Generalized deformed oscillator and nonlinear algebras

Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials

“…In many cases the algebra is no longer a Lie algebra and many examples of polynomial algebras were obtained in quantum mechanics. [12][13][14][15][16][17][18][19][20][21][22] Daskaloyannis 17 studied the case of the quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic ͑associative͒ algebras of quantum superintegrable systems. He showed how the quadratic algebras provide a method to obtain the energy spectrum.…”

“…He used realizations in terms of deformed oscillator algebras. 18 Potentials with a third order integral can be investigated using these techniques.…”

Generalized Coherent States Associated with the Cλ-Extended Oscillator

“…The algebra (2.14)-(2.16) has a Casimir operator, which can be written as 17) where h(J 0 , P µ ) is a λth-degree polynomial in J 0 with P µ -dependent coefficients, h(J 0 , P µ ) = λ i=0 t i (P µ )J i 0 , whose explicit form is given in [33]. Each F µ subspace of F carries a unitary irreducible representation (unirrep) of the SGA, characterized by the eigenvalue c µ of C and by the lowest eigenvalue of J 0 , namely…”

“…For λ = 2, α = 1 (and µ = 0), corresponding to the Perelomov su(1,1) CS |z; 0; 1 , defined in (3.6), it is well known that there exists a positive weight function h (1) 0 (y) on (0, 1), solution of (4.4) and given by [9] 17) provided the conditionβ 1 > 1 (i.e., α 0 > 1) is satisfied. The problem to be solved is therefore finding a solution of (4.7), positive on (0, 1) and vanishing on (1, ∞), for r = 0 and α ≥ 2.…”

“…The latter is defined in terms of creation, annihilation, and number operators, a † = f (N b )b † , a = bf (N b ), N = N b , satisfying the commutation relations [17,18,19,20] N, a † = a † , [N, a] = −a, a, a † = G(N), (1.2) where f is some Hermitian operator-valued function of the number operator and G(N) = (N + 1)f 2 (N + 1) − Nf 2 (N). Nonlinear CS defined as eigenstates of a [18,20,21,22,23], of a 2 [24], or of an arbitrary power a λ (λ > 2) [25] have, for instance, been considered in connection with nonclassical properties.…”