On the wave functions of a covariant linear oscillator

Onq-extended eigenvectors of the integral and finite Fourier transforms

“…. , N−1}, the linear combinations f (n) m (q) in (15) are eigenvectors of the finite Fourier transform A (N) , with the eigenvalue λ n = i n .…”

A q-oscillator Green’s function

“…Recently Atakishiev, Frank and Wolf introduced a simple difference realization of the Heysenberg q-algebra [4].They also studied the corresponding q-oscillator Hamiltonian and its e-functions in terms of the q −1 -Hermite polynomials.The q-annihilation and creation operators acting on the smooth functions f (ξ) with ξ ∈ (−∞, ∞) are given by…”

“…It is the purpose of this note to obtain the Green function for one of the q-oscillators. The q-oscillator we deal with is the one which is solved in terms of the continuous q −1 -Hermite polynomials [4].…”

“…In Section II we briefly review the q-oscillator realization of Ref. 4. Section III is devoted to the derivation of the q-oscillator Green function which is the deformation of the well known Feynman formula.…”

On a q-extension of the linear harmonic oscillator with the continuous orthogonality property on ℝ

“…Let us point out here that there are several publications (see [2]- [10] and references therein) devoted to the study of explicit realizations, which represent q-extensions of the Hermite functions (or the wave functions of the linear harmonic oscillator) H n (x) e −x 2 /2 . But none of these realizations satisfies all of the aforementioned requirements: the continuous weight functions in [2,4,7] are supported on the finite intervals; the continuous weight functions in [3,8] are not positive; the q-extensions in [2], [4]- [9] are not expressed in terms of polynomials in the independent variable; and, finally, the orthogonality relations in [5]- [7], [10] are discrete.…”