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Boris Mityagin

  1. Eigensystem of an $L^2$-perturbed harmonic oscillator is an unconditional basis.

    Tue, 12/15/2009 - 16:01 — SomNambul
    Authors: Boris Mityagin, James Adduci
    Subjects: Spectral Theory
    Abstract

    We prove the following. For any complex valued $L^2$-function $b(x)$ the
    spectrum of a perturbed harmonic oscillator operator $L = -\frac{d^2}{dx^2} +
    x^2 + b(x)$ in $L^2(\mathbb{R}^1)$ is discrete and eventually simple. Its SEAF
    (system of eigen- and associated functions) is an unconditional basis in
    $L^2(\mathbb{R}^1)$.

  2. Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials.

    Wed, 11/18/2009 - 16:01 — SomNambul
    Authors: Plamen Djakov, Boris Mityagin
    Subjects: Spectral Theory
    Abstract

    We consider the Hill operator $$ Ly = - y^{\prime \prime} + v(x)y, \quad 0
    \leq x \leq \pi, $$ subject to periodic or antiperiodic boundary conditions,
    with potentials $v$ which are trigonometric polynomials with nonzero
    coefficients, of the form

    (i) $ ae^{-2ix} +be^{2ix}; $

    (ii) $ ae^{-2ix} +Be^{4ix}; $

    (iii) $ ae^{-2ix} +Ae^{-4ix} + be^{2ix} +Be^{4ix}. $

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