We calculate the transmission coefficient for electrons passing through the helically shaped potential barrier which can be, for example, produced by DNA molecules. Introduction. In recent years, it was discovered that electron transmission through ordered thin films of chiral molecules is highly spin selective 1-5 . This effect was termed as chiral-induced spin selectivity (CISS) effect 4 (for details see review 6 and Refs. therein). In experiments 2,4 , the transmission of photoelectrons through self-assembled monolayers (SAMs) of doublestranded DNA (dsDNA) on gold has been studied. The spin polarization (SP) of electrons ejected from Au substrate and transmitted through SAM of dsDNA was measured and the strong SP, which is defined as 4 P = (I ↑ − I ↓ )/(I ↑ + I ↓ ) was observed. Here I ↑ and I ↓ are the intensities of the signals corresponding to the SP oriented parallel and antiparallel to the electrons' velocity, respectively.Recently, different models have been proposed 7-10 to explain experimental results. A scattering theory in the first Born approximation has been applied to obtain the SP in the differential cross section of electrons moving through chiral molecules with energies above the vacuum level 8 . The model of point charges placed along a helical line is considered in a tight binding approximation for electronic structure of the helix and the transmission of distinct electron spin state is computed by the Landauer formulation 9 . The SP conductance through a metal-DNA-metal structure is calculated in a tight binding picture 10 . Although all these studies differ in details, they possess similar physical basis. Each of them is based on accounting of the spin-orbit interaction (SOI) of an electron that is moving through a helical potential. In other words, describing different aspects of the problem, approaches are based on the Schrödinger equation 11

We study static solutions of a D-dimensional modified nonlinear Schrödinger equation (MNLSE) which was shown to describe, in two dimensions, the self-trapped (spontaneously localized) electron states in a discrete isotropic electron-phonon lattice [1,2]. We show that this MNLSE, unlike the conventional nonlinear Schrödinger equation, possesses static localized solutions at any dimensionality when the effective nonlinearity parameter is larger than a certain critical value which depends on the dimensionality of the system under study. We investigate various properties of the equation analytically, using scaling transformations, within the variational scheme and numerically, and show that the results of these studies agree qualitatively and quantitatively. In particular, we prove that, for various values of D, when the coupling constant is larger than a certain critical value (which depends on D), this equation has two solutions, a stable (metastable) and an unstable one. We show that the solutions can be well approximated by a Gaussian ansatz and we also show that, in two dimensions, the equation possesses solutions with a nonzero angular momentum.

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We investigate some aspects of the soliton dynamics in an α-helical protein macromolecule within the steric Davydov-Scott model. Our main objective is to elucidate the important role of the helical symmetry in the formation, stability and dynamical properties of Davydov's solitons in an α-helix. We show, analytically and numerically, that the corresponding system of nonlinear equations admits several types of stationary soliton solutions and that solitons which preserve helical symmetry are dynamically unstable: once formed, they decay rapidly when they propagate. On the other hand, the soliton which spontaneously breaks the local translational and helical symmetries possess the lowest energy and is a robust localized entity. We also demonstrate that this soliton is the result of an hybridization of the quasiparticle states from the two lowest degenerate bands and has an inner structure which can be described as a modulated multi-hump amplitude distribution of excitations on individual spines. The complex and composite structure of the soliton manifests itself distinctly when the soliton is moving and some interspine oscillations take place. Such a soliton structure and the interspine oscillations have previosly been observed numerically in [A.C. Scott, Phys.Rev. A 26 578 (1982)]. Here we argue that the solitons studied by A. Scott, are hybrid solitons and that the oscillations arise due to the helical symmetry of the system and result from the motion of the soliton along the α-helix. The frequency of the interspine oscillations is shown to be proportional to the soliton velocity.

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