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Scattering on a chain of zero-range potentials with internal structure in the stochastic magnetic field

Nanostructures having the form of chains of short-range potentials are widely used in nanotechnologies. Berezin and Faddeev (see [1]) showed that the Hamiltonian with zero-range Fermi type potential is just an extension of a suitable defined symmetric operator. Later, it has been shown by Pavlov [2], that the structure of the standard point interaction models can be enriched substantially when the self-adjoint extensions are constructed in a wider Hilbert space. This idea yields various models of zero-range interaction with an additional internal structure (see, e.g., [3–5]). In papers [6,7] this method has been used to construct and investigate explicitly solvable models of the scattering of the neutron on a point nucleus, whose internal structure depends on a stochastic magnetic field, and of the scattering of acoustic waves on a stochastic point defect with an internal structure. In the present paper, an exactly solvable model of the neutron scattering on the one-dimensional infinite chain embedded into the three dimensional configuration space R. We will suppose that the chain is inserted into the stochastic magnetic field. The nuclei in the chain are assumed to be equivalent with the internal structure dependent on the magnetic field. This model corresponds to the case, when the whole chain belongs to one magnetic domain. In the absence of the stochasticity, such model was investigated by Albeverio, Gesztesy, Hoegh-Krohn, Holden [8], Karpeshina [9], Subramanian [10], Kurasov and Pavlov [11]. It was shown that the spectrum of the related operator is purely continuous and consists of two branches: 1) Scattered waves branch σs; corresponding eigenfunctions are defined by free waves reflected by the lattice. This branch coincides with the spectrum of the free Laplacian −∆ in L2 ( R ) . 2) Waveguide branch σw; corresponding eigenfunctions are localized in a vicinity of the lattice. In the discussing periodic case these functions are of the Bloch type. It will be shown that the spectral properties of the problem with the stochastic filed are related to the properties of the problems without any stochasticity.