Generalized Heisenberg algebras and Fibonacci series

Abstract

We have constructed a Heisenberg-type algebra generated by the Hamiltonian, the step operators and an auxiliar operator. This algebra describes quantum systems having eigenvalues of the Hamiltonian depending on the eigenvalues of the two previous levels. This happens, for example, for systems having the energy spectrum given by Fibonacci sequence. Moreover, the algebraic structure depends on two functions f (x) and g(x). When these two functions are linear we classify, analysing the stability of the fixed points of the functions, the possible representations for this algebra.

Quantum Groups GL p,q (2)- and $\mathit{SU}_{q_{1}/q_{2}}(2)$ -Invariant Bosonic Gases: A Comparative Study

“…5, the thermo-statistical results for GL p,q (2)-and SU q 1 /q 2 (2)-invariant boson models having the model Hamiltonians in (17) and (23) are obviously different, since the nature of the two-parameter deformations of the quantum group invariant bosonic oscillator algebras for these models is quite different on the algebraic basis. The differences between these two-parameter realizations result not only from the defining commutation relations of both of the bosonic oscillator algebras in (1) and (2), but also from the Fock space representation properties of the two algebras.…”

“…Using the GL p,q (2)-invariant Hamiltonian in (17), the grand partition function in the high-temperature limit is…”

“…They have found many applications in a wide spectrum of research in physics. For instance, there are some possible connections between one-parameter deformed quantum groups and generalized statistical mechanics [7][8][9][10][11][12][13][14][15][16][17]. However, in the last few years, a considerable effort has been spent to studies on thermo-statistical properties of two-parameter generalized bosonic and fermionic oscillator gas models which are invariant under the actions of some specific quantum groups.…”

A New Generalization of Fibonacci Sequence & Extended Binet's Formula

“…The first few terms of the Fibonacci sequence are: 0, 1, 1, 2, 3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584, . That is, F n D F n 1 C F n 2 for all n 2.…”

Generalized Heisenberg algebras andk-generalized Fibonacci numbers

“…This algebra, called generalized Heisenberg algebra, depends on an analytical function f and the eigenvalues α n of the Hamiltonian are given by the one-step recurrence α n+1 = f (α n ). This structure has been used in different physical situations, see the references given in the recent paper [1]. In the same paper [1] de Souza et al introduced an extended two-step Heisenberg algebra having many interesting properties.…”