In this paper we perform an exact study of ``Quantum Fidelity'' (also called Loschmidt Echo) for the time-periodic quantum Harmonic Oscillator of Hamiltonian : $$ \hat H\_{g}(t):=\frac{P^2}{2}+ f(t)\frac{Q^2}{2}+\frac{g^2}{Q^2} $$ when compared with the quantum evolution induced by $\hat H\_{0}(t)$ ($g=0$), in the case where $f$ is a $T$-periodic function and $g$ a real constant. The reference (initial) state is taken to be an arbitrary ``generalized coherent state'' in the sense of Perelomov. We show that, starting with a quadratic decrease in time in the neighborhood of $t=0$, this quantum fidelity may recur to its initial value 1 at an infinite sequence of times {$t\_{k}$}. We discuss the result when the classical motion induced by Hamiltonian $\hat H\_{0}(t)$ is assumed to be stable versus unstable. A beautiful relationship between the quantum and the classical fidelity is also demonstrated.
Generalized KS transformations, $ND$ singular oscillator and generalized MICZ-Kepler system
在广义Kustaanheimo-Stiefel（KS）变换的框架下，给出了对偶（拟）精确可解模型及其隐对称代数的个数在不同分析层次上的描述.证明了N维奇异振子与（n+1）维广义MICZ-Kepler系统是对偶的，其对偶变换是KS变换的推广.在二重坐标系、球坐标系和抛物坐标系下，用分离变量法给出了这些问题的薛定谔方程的可解性。在SU（1，1）代数加法规则的框架下，二次Hahn代数QH（3）作为一个隐对称，不随原实空间${\rm I \!R}^{N}$分解成不同分量而改变.在Howe对偶的意义下，用交换子方法也清楚地证明了Higgs/Hahn代数等隐对称代数。对二维奇异振子进行降维处理，其中每个变量都是n维超半径r。本文还讨论了（N=2n）维奇异振子与（n+1）维广义MICZ-Kepler系统在拟精确（QE）问题类中的对偶联系。在广义KS变换的框架下，给出并分析了各向异性和非线性非谐项对谐振子模型及其对偶模型的某些推广。本文讨论了上述问题的薛定谔方程在QE类中的精确解析解，并给出了四类对偶拟精确可解模型的精确解析解。特别地，给出了在低维情况下与类似结果的比较及其推广.
Wallis Products from the Four-Dimensional Singular Harmonic Oscillator
School of Materials and New Energy, Xingzhi College, South China Normal University（华南师范大学行知学院材料与新工程学院）；State Key Laboratory of Nuclear Physics and Technology, Institute of Quantum Matter, South China Normal University（华南师范大学量子物质研究院核物理与技术国家重点实验室）
Mixed mode oscillations in the Bonhoeffer-van der Pol oscillator with weak periodic perturbation
在K. Shimizu et al.（2011）我们考虑了具有非自治周期扰动的Bonhoven-van der Pol振荡器。我们表明，该文件中报道的混合模式振荡的存在下，可以使用奇异摄动的几何理论来解释。所考虑的模型可以重写为一个4维（局部3维）自治系统，在一定条件下有一个折叠的鞍结奇点，另外可以被视为一个三时间尺度之一。
