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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 $cq_n^{-varkappa/2}$, $c>0$.
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 $var
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 potentia
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 s
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 d
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 partic