arXiv subjects

Tom VandenBoom

Publications and source records attributed to Tom VandenBoom.

7 recordsLinked to original sources

arXiv2021

Localization and Cantor spectrum for quasiperiodic discrete Schrödinger operators with asymmetric, smooth, cosine-like sampling functions

We prove Cantor spectrum and almost-sure Anderson localization for quasiperiodic discrete Schrödinger operators $H = \varepsilonΔ+ V$ with potential $V$ sampled with Diophantine frequency $α$ from an asymmetric, smooth, cosine-like function $v \in C^2(\mathbb{T},[-1,1])$ for sufficiently small interaction $\varepsilon \leq \varepsilon_0(v,α)$. We prove this result via an inductive analysis on scales, whereby we show that locally the Rellich functions of Dirichlet restrictions of $H$ inherit the cosine-like structure of $v$ and are uniformly well-separated.

math.SP↗

arXiv2019

Finite-gap CMV matrices: Periodic coordinates and a Magic Formula

We prove a bijective unitary correspondence between 1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum and 2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as spectrally-dependent operator Möbius transforms of certain generating CMV matrices which are periodic up to a rotational phase; for this reason we call them "MCMV". Such matrices are related to a choice of orthogonal rational functions on the unit circle, and their correspondence to the isospectral torus follows from a functional model in analog to that of GMP matrices. As a corollary of our construction we resolve a conjecture of Simon; namely, that Caratheodory functions associated to such CMV matrices arise as quadratic irrationalities.

math.SP↗

arXiv2019

Positive Lyapunov Exponents and a Large Deviation Theorem for Continuum Anderson Models, Briefly

In this short note, we prove positivity of the Lyapunov exponent for 1D continuum Anderson models by leveraging some classical tools from inverse spectral theory. The argument is much simpler than the existing proof due to Damanik--Sims--Stolz, and it covers a wider variety of random models. Along the way we note that a Large Deviation Theorem holds uniformly on compacts.

math.SP↗

arXiv2018

KdV hierarchy via Abelian coverings and operator identities

We establish precise spectral criteria for potential functions $V$ of reflectionless Schrödinger operators $L_V = -\partial_x^2 + V$ to admit solutions to the Korteweg de-Vries (KdV) hierarchy with $V$ as an initial value. More generally, our methods extend the classical study of algebro-geometric solutions for the KdV hierarchy to noncompact Riemann surfaces by defining generalized Abelian integrals and analogues of the Baker-Akhiezer function on infinitely connected domains with a uniformly thick boundary satisfying a fractional moment condition.

math.SP↗

arXiv2017

Reflectionless discrete Schrödinger operators are spectrally atypical

We prove that, if an isospectral torus contains a discrete Schrödinger operator with nonconstant potential, the shift dynamics on that torus cannot be minimal. Consequently, we specify a generic sense in which finite unions of nondegenerate closed intervals having capacity one are not the spectrum of any reflectionless discrete Schrödinger operator. We also show that the only reflectionless discrete Schrödinger operators having zero, one, or two spectral gaps are periodic.

math.SP↗

arXiv2017

Almost Periodicity in Time of Solutions of the Toda Lattice

We study an initial value problem for the Toda lattice with almost periodic initial data. We consider initial data for which the associated Jacobi operator is absolutely continuous and has a spectrum satisfying a Craig-type condition, and show the boundedness and almost periodicity in time and space of solutions.

math.SP↗

arXiv2017

Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent

We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg's theorem. That is, a Schrödinger operator in $\ell^2(\mathbb{Z})$ whose potential is given by independent identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions and its unitary group exhibits exponential off-diagonal decay, uniformly in time. This is achieved by way of a new result: for the Anderson model, one typically has Lyapunov behavior for all generalized eigenfunctions. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogs of these models.

math-ph↗