Solvability and PT-symmetry in a double-well model with point interactions
Miloslav Znojil, Vit Jakubsky
备注：
24 p., written for the special journal issue "Singular Interactions in Quantum Mechanics: Solvable Models". Will be also presented to the int. conference "Pseudo-Hermitian Hamiltonians in Quantum Physics III" (Instanbul, Koc University, June 20 - 22, 2005) http://home.ku.edu.tr/~amostafazadeh/workshop/workshop.htm
We show that and how point interactions offer one of the most suitable guides towards a quantitative analysis of properties of certain specific non-Hermitian (usually called PT-symmetric) quantum-mechanical systems. A double-well model is chosen, an easy solvability of which clarifies the mechanisms of the unavoided level crossing and of the spontaneous PT-symmetry breaking. The latter phenomenon takes place at a certain natural boundary of the domain of the "acceptable" parameters of the model. Within this domain the model mediates a nice and compact explicit illustration of the not entirely standard probabilistic interpretation of the physical bound states in the very recently developed (so called PT symmetric or, in an alternative terminology, pseudo-Hermitian) new, fairly exciting and very quickly developing branch of Quantum Mechanics.
Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart
我们已经对平面摆本征问题进行了对称性分析，目的是解释其条件准精确可解性（C-QES）与其本征能表面的拓扑结构之间的关系，这在我们的早期工作中建立[Frontiers in Physical Chemistry and化学物理2，1-16，（2014）]。目前的分析表明，这种关系可以追溯到三对角矩阵的结构，代表了自适应摆哈密顿量，以及使我们能够确定更多的-四十个总的确切-解析解。此外，平面摆的双曲对应物的类似分析，即Razavy问题，也被证明是C-QES [American Journal of Physics 48，285（1980）]，证实了它与摆本征问题是反等谱的。对这两个本征问题的关键重要性被证明是拓扑指数$κ$，因为它确定了由无量纲相互作用参数$η$和$ζ$所跨越的本征能表面的交点（真正的和避免的）的轨迹。它还封装的条件下，获得两个特征值问题的解析解，并提供了解析解的数量。在给定的$κ$处，反等谱性仅针对单个状态发生（即，不适用于双峰），像C-QES仅适用于$κ$的整数值，并且仅适用于摆哈密顿量和Razavy哈密顿量的最低本征值，对于后者，本征值的顺序颠倒。对于所有其他状态，摆动和Razavy光谱实际上在性质上变得不同，因为更高的摆动状态出现为双峰，而所有更高的Razavy状态都是单峰。
Quantum solvability of quadratic Li'enard type nonlinear oscillators possessing maximal Lie point symmetries: An implication of arbitrariness of ordering parameters
