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Ilya Kachkovskiy

Publications and source records attributed to Ilya Kachkovskiy.

At least 19 recordsLinked to original sources
arXiv2026

On almost commuting matrices with respect to the normalized Hilbert--Schmidt norm

In this paper, we consider the Rosenthal -- Halmos problem of almost commuting matrices with respect to the normalized Hilbert -- Schmidt norm $\|\cdot\|_{2,d}=d^{-1/2}\|\cdot\|_2$. We show that if $X$ and $Y$ are self-adjoint matrices with $\|X\|\le 1$, $\|Y\|\le 1$, then there exist commuting self-adjoint matrices $X',Y'$ such that $\|X-X'\|_{2,d}+\|Y-Y'\|_{2,d}\le 5\|[X,Y]\|_{2,d}^{1/3}$, and $[X,X']=0$. Within these constraints, the exponent $1/3$ cannot be improved.

math.SP↗
arXiv2026

On almost commuting unitary matrices

A question going back to Halmos asks when two approximately commuting matrices of a certain kind are close to exactly commuting matrices of the same kind. It has long been known that there is a winding number obstruction for approximately commuting unitary matrices to be close, in a dimension-independent way, to genuinely commuting unitary matrices. In this paper, under the vanishing of the said obstruction, we obtain effective bounds for the distance to commuting unitary matrices in terms of the commutator of the original matrices.

math.OA↗
arXiv2025

Perturbative diagonalization and spectral gaps of quasiperiodic operators on $\ell^2(\mathbb Z^d)$ with monotone potentials

We obtain a perturbative proof of localization for quasiperiodic operators on $\ell^2(\Z^d)$ with one-dimensional phase space and monotone sampling functions, in the regime of small hopping. The proof is based on an iterative scheme which can be considered as a local (in the energy and the phase) and convergent version of KAM-type diagonalization, whose result is a covariant family of uniformly localized eigenvalues and eigenvectors. We also proof that the spectra of such operators contain infinitely many gaps.

math.SP↗
arXiv2025

On gaps in the spectra of quasiperiodic Schrödinger operators with discontinuous monotone potentials

We show that, for one-dimensional discrete Schrödinger operators, stability of Anderson localization under a class of rank one perturbations implies absence of intervals in spectra. The argument is based on well-known result of Gordon and del Rio--Makarov--Simon, combined with a way to consider perturbations whose ranges are not necessarily cyclic. The main application of the results is showing that a class of quasiperiodic operators with sawtooth-like potentials, for which such a version of stable localization is known, has Cantor spectra. We also obtain several results on gap filling under rank one perturbations for some general (not necessarily monotone) classes of quasiperiodic operators with discontinuous potentials.

math.SP↗
arXiv2024

Sharp arithmetic delocalization for quasiperiodic operators with potentials of semi-bounded variation

We obtain the sharp arithmetic Gordon's theorem: that is, absence of eigenvalues on the set of energies with Lyapunov exponent bounded by the exponential rate of approximation of frequency by the rationals, for a large class of one-dimensional quasiperiodic Schrödinger operators, with no (modulus of) continuity required. The class includes all unbounded monotone potentials with finite Lyapunov exponents and all potentials of bounded variation. The main tool is a new uniform upper bound on iterates of cocycles of bounded variation.

math.SP↗
arXiv2022

Convergence of perturbation series for unbounded monotone quasiperiodic operators

We consider a class of unbounded quasiperiodic Schrödinger-type operators on $\ell^2(\mathbb Z^d)$ with monotone potentials (akin to the Maryland model) and show that the Rayleigh--Schrödinger perturbation series for these operators converges in the regime of small kinetic energies, uniformly in the spectrum. As a consequence, we obtain a new proof of Anderson localization in a more general than before class of such operators, with explicit convergent series expansions for eigenvalues and eigenvectors. This result can be restricted to an energy window if the potential is only locally monotone and one-to-one. A modification of this approach also allows the potential to be non-strictly monotone and have a flat segment, under additional restrictions on the frequency.

