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Register a new userBand edge limit of the scattering matrix for quasi-one-dimensional discrete Schrodinger operators
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Added by
Hermann Schulz-Baldes
Publication date
2020
fields
Physics
and research's language is
English
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This paper is about the scattering theory for one-dimensional matrix Schrodinger operators with a matrix potential having a finite first moment. The transmission coefficients are analytically continued and extended to the band edges. An explicit expression is given for these extensions. The limits of the reflection coefficients at the band edges is also calculated.
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Explicit formulas for the analytic extensions of the scattering matrix and the time delay of a quasi-one-dimensional discrete Schrodinger operator with a potential of finite support are derived. This includes a careful analysis of the band edge singularities and allows to prove a Levinson-type theorem. The main algebraic tool are the plane wave transfer matrices.
For the solution $q(t)=(q_n(t))_{ninmathbb Z}$ to one-dimensional discrete Schrodinger equation $${rm i}dot{q}_n=-(q_{n+1}+q_{n-1})+ V(theta+nomega) q_n, quad ninmathbb Z,$$ with $omegainmathbb R^d$ Diophantine, and $V$ a small real-analytic function on $mathbb T^d$, we consider the growth rate of the diffusion norm $|q(t)|_{D}:=left(sum_{n}n^2|q_n(t)|^2right)^{frac12}$ for any non-zero $q(0)$ with $|q(0)|_{D}<infty$. We prove that $|q(t)|_{D}$ grows {it linearly} with the time $t$ for any $thetainmathbb T^d$ if $V$ is sufficiently small.
We demonstrate how the Moutard transformation of two-dimensional Schrodinger operators acts on the Faddeev eigenfunctions on the zero energy level and present some explicitly computed examples of such eigenfunctions for smooth fast decaying potentials of operators with non-trivial kernel and for deformed potentials which correspond to blowing up solutions of the Novikov-Veselov equation.
We study the spectrum and dynamics of a one-dimensional discrete Dirac operator in a random potential obtained by damping an i.i.d. environment with an envelope of type $n^{-alpha}$ for $alpha>0$. We recover all the spectral regimes previously obtained for the analogue Anderson model in a random decaying potential, namely: absolutely continuous spectrum in the super-critical region $alpha>frac12$; a transition from pure point to singular continuous spectrum in the critical region $alpha=frac12$; and pure point spectrum in the sub-critical region $alpha<frac12$. From the dynamical point of view, delocalization in the super-critical region follows from the RAGE theorem. In the critical region, we exhibit a simple argument based on lower bounds on eigenfunctions showing that no dynamical localization can occur even in the presence of point spectrum. Finally, we show dynamical localization in the sub-critical region by means of the fractional moments method and provide control on the eigenfunctions.
We consider Schrodinger operators on [0,infty) with compactly supported, possibly complex-valued potentials in L^1([0,infty)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances determines the potential uniquely. From the physical point of view one expects that large resonances are increasingly insignificant for the reconstruction of the potential from the data. In this paper we prove the validity of this statement, i.e., we show conditional stability for finite data. As a by-product we also obtain a uniqueness result for the inverse resonance problem for complex-valued potentials.
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