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Quasi-exactly solvable Bose systems. (English) Zbl 1061.81074
Shabat, A.B.(ed.) et al., New trends in integrability and partial solvability. Proceedings of the NATO Advanced Research Workshop, Cadiz, Spain, June 12–16, 2002. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1835-5/hbk). NATO Science Series II: Mathematics, Physics and Chemistry 132, 105-114 (2004).
Summary: We extend the notion of quasi-exactly solvable (QES) models to Bose systems. We obtain conditions under which algebraization of the part of the spectrum occurs. In some particular cases simple exact expressions for several energy levels of an anharmonic Bose oscillator are obtained explicitly. The corresponding results do not exploit perturbation theory and include strong coupling regime. A generic Hamiltonian under discussion cannot, in contrast to QES potential models, be expressed as a polynomial in generators of $$\text{sl}_2$$ algebra. The suggested approach is extendable to many-particle Bose systems with interaction.
For the entire collection see [Zbl 1050.35003].

##### MSC:
 81U15 Exactly and quasi-solvable systems arising in quantum theory 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 81S05 Commutation relations and statistics as related to quantum mechanics (general) 81V70 Many-body theory; quantum Hall effect