Use of coherent states in operator algebra

S. Sivakumar

New quantal states which interpolate between the coherent states of the Heisenberg–Weyl ( W 3 ) and SU (1, 1) algebras are introduced. The interpolating states are coherent states of a closed and symmetric algebra containing a parameter k which takes values from zero to one. The elements of the algebra are realized in terms of operator-valued functions of creation and annihilation operators. The realization is such that the operators become those of the W 3 algebra when k = 0 and of the SU (1, 1) algebra when k is unity. The overcompleteness of the interpolating coherent states is established by constructing suitable integration measures. Differential operator representations for the elements of the algebra are given along with the relevant spaces of entire functions. A nonsymmetric set of operators to realize the W 3 algebra is presented and the relevant coherent states are studied. Physical properties such as squeezing and photon statistics of the interpolating coherent states are investigated.

Élliott H. Lieb, Jan Philip Solovej

Won Sang Chung, Mahouton Norbert Hounkonnou, Sama Arjika

This paper addresses a construction of new $q-$Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three-term recursive relation as well as the second-order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self-adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem.

Élliott H. Lieb, Jan Philip Solovej

Michel V. Romerio

Won Sang Chung, Mahouton Norbert Hounkonnou, Sama Arjika

This paper addresses a construction of new q-Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three-term recursive relation as well as the second-order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self-adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem. The phase collapse in a q-deformed boson system is studied.