标题： 完全隐形的$\mathcal{PT}$-对称零隙系统、共形场理论孤子以及奇异的非线性超对称性 显示英文标题

标题： Perfectly invisible $\mathcal{PT}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry

摘要： 我们研究了一类特殊的$\mathcal{PT}$-对称量子模型，这些模型是完全不可见的零能隙系统，在散射态连续谱的边缘具有唯一的束缚态。 该族包括 $\mathcal{PT}$-正则化的两粒子 Calogero 系统（de Alfaro-Fubini-Furlan 的共形量子力学模型）及其有理扩展，其势满足 KdV 层次方程，并表现出极端波的典型行为。 我们证明了 Calogero 子族中最简单的两个哈密顿量决定了围绕由场论中的 Liouville 和$SU(3)$共形 Toda 系统中传播波形成的$\mathcal{PT}$-正则化 kink 的涨落谱。 量子系统的奇特性质反映在其相关的奇异非线性超对称性中，这种超对称性在未破裂或部分破裂相中显现。 传统的$\mathcal{N}=2$超对称性在这里被扩展为涉及两个由子系统 Lax-Novikov 积分构成的玻色生成元的$\mathcal{N}=4$非线性超对称性，其中一个生成元是超代数的中心荷。 Jordan 态在构造中显示出重要作用。 显示英文摘要

摘要： We investigate a special class of the $\mathcal{PT}$-symmetric quantum models being perfectly invisible zero-gap systems with a unique bound state at the very edge of continuous spectrum of scattering states. The family includes the $\mathcal{PT}$-regularized two particle Calogero systems (conformal quantum mechanics models of de Alfaro-Fubini-Furlan) and their rational extensions whose potentials satisfy equations of the KdV hierarchy and exhibit, particularly, a behaviour typical for extreme waves. We show that the two simplest Hamiltonians from the Calogero subfamily determine the fluctuation spectra around the $\mathcal{PT}$-regularized kinks arising as traveling waves in the field-theoretical Liouville and $SU(3)$ conformal Toda systems. Peculiar properties of the quantum systems are reflected in the associated exotic nonlinear supersymmetry in the unbroken or partially broken phases. The conventional $\mathcal{N}=2$ supersymmetry is extended here to the $\mathcal{N}=4$ nonlinear supersymmetry that involves two bosonic generators composed from Lax-Novikov integrals of the subsystems, one of which is the central charge of the superalgebra. Jordan states are shown to play an essential role in the construction.