A set of generalized squeezed–coherent states for the finite oscillator is obtained. These states are given as linear combinations of the mode eigenstates with amplitudes determined by matrix elements of exponentials in the generators. These matrix elements are given in the ( N + 1)-dimensional basis of the finite oscillator eigenstates and are seen to involve 3 × 3 matrix multi-orthogonal polynomials Q n ( k ) in a discrete variable k which have the Krawtchouk and vector-orthogonal polynomials as their building blocks. The algebraic setting allows for the characterization of these polynomials and the computation of mean values in the squeezed–coherent states. In the limit where N goes to infinity and the discrete oscillator approaches the standard harmonic oscillator, the polynomials tend to 2 × 2 matrix orthogonal polynomials and the squeezed–coherent states tend to those of the standard oscillator.A set of generalized squeezed–coherent states for the finite oscillator is obtained. These states are given as linear combinations of the mode eigenstates with amplitudes determined by matrix elements of exponentials in the generators. These matrix elements are given in the ( N + 1)-dimensional basis of the finite oscillator eigenstates and are seen to involve 3 × 3 matrix multi-orthogonal polynomials Q n ( k ) in a discrete variable k which have the Krawtchouk and vector-orthogonal polynomials as their building blocks. The algebraic setting allows for the characterization of these polynomials and the computation of mean values in the squeezed–coherent states. In the limit where N goes to infinity and the discrete oscillator approaches the standard harmonic oscillator, the polynomials tend to 2 × 2 matrix orthogonal polynomials and the squeezed–coherent states tend to those of the standard oscillator.

arXiv:1202.3410Journal of Physics A Mathematical and TheoreticalMay 4, 20123 citations

VECTOR COHERENT STATES OVER MATRIX DOMAINS

S. Twareque Ali

International Workshop on Contemporary Problems in Mathematical Physics (COPROMAPH)Dec 1, 20040 citations

VECTOR COHERENT STATE REPRESENTATIONS

DAVID J. ROWE

May 1, 19940 citations

Adiabatic Decoupling and Time Evolution of Coherent States for Multi-component Systems

Didier Robert, Monique Combescure

Coherent states are well adapted to understand adiabatic decoupling for multi-components systems like the Dirac operator or systems obtained by reduction in the Born-Oppenheimer approximation [58]. Hagedorn [130] has done an accurate analysis of this using a variant of coherent states. The new fact for systems is that now states are vectors valued in $\mathbb C^m$ for $m\ge 2$ and the eigenvalues of the matrix of the system may cross so that one cannot diagonalize it smoothly. In the case when the eigenvalues are not crossing one can construct an adiabatic modification of the spectral projectors on the modes and compute the time evolution of coherent states following the same strategy as in the scalar case. Here we shall consider only non crossing cases where we can prove an adiabatic decoupling between the modes. For the more difficult crossing case we refer to the literature [130] where non-adiabatic transitions can be computed [101, 129] for more recent results.Coherent states are well adapted to understand adiabatic decoupling for multi-components systems like the Dirac operator or systems obtained by reduction in the Born-Oppenheimer approximation [58]. Hagedorn [130] has done an accurate analysis of this using a variant of coherent states. The new fact for systems is that now states are vectors valued in $\mathbb C^m$ for $m\ge 2$ and the eigenvalues of the matrix of the system may cross so that one cannot diagonalize it smoothly. In the case when the eigenvalues are not crossing one can construct an adiabatic modification of the spectral projectors on the modes and compute the time evolution of coherent states following the same strategy as in the scalar case. Here we shall consider only non crossing cases where we can prove an adiabatic decoupling between the modes. For the more difficult crossing case we refer to the literature [130] where non-adiabatic transitions can be computed [101, 129] for more recent results.

Theoretical and mathematical physicsJan 1, 20210 citations

Correlated coherent states

V. V. Dodonov, E.V. Kurmyshev, Vladimir I. Man’ko

Jan 1, 198610 citations

\mathbb C^m

for

m\ge 2

and the eigenvalues of the matrix of the system may cross so that one cannot diagonalize it smoothly. In the case when the eigenvalues are not crossing one can construct an adiabatic modification of the spectral projectors on the modes and compute the time evolution of coherent states following the same strategy as in the scalar case. Here we shall consider only non crossing cases where we can prove an adiabatic decoupling between the modes. For the more difficult crossing case we refer to the literature [130] where non-adiabatic transitions can be computed [101, 129] for more recent results.