The $\mathfrak{su}(2)$ Krawtchouk oscillator model under the ${\cal C}{\cal P}$ deformed symmetry
E. I. Jafarov, A. M. Jafarova, J. Van der Jeugt
备注：
Contribution to the 30th International Colloquium on Group Theoretical Methods in Physics (Ghent, Belgium, 2014). To be published in Journal of Physics: Conference Series
We define a new algebra, which can formally be considered as a ${\cal C}{\cal P}$ deformed $\mathfrak{su}(2)$ Lie algebra. Then, we present a one-dimensional quantum oscillator model, of which the wavefunctions of even and odd states are expressed by Krawtchouk polynomials with fixed $p=1/2$, $K_{2n}(k;1/2,2j)$ and $K_{2n}(k-1;1/2,2j-2)$. The dynamical symmetry of the model is the newly introduced $\mathfrak{su}(2)_{{\cal C}{\cal P}}$ algebra. The model itself gives rise to a finite and discrete spectrum for all physical operators (such as position and momentum). Among the set of finite oscillator models it is unique in the sense that any specific limit reducing it to a known oscillator models does not exist.
The oscillator Racah algebra $\mathcal{R}_n(\mathfrak{h})$ is realized by the intermediate Casimir operators arising in the multifold tensor product of the oscillator algebra $\mathfrak{h}$. An embedding of the Lie algebra $\mathfrak{sl}_{n-1}$ into $\mathcal{R}_n(\mathfrak{h})$ is presented. It relates the representation theory of the two algebras. We establish the connection between recoupling coefficients for $\mathfrak{h}$ and matrix elements of $\mathfrak{sl}_n$-representations which are both expressed in terms of multivariate Krawtchouk polynomials of Griffiths type.
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