Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1807.06895

Darboux transformations and second order difference equations

Alina Dobrogowska·David J. Fernández C
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we implement the Darboux transformation, as well as an analogue of Crum's theorem, for a discrete version of Schrödinger equation. The technique is based on the use of first order operators intertwining two difference operators of second order. This method, which has been applied successfully for differential cases, leads also to interesting non trivial results in the discrete case. The technique allows us to construct the solutions for a wide class of difference Schrödinger equations. The exact solutions for some special potentials are also found explicitly.

Explore related subjects

math.DSmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alina Dobrogowska, David J. Fernández C. 2018-07-18. Darboux transformations and second order difference equations. https://arxiv.org/abs/1807.06895

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Amenable graphs and the spectral radius of extensions of Markov maps

We discuss relations between the amenability of a graph and spectral properties of a random walk driven by a dynamical system. In order to include graphs which are not locally compact, we introduce the concept of amenability of weighted graphs, which generalises the usual notion as the new definition is shown to be equivalent to Folner's condition. As a first result, we obtain the following generalisation of Kesten's amenability criterion to graphs and non-independent increments: If the random walk is driven by a full-branched Gibbs-Markov map, the graph is amenable with respect to the weight induced by the random walk if and only if the spectral radius of the associated Markov operator is equal to one. By employing inducing schemes, one then obtains criteria for amenability through Markov maps with less regularity. We conclude the paper with the following applications to Schreier graphs. If the random walk is driven by a uniformly expanding map with non-Markovian increments or a Sinai billiard, then, under certain conditions, the Schreier graph is amenable if the probability of a return in time n does not decay exponentially in n. Furthermore, in the context of geometrically finite Kleinian groups, one obtains a version of Brooks's amenability criterion for not necessarily normal subgroups.

math.DS↗
arXiv2026

Measures of maximal entropy for $C^\infty$ three-dimensional flows

We prove that every $C^\infty$ non-singular flow with positive entropy on a compact three-dimensional manifold without boundary admits finitely many ergodic measures of maximal entropy. This result extends the notable work of Buzzi-Crovisier-Sarig (\emph{Ann. of Math.}, 2022) on surface diffeomorphisms. Our approach differs by addressing the continuity of Lyapunov exponents and the uniform largeness of Pesin sets for measures of maximal entropy. Furthermore, it provides an alternative proof for the case of surface diffeomorphisms.

math.DS↗
arXiv2026

The role of coupling and timescales for interacting tipping elements

Sudden and abrupt changes can occur in a nonlinear system within many fields of science when such a system crosses a tipping point; rapid changes of the system can then occur in response to slow changes in an external forcing. These can occur when time-varying inputs cross a bifurcation. If an upstream system loses stability in this way, it may cause a downstream system influenced by it to tip. This especially happens if the downstream system evolves on a much faster timescale, in what we call an accelerating cascade of tipping elements. In this paper, we identify the conditions on coupling and timescales of these systems that result in such tipping cascades and suggest a taxonomy to classify the different possible timings of tipping sequences. We also present a prototypical example of a unidirectionally coupled pair of simple tipping elements with hysteresis. This allows us to map out the various types of response as a function of system parameters and to link it to bifurcations of the underlying system that may have multiple timescales.

math.DS↗