The B2 Harmonic Oscillator with Reflections and Superintegrability

IF 0.9 3区 物理与天体物理 Q2 MATHEMATICS Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-10-25 DOI:10.3842/SIGMA.2023.025
C. Dunkl
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引用次数: 2

Abstract

The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group B2, 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.
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具有反射和超可积性的B2谐振子
二维量子谐振子用与作为正方形的对称群的Coxeter群B2的作用相关联的反射项来修改。角动量算子也通过反射进行修改。已知波函数是由雅可比多项式和拉盖尔多项式建立起来的。本文引入了一个四阶微分差分算子,它和哈密顿算子交换,而不和角动量算子交换;超可集成性的一个具体例子。明确地描述了算子在波函数的通常正交基上的作用。波函数根据组的表示进行分类:一阶四个和二阶一个。恒等式表示包含群下不变的波函数。本文首先简要讨论了与有限反射群相关的修正哈密顿量,以及相关的升降算子。特别地,对称群的哈密顿量描述了具有谐波约束的线上相同粒子的Calogero Sutherland模型。
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来源期刊
CiteScore
1.80
自引率
0.00%
发文量
87
审稿时长
4-8 weeks
期刊介绍: Scope Geometrical methods in mathematical physics Lie theory and differential equations Classical and quantum integrable systems Algebraic methods in dynamical systems and chaos Exactly and quasi-exactly solvable models Lie groups and algebras, representation theory Orthogonal polynomials and special functions Integrable probability and stochastic processes Quantum algebras, quantum groups and their representations Symplectic, Poisson and noncommutative geometry Algebraic geometry and its applications Quantum field theories and string/gauge theories Statistical physics and condensed matter physics Quantum gravity and cosmology.
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