量子代数U_q（su_{1,1}）表示中的Hamilton型算子
Hamiltonian Type Operators in Representations of the Quantum Algebra U_q(su_{1,1})
摘要：
研究了量子代数U_q（su_{1,1}）的离散级数表示的几类对称算子，它们可以作为各种物理系统的哈密顿量.这些算子（本征函数、谱、重叠系数等）的对角化问题通过用已知的q-正交多项式族表示它们的重叠系数来求解。我们考虑有界和无界算子。在后一种情况下，它们不是自伴的，并且有亏指数（1，1），这意味着它们有无穷多个自伴扩张。我们发现可能的点谱集（这取决于所考虑的表示空间），这样的对称算子之一，通过使用的正交关系的q-拉盖尔多项式。在另一种情况下，我们得到了{}_3 ∈_1-超几何多项式的新的正交关系.许多新的实现的离散级数表示的构造，这是从所考虑的运营商的对角化。特别是，一个新的正交函数系的离散集上出现。
英文摘要：
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 have infinitely many selfadjoint extensions. We find possible sets of point spectrum (which depends on the representation space under consideration) for one of such symmetric operators by using the orthogonality relations for q-Laguerre polynomials. In another case we are led to new orthogonality relations for {}_3ϕ_1-hypergeometric polynomials. Many new realizations for the discrete series representations are constructed, which follows from the diagonalization of the operators considered. In particular, a new system of orthogonal functions on a discrete set is shown to emerge.
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