Quasi-exactly solvable models in nonlinear optics

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We study a large class of models with an arbitrary (finite) number of degrees of freedom, described by Hamiltonians which are polynomial in bosonic creation and annihilation operators, and including as particular cases nth harmonic generation and photon cascades. For each model, we construct a complete set of commuting integrals of motion of the Hamiltonian, fully characterize the common eigenspaces of the integrals of motion and show that the action of the Hamiltonian on these common eigenspaces can be represented by a quasiexactly solvable reduced Hamiltonian, whose expression in terms of the usual generators of sl_2 is computed explicitly.
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[1] Zaslavskii O B 1990 Phys. Lett. A 149 365 [2] Zaslavskii O B and Ulyanov V V 1984 Zh. Eksp. Teor. Fiz. 87 1724 (Engl. transl. 1984 Sov. Phys.–JETP 60 991) [3] Zaslavskii O B and Ulyanov V V 1987 Teor. Mat. Fiz. 71 260 (Engl. transl. 1987 Theor. Math. Phys. 71 520) [4] Turbiner A V 1988 Commun. Math. Phys. 118 467 [5] Shifman M A 1989 Int. J. Mod. Phys. A 4 2897 [6] Ushveridze A G 1994 Quasi-Exactly Solvable Models in Quantum Mechanics (Bristol: Institute of Physics Publishing) [7] González-López A, Kamran N and Olver P J 1994 Contemp. Math. 160 113 [8] Álvarez G and Álvarez-Estrada R F 1995 J. Phys. A: Math. Gen. 28 5767 [9] Álvarez G and Álvarez-Estrada R F 2001 J. Phys. A: Math. Gen. 34 10045 [10] Dolya S N and Zaslavskii O B 2000 J. Phys. A: Math. Gen. 33 L369 [11] Dolya S N and Zaslavskii O B 2001 J. Phys. A: Math. Gen. 34 5955 [12] Bajer J and Miranowicz A 2000 J. Opt. B: Quantum Semiclass. Opt. 2 L10 [13] Klimov A B and Sánchez-Soto L L 2000 Phys. Rev. A 61 063802 [14] Karassiov V P, Gusev A A and Vinitsky S I 2001 Preprint quant-ph/0105152 [15] Galindo A and Pascual P 1990 Quantum Mechanics I (Berlin: Springer) [16] Arscott F M 1964 Periodic Differential Equations (Oxford: Pergamon) [17] Turbiner A 1992 J. Phys. A: Math. Gen. 25 L1087 [18] Finkel F and Kamran N 1998 Adv. Appl. Math. 20 300