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Register a new userDarboux-Moutard transformations and Poincare-Steklov operators
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Added by
Iskander A. Taimanov
Publication date
2018
fields
Physics
and research's language is
English
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Formulas relating Poincare-Steklov operators for Schroedinger equations related by Darboux-Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary.
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We construct a Darboux transformation for a class of two-dimensional Dirac systems at zero energy. Our starting equation features a position-dependent mass, a matrix potential, and an additional degree of freedom that can be interpreted either as a magnetic field perpendicular to the plane or a generalized Dirac oscillator interaction. We obtain a number of Darbouxtransformed Dirac equations for which the zero energy solutions are exactly known.
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P.G. Grinevich L.D. Landau Institute forn Theoretical Physics
, Lomonosov Moscow State University
2015
We construct a Moutard-type transform for the generalized analytic functions. The first theorems and the first explicit examples in this connection are given.
We consider the Steklov zeta function $zeta$ $Omega$ of a smooth bounded simply connected planar domain $Omega$ $subset$ R 2 of perimeter 2$pi$. We provide a first variation formula for $zeta$ $Omega$ under a smooth deformation of the domain. On the base of the formula, we prove that, for every s $in$ (--1, 0) $cup$ (0, 1), the difference $zeta$ $Omega$ (s) -- 2$zeta$ R (s) is non-negative and is equal to zero if and only if $Omega$ is a round disk ($zeta$ R is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality $zeta$ $Omega$ (s) -- 2$zeta$ R (s) $ge$ 0 for s $in$ (--$infty$, --1] $cup$ (1, $infty$); the latter fact was proved in our previous paper [2018] in a different way. We also provide an alternative proof of the equality $zeta$ $Omega$ (0) = 2$zeta$ R (0) obtained by Edward and Wu [1991].
We derive a permutability theorem for the Christoffel, Goursat and Darboux transformations of isothermic surfaces. As a consequence we obtain a simple proof of a relation between Darboux pairs of minimal surfaces in Euclidean space, curved flats in the 2-sphere and flat fronts in hyperbolic space.
We consider the zeta function $zeta_Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $Omega$ bounded by a smooth closed curve of perimeter $2pi$. We prove that $zeta_Omega(0)ge zeta_{mathbb{D}}(0)$ with equality if and only if $Omega$ is a disk where $mathbb{D}$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $sle-1$ the estimate $zeta_Omega(s)ge zeta_{mathbb{D}}(s)$ holds with equality if and only if $Omega$ is a disk. We then bring examples of domains $Omega$ close to the unit disk where this estimate fails to be extended to the interval $(0,2)$. Other computations related to previous works are also detailed in the remaining part of the text.
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