Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNon-periodic one-gap potentials in quantum mechanics
75
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We describe a broad class of bounded non-periodic potentials in one-dimensional stationary quantum mechanics having the same spectral properties as periodic potentials. The spectrum of the corresponding Schroedinger operator consists of a finite or infinite number of allowed bands separated by gaps. In this letter we consider the simplest class of potentials, whose spectra consist of an interval on the negative semiaxis and the entire positive axis. The potentials are reflectionless, and a particle with positive energy moves freely in both directions. The potential is constructed as a limit of Bargmann potentials and is determined by a Riemann-Hilbert problem, which is equivalent to a pair of singular integral equations that can be efficiently solved using numerical techniques.
rate research
Read More
Kahan discretization is applicable to any system of ordinary differential equations on $mathbb R^n$ with a quadratic vector field, $dot{x}=f(x)=Q(x)+Bx+c$, and produces a birational map $xmapsto widetilde{x}$ according to the formula $(widetilde{x}-x)/epsilon=Q(x,widetilde{x})+B(x+widetilde{x})/2+c$, where $Q(x,widetilde{x})$ is the symmetric bilinear form corresponding to the quadratic form $Q(x)$. When applied to integrable systems, Kahan discretization preserves integrability much more frequently than one would expect a priori, however not always. We show that in some cases where the original recipe fails to preserve integrability, one can adjust coefficients of the Kahan discretization to ensure its integrability.
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the periodic (and slightly more generally of the quasi-periodic finite-gap) Toda lattice for decaying initial data in the soliton region. In addition, we show how to reduce the problem in the remaining region to the known case without solitons.
In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre Unitary ensemble, from the point of view of Sakais geometric theory of Painleve equations. On one hand, this gives us one more detailed example of the appearance of discrete Painleve equations in the theory of orthogonal polynomials. On the other hand, it serves as a good illustration of the effectiveness of a recently proposed procedure on how to reduce such recurrences to some canonical discrete Painleve equations.
In a recent paper we have considered the long time asymptotics of the periodic Toda lattice under a short range perturbation and we have proved that the perturbed lattice asymptotically approaches a modulated lattice. In the present paper we capture the higher order asymptotics, at least away from some resonance regions. In particular we prove that the decay rate is $O(t^{-1/2})$. Our proof relies on the asymptotic analysis of the associated Riemann-Hilbert factorization problem, which is here set on a hyperelliptic curve. As in previous studies of the free Toda lattice, the higher order asymptotics arise from local Riemann-Hilbert factorization problems on small crosses centered on the stationary phase points. We discover that the analysis of such a local problem can be done in a chart around each stationary phase point and reduces to a Riemann--Hilbert factorization problem on the complex plane. This result can then be pulled back to the hyperelliptic curve.
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. More 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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
