Quantum symmetries and solvability

显式构造了无穷多个精确可解的一维量子力学系统，并详细讨论了它们的量子对称性和可解性。通过对原始本征函数进行离散对称变换得到(伪)虚态波函数，通过对虚态进行多次达布变换得到多指标正交多项式。许多Wronskian恒等式和Casoratian恒等式是通过变形得到的，原始系统是普通量子力学中的(径向)谐振子、Poschl-Teller、Morse、Ekart、Coulomb势，以及离散量子力学中的Wilson、Askey-Wilson和(q-)racah多项式系统。具有较高表观奇异度的多指标拉盖尔多项式和雅可比多项式也与相应的位势一起被构造。

Infinitely many exactly solvable 1-d quantum mechanical systems are constructed explicitly and their quantum symmetries and solvability are discussed in detail. (Pseudo) virtual state wavefunctions are obtained by applying discrete symmetry transformations to the original eigenfunctions.Multi-indexed orthogonal polynomials are obtained by using multiple Darboux transformations in terms of virtual states. Many Wronskian and Casoratian identities are derived through deformations in terms of pseudo virtual states.The original systems are (radial) harmonic oscillator, Poschl-Teller, Morse, Eckart, Coulomb potentials from ordinary quantum mechanics and Wilson, Askey-Wilson and (q-)Racah polynomial systems from discrete quantum mechanics. Multi-indexed Laguerre and Jacobi polynomials having higher degree of apparent singularities are also constructed together with the corresponding potentials.