On the spectrum of a multidimensional periodic magnetic Shrödinger operator with a singular electric potential

We prove absolute continuity of the spectrum of a periodic $n$-dimensional Schrödinger operator for $n\geqslant 4$. Certain conditions on the magnetic potential $A$ and the electric potential $V+\sum f_j\delta_{S_j}$ are supposed to be fulfilled. In particular, we can assume that the following conditions are satisfied. (1) The magnetic potential $A\colon{\mathbb{R}}^n\to{\mathbb{R}}^n$ either has an absolutely convergent Fourier series or belongs to the space $H^q_{\mathrm{loc}}({\mathbb{R}}^n;{\mathbb{R}}^n)$, $2q>n-1$, or to the space $C({\mathbb{R}}^n;{\mathbb{R}}^n)\cap H^q_{\mathrm{loc}}({\mathbb{R}}^n;{\mathbb{R}}^n)$, $2q>n-2$. (2) The function $V\colon{\mathbb{R}}^n\to\mathbb{R}$ belongs to Morrey space ${\mathfrak{L}}^{2,p}$, $p\in \big(\frac{n-1}{2},\frac{n}{2}\big]$, of periodic functions (with a given period lattice), and $$\lim\limits_{\tau\to+0}\sup\limits_{0<r\leqslant\tau}\sup\limits_{x\in{\mathbb{R}}^n}r^2\bigg(\big(v(B^n_r)\big)^{-1}\int_{B^n_r(x)}|{\mathcal{V}}(y)|^pdy\bigg)^{1/p}\leqslant C,$$ where $B^n_r(x)$ is a closed ball of radius $r>0$ centered at a point $x\in{\mathbb{R}}^n$, $B^n_r=B^n_r(0)$, $v(B^n_r)$ is volume of the ball $B^n_r$, $C=C(n,p;A)>0$. (3) $\delta_{S_j}$ are $\delta$-functions concentrated on (piecewise) $C^1$-smooth periodic hypersurfaces $S_j$, $f_j\in L^p_{\mathrm{loc}}(S_j)$, $j=1,\ldots,m$. Some additional geometric conditions are imposed on the hypersurfaces $S_j$, and these conditions determine the choice of numbers $p\geqslant n-1$. In particular, let hypersurfaces $S_j$ be $C^2$-smooth, the unit vector $e$ be arbitrarily taken from some dense set of the unit sphere $S^{n-1}$ dependent on the magnetic potential $A$, and the normal curvature of the hypersurfaces $S_j$ in the direction of the unit vector $e$ be nonzero at all points of tangency of the hypersurfaces $S_j$ and the lines $\{x_0+te\colon t\in\mathbb{R}\}$, $x_0\in{\mathbb{R}}^n$. Then we can choose the number $p>\frac{3n}{2}-3$, $n\geqslant 4$.

Dilations of $q$-Commuting Unitaries

Let $q = e^{i \theta } \in \mathbb{T}$ (where $\theta \in \mathbb{R}$), and let $u,v$ be $q$-commuting unitaries, that is, $u$ and $v$ are unitaries such that $vu = quv$. In this paper, we find the optimal constant $c = c_{\theta }$ such that $u,v$ can be dilated to a pair of operators $c U, c V$, where $U$ and $V$ are commuting unitaries. We show that $$\begin{equation*} c_{\theta} = \frac{4}{\|u_{\theta}+u_{\theta}^*+v_{\theta}+v_{\theta}^*\|}, \end{equation*}$$where $u_{\theta }, v_{\theta }$ are the universal $q$-commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating $q$-commuting unitaries to scalar multiples of $q^{\prime}$-commuting unitaries. The techniques that we develop allow us to give new and simple “dilation theoretic” proofs of well-known results regarding the continuity of the field of rotations algebras. In particular, for the so-called “almost Mathieu operator” $h_{\theta } = u_{\theta }+u_{\theta }^*+v_{\theta }+v_{\theta }^*$, we recover the fact that the norm $\|h_{\theta }\|$ is a Lipschitz continuous function of $\theta $, as well as the result that the spectrum $\sigma (h_{\theta })$ is a $\frac{1}{2}$-Hölder continuous function in $\theta $ with respect to the Hausdorff metric. In fact, we obtain this Hölder continuity of the spectrum for every self-adjoint *-polynomial $p(u_{\theta },v_{\theta })$, which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.

Path dependence and evolution of Russia’s territorial production systems

The article examines the influence of natural history and socio-cultural processes on the formation of industrial areas within the borders of modern Russia and partially the Russian Empire and the USSR. Basically, the evolutionary approach is used in the version of cyclic genetic dynamics. To assess the long-term effects of the “path dependence”, we analyze the 150-year cycle of industrialization within the borders of Historical Russia. To do this, we use the mechanisms of spatial localization of Kondratiev cycles and associated technological structures of the first (first-second), second (third-fourth)) and the third (fifth) industrial revolution. As an information base, data on the number of people employed and the volume of industrial production (in comparable prices) are used, and for the XX century, data on the dynamics of fixed assets of industry are also used. The relationship between modernization schemes on the old basis (within the boundaries of old industrial areas) and in the areas of pioneer development is revealed. The stability of industry specializations within the grid of industrial areas is considered separately. Based on the results obtained, the conclusion is made about the stability of old industrial cores of localization of industrial potential, the continuity of the spectrum of their specializations (not the replacement of old industries with new ones, but the addition). There is also a repetition in the regions of the new development of the General scheme of the chosen path and the inherited track.

Some results about spectral continuity and compact perturbations

In this paper, we are interested in the continuity of the spectrum and some of its parts in the setting of Hilbert spaces. We study the continuity of the spectrum in the class of operators {T}+K(H), where K(H) denote the ideal of compact operators. Also, we give conditions in order to transfer the continuity of spectrum from T to T + K, where K ? K(H). Then, we characterize those operators for which the continuity of spectrum is stable under compact perturbations.

Decay rates at infinity for solutions to periodic Schrödinger equations

We consider the equation Δu = Vu in the half-space ${\open R}_ + ^d $, d ⩾ 2 where V has certain periodicity properties. In particular, we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schrödinger operators. The equation Δu = Vu is studied as part of a broader class of elliptic evolution equations.

Continuity of the spectrum of quasi-periodic Schrödinger operators with finitely differentiable potentials

In this paper, we consider the spectrum of discrete quasi-periodic Schrödinger operators on $\ell ^{2}(\mathbb{Z})$ with the potentials $v\in C^{k}(\mathbb{T})$. For sufficiently large $k$, we show that the Lebesgue measure of the spectrum at irrational frequencies is the limit of the Lebesgue measure of the spectrum of its periodic approximants. This gives a partial answer to the problem proposed in Jitomirskaya and Mavi [Continuity of the measure of the spectrum for quasiperiodic schrödinger operator with rough potentials. Comm. Math. Phys.325 (2014), 585–601]. Our results are based on a generalization of the rigidity theorem in Avila and Krikorian [Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. of Math. (2)164 (2006), 911–940]; more precisely, we prove that in the $C^{k}$ case, for almost every frequency $\unicode[STIX]{x1D6FC}\in \mathbb{R}\backslash \mathbb{Q}$ and for almost every $E$, the corresponding quasi-periodic Schrödinger cocycles are either reducible or non-uniformly hyperbolic.