论文标题
保形量子力学的对称发生器的光谱特性：一种路径综合方法
Spectral Properties of the Symmetry Generators of Conformal Quantum Mechanics: A Path-Integral Approach
论文作者
论文摘要
使用路径综合方法来研究SO（2,1）共形量子力学（CQM）的SO（2,1）对称性的光谱特性。特别是，我们考虑与逆平方电位的弱耦合方案相对应的CQM版本。我们开发了一个通用框架来表征通用的对称发生器$ g $（汉密尔顿$ h $的线性组合，特殊的保形操作员$ k $和扩张运算符$ d $），从中可以从中遵循路径综合传播器，从而导致完整的光谱分解。这是针对三类操作员完成的：椭圆，抛物线和双曲线。我们还重点介绍了具有连续光谱及其量子力学解释的双曲线算子的新结果。为连续光谱运算符的特征系统开发的光谱技术可以推广到其他操作员问题。
A path-integral approach is used to study the spectral properties of the generators of the SO(2,1) symmetry of conformal quantum mechanics (CQM). In particular, we consider the CQM version that corresponds to the weak-coupling regime of the inverse square potential. We develop a general framework to characterize a generic symmetry generator $G$ (linear combinations of the Hamiltonian $H$, special conformal operator $K$, and dilation operator $D$), from which the path-integral propagators follow, leading to a complete spectral decomposition. This is done for the three classes of operators: elliptic, parabolic, and hyperbolic. We also highlight novel results for the hyperbolic operators, with a continuous spectrum, and their quantum-mechanical interpretation. The spectral technique developed for the eigensystem of continuous-spectrum operators can be generalized to other operator problems.
