15 research outputs found

    Regular Spacings of Complex Eigenvalues in the One-dimensional non-Hermitian Anderson Model

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    We prove that in dimension one the non-real eigenvalues of the non-Hermitian Anderson (NHA) model with a selfaveraging potential are regularly spaced. The class of selfaveraging potentials which we introduce in this paper is very wide and in particular includes stationary potentials (with probability one) as well as all quasi-periodic potentials. It should be emphasized that our approach here is much simpler than the one we used before. It allows us a) to investigate the above mentioned spacings, b) to establish certain properties of the integrated density of states of the Hermitian Anderson models with selfaveraging potentials, and c) to obtain (as a by-product) much simpler proofs of our previous results concerned with non-real eigenvalues of the NHA model.Comment: 21 pages, 1 figur

    Random walks in a random environment on a strip: a renormalization group approach

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    We present a real space renormalization group scheme for the problem of random walks in a random environment on a strip, which includes one-dimensional random walk in random environment with bounded non-nearest-neighbor jumps. We show that the model renormalizes to an effective one-dimensional random walk problem with nearest-neighbor jumps and conclude that Sinai scaling is valid in the recurrent case, while in the sub-linear transient phase, the displacement grows as a power of the time.Comment: 9 page

    Spectral Curves and Localization in Random Non-Hermitian Tridiagonal Matrices

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    Eigenvalues and eigenvectors of non-Hermitian tridiagonal periodic random matrices are studied by means of the Hatano-Nelson deformation. The deformed spectrum is annular-shaped, with inner radius measured by the complex Thouless formula. The inner bounding circle and the annular halo are stuctures that correspond to the two-arc and wings observed by Hatano and Nelson in deformed Hermitian models, and are explained in terms of localization of eigenstates via a spectral duality and the Argument principle.Comment: 5 pages, 9 figures, typographical error corrected in reference

    Spectra of pinned charge density waves with background current

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    We develop techniques which allow us to calculate the spectra of pinned charge density waves with background current

    Enumeration of simple random walks and tridiagonal matrices

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    We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the nn-th power of a tridiagonal matrix and the enumeration of weighted paths of nn steps allows an easier combinatorial enumeration of the paths. It also seems promising for the theory of tridiagonal random matrices .Comment: several ref.and comments added, misprints correcte

    Non-Hermitian Delocalization and Eigenfunctions

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    Recent literature on delocalization in non-Hermitian systems has stressed criteria based on sensitivity of eigenvalues to boundary conditions and the existence of a non-zero current. We emphasize here that delocalization also shows up clearly in eigenfunctions, provided one studies the product of left- and right-eigenfunctions, as required on physical grounds, and not simply the squared modulii of the eigenfunctions themselves. We also discuss the right- and left-eigenfunctions of the ground state in the delocalized regime and suggest that the behavior of these functions, when considered separately, may be viewed as ``intermediate'' between localized and delocalized.Comment: 8 pages, 11 figures include

    Density of states in the non-hermitian Lloyd model

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    We reconsider the recently proposed connection between density of states in the so-called ``non-hermitian quantum mechanics'' and the localization length for a particle moving in random potential. We argue that it is indeed possible to find the localization length from the density of states of a non-hermitian random ``Hamiltonian''. However, finding the density of states of a non-hermitian random ``Hamiltonian'' remains an open problem, contrary to previous findings in the literature.Comment: 6 pages, RevTex, two-column

    Schrödinger operators in the twentieth century

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