Bogolyubov Institute for Theoretical Physics,

Kyiv 03143, UKRAINE

and

Instituto de Matematicas, UNAM,

CP 62210 Cuernavaca, Morelos, MEXICO

E-mail: aklimyk@bitp.kiev.ua, anatoliy@matcuer.unam.mx

**Hamiltonian type operators in representations of the
quantum algebra U_{q}(su_{1,1})**

**Abstract:**

We study some classes of symmetric operators for the discrete series representations of the quantum algebra U_{q}(su_{1,1}), which may serve as Hamiltonians of various physical systems. The problem of diagonalization of these operators (eigenfunctions, spectra, overlap coefficients, etc.) is solved by expressing their overlap coefficients in terms of the known families of q-orthogonal polynomials. We consider both bounded and unbounded operators. In the latter case they are not selfadjoint and have deficiency indices (1,1), which means that they may have infinitely many selfadjoint extensions. We find possible sets of
point spectrum for one of such operators by using the orthogonality relations for q-Laguerre polynomials. In the other case, we are led to q-orthogonal polynomials related to the Al-Salam-Chihara polynomials. Many new realizations for the discrete series representations are constructed,
which follow from the diagonalization of the operators considered.