Higher rank quadratic algebra of the $N$-dimensional Smorodinsky-Winternitz model

A superintegrable system is a system that allows more integrals of motion than degrees of freedom, i.e., in a classical mechanics, an n-dimensional Hamiltonian system with Hamiltonian

\begin{eqnarray*}

H=\frac{1}{2}g_{ik}p_i p_j+V(\vec{x},\vec{p}),\quad X_a=f_a(\vec{x},\vec{p}), a=1,\dots, n-1,

\end{eqnarray*}

is called completely integrable (Liouville integrable) if it allows $n$ integrals of motion (including $H$) that are well-defined functions on phase space, are in involution

\begin{eqnarray*}

\{H,X_a\}_{p}=0,\quad \{X_a, X_b\}_{p}=0, a, b=1,\dots, n-1

\end{eqnarray*}

and are functionally independent. The system is superintegrable if it is integrable and allows additional integrals of motion $Y_b(\vec{x},\vec{p})$, $\{H,Y_b\}_{p}=0, b=1,2,\dots, k$, $1\leq k\leq n-1$, that are also well-defined functions on phase space and the integrals $\{H, X_1,\dots, X_{n-1}, Y_n,\dots, Y_{n+k}\}$ are functionally independent.

In this talk, I present higher rank quadratic algebra of the $N$-dimensional quantum Smorodinsky-Winternitz system, which is a maximally superintegrable and exactly solvable model. It is shown that the model is multiseparable and the wave function can be expressed in terms of Laguerre and Jacobi polynomials. We present a complete symmetry algebra ${\cal SW}(N)$ of the system, which it is a higher-rank quadratic one containing Racah algebra ${\cal R}(N)$ as subalgebra. The substructures of distinct quadratic Q(3) algebras and their related Casimirs are also studied. The energy spectrum of the $N$-dimensional Smorodinsky-Winternitz system is obtained algebraically via the different set of subalgebras based on the Racah algebra ${\cal R}(N)$.