Abstract
We investigate a special class of the mathcal{P}mathcal{T} -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{P}mathcal{T} -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{P}mathcal{T} -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.
Highlights
PT -symmetric quantum systems [1] have many interesting properties and attract a lot of attention, for reviews see [2, 3]
We discuss how the Hamiltonians H1α and H2α governing the dynamics in conformal invariant PT -symmetric quantum mechanical systems determine the fluctuation spectra around the singular kinks arising as traveling waves in the Liouville and SU (3) conformal Toda systems in field theory. This is remarkable because the conformal group which is infinite dimensional in the two-dimensional Minkowskian setting contracts to a finite subgroup in conformal quantum mechanics, see [44, 45]
(3.9), descends this way to a spectral problem in conformal quantum mechanics. We infer this link in the opposite direction as compared to the References [44, 45] where an ascending path is travelled from conformal quantum mechanics to Virasoro algebra
Summary
PT -symmetric quantum systems [1] have many interesting properties and attract a lot of attention, for reviews see [2, 3]. In the present paper we construct and investigate a special class of the perfectly invisible PT symmetric zero-gap quantum mechanical systems related to the KdV hierarchy They represent the PT -regularized two-particle Calogero systems (conformal quantum mechanics models of de Alfaro-Fubini-Furlan [29]) and their rational extensions, which have the unique bound state at the very edge of the continuous spectrum of scattering states, and are characterized by transmission amplitude equal to one. 2 we construct the indicated class of the quantum systems by applying the appropriate Darboux-Crum transformations to a free particle We investigate their relationship with the stationary and non-stationary equations of the KdV hierarchy, and describe the properties of the higher-derivative Lax-Novikov integrals in such systems. In Appendix we briefly discuss a quantum scattering problem on the half-line
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