We take a closer look at the Riemann-Hilbert problem associated to one-gap solutions of the Korteweg-de Vries equation. To gain more insight, we reformulate it as a scalar Riemann-Hilbert problem on the torus. This enables us to derive deductively th
e model vector-valued and singular matrix-valued solutions in terms of Jacobi theta functions. We compare our results with those obtained in recent literature.
We study perturbations of the self-adjoint periodic Sturm--Liouville operator [ A_0 = frac{1}{r_0}left(-frac{mathrm d}{mathrm dx} p_0 frac{mathrm d}{mathrm dx} + q_0right) ] and conclude under $L^1$-assumptions on the differences of the coefficient
s that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.
We derive dispersion estimates for solutions of a one-dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. To this end we also provide new results concerning
scattering for the corresponding perturbed Dirac operators which are of independent interest. Most notably, we show that the reflection and transmission coefficients belong to the Wiener algebra.
We derive a dispersion estimate for one-dimensional perturbed radial Schrodinger operators where the angular momentum takes the critical value $l=-frac{1}{2}$. We also derive several new estimates for solutions of the underlying differential equation
and investigate the behavior of the Jost function near the edge of the continuous spectrum.
We prove that a solution of the Toda lattice cannot decay too fast at two different times unless it is trivial. In fact, we establish this result for the entire Toda and Kac-van Moerbeke hierarchies.
We prove that a solution of the Schrodinger-type equation $mathrm{i}partial_t u= Hu$, where $H$ is a Jacobi operator with asymptotically constant coefficients, cannot decay too fast at two different times unless it is trivial.
The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive es
timates for a certain class of Schrodinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrodinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi polynomials to establish some new dispersive estimates. In turn, the optimal dispersive decay estimates lead to new Bernstein-type inequalities.
We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Dirac equation. To this end we develop basic scattering theory and establish a limiting absorption principle for discrete perturbed Dirac operators.
Volatile organic compounds emitted by a human body form a chemical signature capable of providing invaluable information on the physiological status of an individual and, thereby, could serve as signs-of-life for detecting victims after natural or ma
n-made disasters. In this review a database of potential biomarkers of human presence was created on the basis of existing literature reports on volatiles in human breath, skin emanation, blood, and urine. Approximate fluxes of these species from the human body were estimated and used to predict their concentrations in the vicinity of victims. The proposed markers were classified into groups of different potential for victim detection. The major classification discriminants were the capability of detection by portable, real-time analytical instruments and background levels in urban environment. The data summarized in this review are intended to assist studies on the detection of humans via chemical analysis and accelerate investigations in this area of knowledge.
We consider the stability of the periodic Toda lattice (and slightly more generally of the algebro-geometric finite-gap lattice) under a short range perturbation. We prove that the perturbed lattice asymptotically approaches a modulated lattice. Mo
re precisely, let $g$ be the genus of the hyperelliptic curve associated with the unperturbed solution. We show that, apart from the phenomenon of the solitons travelling on the quasi-periodic background, the $n/t$-pane contains $g+2$ areas where the perturbed solution is close to a finite-gap solution in the same isospectral torus. In between there are $g+1$ regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free lattice ($g=0$) the isospectral torus consists of just one point and we recover the known result. Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve. Our method relies on the equivalence of the inverse spectral problem to a matrix Riemann--Hilbert problem defined on the hyperelliptic curve and generalizes the so-called nonlinear stationary phase/steepest descent method for Riemann--Hilbert problem deformations to Riemann surfaces.
