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The paper is devoted to the following question: consider two self-adjoint $n\times n$-matrices $H_1,H_2$, $\|H_1\|\le 1$, $\|H_2\|\le 1$, such that their commutator $[H_1,H_2]$ is small in some… (More)

We give a simple argument that if a quasiperiodic multi-frequency Schrodinger cocycle is reducible to a constant rotation for almost all energies with respect to the density of states measure, then… (More)

For a wide class of 2D periodic elliptic operators, we show that the global extrema of all spectral band functions are isolated.

- Ilya Kachkovskiy
- 2016

AbstractWe consider isotropic XY spin chains whose magnetic potentials are quasiperiodic and the effective one-particle Hamiltonians have absolutely continuous spectra. For a wide class of such XY… (More)

We establish Anderson localization for quasiperiodic operator families of the form
$$
(H(x)\psi)(m)=\psi(m+1)+\psi(m-1)+\lambda v(x+m\alpha)\psi(m)
$$ for all $\lambda>0$ and all Diophantine… (More)

We obtain an order sharp estimate for the distance from a given bounded operator $A$ on a Hilbert space to the set of normal operators in terms of $\|[A,A^*]\|$ and the distance to the set of… (More)

The functional calculus for normal elements in $C^*$-algebras is an important tool of analysis. We consider polynomials $p(a,a^*)$ for elements $a$ with small self-commutator norm $\|[a,a^*]\| \le… (More)

- Ilya Kachkovskiy
- 2013

The purpose of this thesis is to study the following problem. Suppose that X, Y are bounded self-adjoint operators in a Hilbert space H with their commutator [X, Y ] being small. Such operators are… (More)

- Ilya Kachkovskiy
- 2013

We consider the periodic Schrödinger operator on a d-dimensional cylinder with rectangular section. The electric potential may contain a singular component of the form σ(x, y)δΣ(x,y), where Σ is a… (More)