly solvable models is substantial – much of it in Russian and a good proportion in unpublished preprints – so publication of his own account of the subject is very welcome.
The author describes his own contributions in relationship with those of others and considers in successive chapters the following: 1. The concept of quasi-exact solvability. 2. Examples and properties from one- and many-dimensional quantum mechanics. 3. Multi-parameter spectral equations and their applications to quantum theory. 4. The method for constructing quasi-exactly solvable models with separable variables. 5. The relationship between Gaudin models and quasi-exact solvability. In three appendices the author considers some peripheral issues and methods.
The price of the book is atrocious and even though authors have little say in the pricing of their books, one should expect a succinct and definitive account in such an expensive book – the account given by the author rambles quite a lot.
The book has a fourth appendix which is written by A. González-López, N. Kamran and P. J. Olver. These authors give a relatively succinct summary of their own contributions to Lie-algebraic Hamiltonians and quasi-exact solvability.

81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81R12 Groups and algebras in quantum theory and relations with integrable systems
37A30 Ergodic theorems, spectral theory, Markov operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory