arXiv subjects
Publications and source records attributed to H. Ahmedov.
q-Neumann function of integer order N_n(x,q) is obtained and some of its properties are given. q-Psi function which is used in deriving N_n(x,q) is also introduced and some of its properties are presented.
Green function (which can be called the q-analogous of the Hankel function) on the quantum plane E_q^2= E_q(2)/U(1) is constructed.
2-Dim quantum Poincare` Group E_q(1,1) at roots of unity, its dual U_q(e(1,1)) and some of its homogeneous spaces are introduced. Invariant integrals on E_q(1,1) and its invariant discrete subgroup E(1,1\mid p) are constructed. *-Representations of the quantum algebra U_q(e(1,1)) constructed in the homogeneous space SO(1,1\mid p) are integrated to the pseudo-unitary representations of E_q(1,1) by means of the universal T-matrix. U_q(e(1,1)) is realized on the quantum plane E_q^{(1,1)} and the eigenfunctions of the complete set of observables are obtained in the angular momentum and momentum basis. The matrix elements of the pseudo-unitary irreducible representations are given in terms of the cut off q-exponential and q-Bessel functions whose properties we also investigate.
Following the construction of the invariant integral and the scalar product for the quantum Euclidean group E_q(2), we obtained the full matrix elements of its unitary irreducible representations from SU_q(2) by contraction and then derived the Planscherel measure.
(2+2)-dimensional quantum mechanical q-phase space which is the semi-direct product of the quantum plane E_q(2)/U(1) and its dual algebra e_q(2)/u(1) is constructed. Commutation and the resulting uncertainty relations are studied. ``Quantum mechanical q-Hamiltonian'' of the motion over the quantum plane is derived and the solution of the Schroedinger equation for the q-semiclassical motion governed by the expectation value of that Hamiltonian is solved.
Following the introduction of the invariant distance on the non-commutative C-algebra of the quantum group SU_q(2), the Green function and the Kernel on the q-homogeneous space M=SU(2)_q/U(1) are derived. A path integration is formulated. Green function for the free massive scalar field on the non-commutative Einstein space R^1xM is presented.
Unitary irreducible representation of the group SO(1,2) is obtained in the mixed basis, i.e. between the compact and noncompact basis and the new addition theorems are derived which are required in path integral applications involving the positively signed potential. The Green function for the potential barrier $V=cosh^{-2}ωx$ is evaluated from the path integration over the coset space SO(1,2)/K where K is the compact subgroup. The transition and the reflection coefficients are given. Results for the moving barrier $V=\cosh^{-2}ω(x-g_0t)$ are also presented.
By using the generating function formula for the product of two q-Hermite polynomials q-deformation of the Feynman Green function for the harmonic oscillator is obtained.