Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCentral spectral gaps of the almost Mathieu operator
139
0
0.0
(
0
)
Authors
I. Krasovsky
Ask ChatGPT about the research
No Arabic abstract
We consider the spectrum of the almost Mathieu operator $H_alpha$ with frequency $alpha$ and in the case of the critical coupling. Let an irrational $alpha$ be such that $|alpha-p_n/q_n|<c q_n^{-varkappa}$, where $p_n/q_n$, $n=1,2,dots$ are the convergents to $alpha$, and $c$, $varkappa$ are positive absolute constants, $varkappa<56$. Assuming certain conditions on the parity of the coefficients of the continued fraction of $alpha$, we show that the central gaps of $H_{p_n/q_n}$, $n=1,2,dots$, are inherited as spectral gaps of $H_alpha$ of length at least $cq_n^{-varkappa/2}$, $c>0$.
rate research
Read More
Let $Gamma$ be an arbitrary $mathbb{Z}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $mathcal{H}_varepsilon$ on $Gamma$ with the action $-varepsilon^{-1}{mathrm{d}^2/mathrm{d} x^2}$ on its edges; here $varepsilon>0$ is a small parameter. Let $minmathbb{N}$. We show that under a proper choice of vertex conditions the spectrum $sigma(mathcal{H}^varepsilon)$ of $mathcal{H}^varepsilon$ has at least $m$ gaps as $varepsilon$ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of $sigma(mathcal{H}^varepsilon)$ as $varepsilonto 0$ can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) $varepsilon$ the precise coincidence of the left endpoints of the first $m$ spectral gaps with predefined numbers.
We consider a 2D Pauli operator with almost periodic field $b$ and electric potential $V$. First, we study the ergodic properties of $H$ and show, in particular, that its discrete spectrum is empty if there exists an almost periodic magnetic potential which generates the magnetic field $b - b_{0}$, $b_{0}$ being the mean value of $b$. Next, we assume that $V = 0$, and investigate the zero modes of $H$. As expected, if $b_{0} eq 0$, then generically $operatorname{dim} operatorname{Ker} H = infty$. If $b_{0} = 0$, then for each $m in {mathbb N} cup { infty }$, we construct almost periodic $b$ such that $operatorname{dim} operatorname{Ker} H = m$. This construction depends strongly on results concerning the asymptotic behavior of Dirichlet series, also obtained in the present article.
This paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set whose boundary carries Dirichlet conditions. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Segal-Bargmann and Hardy spaces associated with the magnetic field.
We develop a Hilbert-space approach to the diffusion process of the Brownian motion in a bounded domain with random jumps from the boundary introduced by Ben-Ari and Pinsky in 2007. The generator of the process is introduced by a diffusion elliptic differential operator in the space of square-integrable functions, subject to non-self-adjoint and non-local boundary conditions expressed through a probability measure on the domain. We obtain an expression for the difference between the resolvent of the operator and that of its Dirichlet realization. We prove that the numerical range is the whole complex plane, despite the fact that the spectrum is purely discrete and is contained in a half-plane. Furthermore, for the class of absolutely continuous probability measures with square-integrable densities we characterise the adjoint operator and prove that the system of root vectors is complete. Finally, under certain assumptions on the densities, we obtain enclosures for the non-real spectrum and find a sufficient condition for the non-zero eigenvalue with the smallest real part to be real. The latter supports the conjecture of Ben-Ari and Pinsky that this eigenvalue is always real.
The spectrum of the non-self-adjoint Zakharov-Shabat operator with periodic potentials is studied, and its explicit dependence on the presence of a semiclassical parameter in the problem is also considered. Several new results are obtained. In particular: (i) it is proved that the resolvent set has two connected components, (ii) new bounds on the location of the Floquet and Dirichlet spectra are obtained, some of which depend explicitly on the value of the semiclassical parameter, (iii) it is proved that the spectrum localizes to a cross in the spectral plane in the semiclassical limit. The results are illustrated by discussing several examples in which the spectrum is computed analytically or numerically.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
