We study a planar model of a non-relativistic electron in periodic magnetic and electric fields that produce a 1D crystal for two spin components separated by a half-period spacing. We fit the fields to create a self-isospectral pair of finite-gap associated Lamé equations shifted for a half-period, and show that the system obtained is characterized by a new type of supersymmetry. It is a special nonlinear supersymmetry generated by three commuting integrals of motion, related to the parityodd operator of the associated Lax pair, that coherently reflects the band structure and all its peculiarities. In the infinite period limit it provides an unusual picture of supersymmetry breaking.
We explain the origin and the nature of a special nonlinear supersymmetry of a reflectionless Pöschl-Teller system by the Aharonov-Bohm effect for a nonrelativistic particle on the AdS 2 . A key role in the supersymmetric structure appearing after reduction by a compact generator of the AdS 2 isometry is shown to be played by the discrete symmetries related to the space and time reflections in the ambient Minkowski space. We also observe that a correspondence between the two quantum non-relativistic systems is somewhat of the AdS/CFT holography nature.
We show that an n-gap periodic quantum system with parity-even smooth potential admits 2 n − 1 isospectral super-extensions. Each is described by a tri-supersymmetry that originates from a higher-order differential operator of the Lax pair and two-term nonsingular decompositions of it; its local part corresponds to a spontaneously partially broken centrally extended nonlinear N = 4 supersymmetry. We conjecture that any finite-gap system having antiperiodic singlet states admits a self-isospectral trisupersymmetric extension with the partner potential to be the original one translated for a half-period. Applying the theory to a broad class of finite-gap elliptic systems described by a two-parametric associated Lamé equation, our conjecture is supported by the explicit construction of the self-isospectral tri-supersymmetric pairs. We find that the spontaneously broken tri-supersymmetry of the self-isospectral periodic system is recovered in the infinite period limit.
We show that confluent Darboux-Crum transformations with emergent Jordan states are an effective tool for the design of optical systems governed by the Helmholtz equation under the paraxial approximation. The construction of generic, asymptotically real and periodic, P Tsymmetric systems with local complex periodicity defects is discussed in detail. We show how the decay rate of the defect is related with the energy of the bound state trapped by the defect. In particular, the bound states in the continuum are confined by the periodicity defects with power law decay. We show that these defects possess complete invisibility; the wave functions of the system coincide asymptotically with the wave functions of the undistorted setting. The general results are illustrated with explicit examples of reflectionless models and systems with one spectral gap. We show that the spectral properties of the studied models are reflected by Lax-Novikov-type integrals of motion and associated supersymmetric structures of bosonized and exotic nature.1 The Hamiltonian H with complex, P T symmetric potential ceases to be self-adjoint with respect to the usual scalar product (f, g) = ∞ −∞ f * (x)g(x)dx. To recover physical relevance of the scalar product and the self-adjointness of the Hamiltonian, it is necessary to make a suitable redefinition of the former one, see e.g. [18], [25].
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