论文标题
$ b_2 $谐波振荡器，具有反射和可整合性
The $B_2$ Harmonic Oscillator with Reflections and Superintegrability
论文作者
论文摘要
二维量子谐波振荡器通过与Coxeter组$ b_2 $的作用相关的反射术语进行了修改，即广场的对称组。角动量操作员还通过反射进行了修改。已知波函数是由雅各比和拉瓜多项式建立的。本文介绍了与哈密顿量通勤的四阶差分差异操作员，但没有与角动量操作员交通；可矫正性的特定实例。明确描述了操作员在通常的波形基础基础上的作用。波形根据该组的表示进行分类：第二学位的四个和第二度。身份表示包含该组下的波形不变。本文首先讨论了与有限反思组相关的改良的汉密尔顿人，以及相关的饲养和降低操作员。特别是，对称组的哈密顿量描述了与谐波限制在线上相同颗粒的Calogero-Sutherland模型。
The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group $B_2$, which is the symmetry group of the square. The angular momentum operator is also modified with reflections. The wavefunctions are known to be built up from Jacobi and Laguerre polynomials. This paper introduces a fourth-order differential-difference operator commuting with the Hamiltonian but not with the angular momentum operator; a specific instance of superintegrability. The action of the operator on the usual orthogonal basis of wavefunctions is explicitly described. The wavefunctions are classified according to the representations of the group: four of degree one and one of degree two. The identity representation encompasses the wavefunctions invariant under the group. The paper begins with a short discussion of the modified Hamiltonians associated to finite reflection groups, and related raising and lowering operators. In particular, the Hamiltonian for the symmetric groups describes the Calogero-Sutherland model of identical particles on the line with harmonic confinement.
