The quantum anharmonic oscillator and quasi-exactly solvable Bose systems

Quasi-exactly solvable quartic Bose Hamiltonians

“…In general, the solutions of the Shrödinger equation contain infinite numbers of quasi-particles and only approximate or numerical methods can be applied to such systems. However, as was shown recently [17], if the coefficients at different powers of a, a + are selected in a proper way, in some cases a finite-dimensional closed subspace is singled out and algebraization of the spectrum occurs similar to what happens in "usual"…”

“…(25) One can check that the solution 1, when u = 1, corresponds to even states for the example considered in eq. (7) of [17] provided in that equation the coefficient A 2 = 0. In a similar way, the case 2 (u = z) corresponds to odd states from the same example.…”

“…In a similar way, the case 2 (u = z) corresponds to odd states from the same example. However, the cases (3)- (5) represent new solutions that were not contained in [17].…”

“…In so doing, the eigenvectors belonging to the subspace under discussion, can be expressed as a finite linear combination of eigenvectors of an harmonic oscillator and, thus, contain a finite number of quasi-particles [17].…”

Solutions of Two-Mode Bosonic and Transformed Hamiltonians

“…The single boson realization of the SU(1, 1) algebra has been studied in [39,40]. In this work we follow a different strategy to obtain the condition for QES of the two-bosonic systems.…”