DNLS-AL链的q对称性与量子二聚体的精确解
q-Symmetries in DNLS-AL chains and exact solutions of quantum dimers
摘要：
研究了离散非线性薛定谔链（DNLS）和Ablowitz-Ladik链（AL）哈密顿量子化模型的动力学对称性。结果表明，对于$n$ -格点，DNLS Hamilton算符的动力学代数由$su(n)$代数给出，而对于AL格点，相应的对称性是量子代数su_q（n）. AL模型中动力学对称性的q-变形是由于每个位置上的升降算子的非正则的类增子结构造成的。分别在DNLS和QAL情形下，用su（n）和su_q（n）代数生成元的Casimir中心元找到了运动不变量.利用对称代数的表示理论，我们专门讨论了$n=2$量子二聚体的情况，并制定了每个二聚体的本征值问题作为一个非线性（q）-自旋模型。随后的三项非线性递归关系进行了分析调查，并确定各自的正交和完整的特征向量基地。通过研究能级与非线性强度的渐近吸引和排斥，分析了QDNLS二聚体中经典自陷的量子表现和QAL二聚体中经典自陷的量子表现.我们的治疗预测的QDNLS-二聚体，相变的行为在本征能量差的对数的变化率，为古典分岔点附近的非线性参数的值。
英文摘要：
Dynamical symmetries of Hamiltonians quantized models of discrete non-linear Schroedinger chain (DNLS) and of Ablowitz-Ladik chain (AL) are studied. It is shown that for $n$-sites the dynamical algebra of DNLS Hamilton operator is given by the $su(n)$ algebra, while the respective symmetry for the AL case is the quantum algebra su_q(n). The q-deformation of the dynamical symmetry in the AL model is due to the non-canonical oscillator-like structure of the raising and lowering operators at each site. Invariants of motions are found in terms of Casimir central elements of su(n) and su_q(n) algebra generators, for the DNLS and QAL cases respectively. Utilizing the representation theory of the symmetry algebras we specialize to the $n=2$ quantum dimer case and formulate the eigenvalue problem of each dimer as a non-linear (q)-spin model. Analytic investigations of the ensuing three-term non-linear recurrence relations are carried out and the respective orthonormal and complete eigenvector bases are determined. The quantum manifestation of the classical self-trapping in the QDNLS-dimer and its absence in the QAL-dimer, is analysed by studying the asymptotic attraction and repulsion respectively, of the energy levels versus the strength of non-linearity. Our treatment predicts for the QDNLS-dimer, a phase-transition like behaviour in the rate of change of the logarithm of eigenenergy differences, for values of the non-linearity parameter near the classical bifurcation point.
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