math.SP↗
arXiv2022

On spectral bands of discrete periodic operators

We consider discrete periodic operator on $\mathbb Z^d$ with respect to lattices $Γ\subset\mathbb Z^d$ of full rank. We describe the class of lattices $Γ$ for which the operator may have a spectral gap for arbitrarily small potentials. We also show that, for a large class of lattices, the dimensions of the level sets of spectral band functions at the band edges do not exceed $d-2$.

math.SP↗
arXiv2021

Perturbative diagonalisation for Maryland-type quasiperiodic operators with flat pieces

We consider quasiperiodic operators on $\mathbb Z^d$ with unbounded monotone sampling functions ("Maryland-type"), which are not required to be strictly monotone and are allowed to have flat segments. Under several geometric conditions on the frequencies, lengths of the segments, and their positions, we show that these operators enjoy Anderson localization at large disorder.

math.SP↗
arXiv2020

Ballistic transport for one-dimensional quasiperiodic Schrödinger operators

In this paper, we show that one-dimensional discrete multi-frequency quasiperiodic Schrödinger operators with smooth potentials demonstrate ballistic motion on the set of energies on which the corresponding Schrödinger cocycles are smoothly reducible to constant rotations. The proof is performed by establishing a local version of strong ballistic transport on an exhausting sequence of subsets on which reducibility can be achieved by a conjugation uniformly bounded in the $\mathrm{C}^{\ell}$-norm. We also establish global strong ballistic transport under an additional integral condition on the norms of conjugation matrices. The latter condition is quite mild and is satisfied in many known examples.

math-ph↗
arXiv2020

On the relation between strong ballistic transport and exponential dynamical localization

We establish strong ballistic transport for a family of discrete quasiperiodic Schrödinger operators as a consequence of exponential dynamical localization for the dual family. The latter has been, essentially, shown by Jitomirskaya and Krüger in the one-frequency setting and by Ge--You--Zhou in the multi-frequency case. In both regimes, we obtain strong convergence of $\frac{1}{T}X(T)$ to the asymptotic velocity operator $Q$, which improves recent perturbative results by Zhao and provides the strongest known form of ballistic motion. In the one-frequency setting, this approach allows to treat Diophantine frequencies non-perturbatively and also consider the weakly Liouville case.

math.SP↗
arXiv2018

Anderson localization for two interacting quasiperiodic particles

We consider a system of two discrete quasiperiodic 1D particles as an operator on $\ell^2(\mathbb Z^2)$ and establish Anderson localization at large disorder, assuming the potential has no cosine-type symmetries. In the presence of symmetries, we show localization outside of a neighborhood of finitely many energies. One can also add a deterministic background potential of low complexity, which includes periodic backgrounds and finite range interaction potentials. Such background potentials can only take finitely many values, and the excluded energies in the symmetric case are associated to those values.

math.SP↗
arXiv2015

All couplings localization for quasiperiodic operators with Lipschitz monotone potentials

We establish Anderson localization for quasiperiodic operator families of the form $$ (H(x)ψ)(m)=ψ(m+1)+ψ(m-1)+λv(x+mα)ψ(m) $$ for all $λ>0$ and all Diophantine $α$, provided that $v$ is a $1$-periodic function satisfying a Lipschitz monotonicity condition on $[0,1)$. The localization is uniform on any energy interval on which Lyapunov exponent is bounded from below.

math.SP↗
arXiv2015

$L^2$-reducibility and localization for quasiperiodic operators

We give a simple argument that if a quasiperiodic multi-frequency Schrödinger cocycle is reducible to a constant rotation for almost all energies with respect to the density of states measure, then the spectrum of the dual operator is purely point for Lebesgue almost all values of the ergodic parameter $θ$. The result holds in the $L^2$ setting provided, in addition, that the conjugation preserves the fibered rotation number. Corollaries include localization for (long-range) 1D analytic potentials with dual ac spectrum and Diophantine frequency as well as a new result on multidimensional localization.

math.SP↗
arXiv2015

On transport properties of isotropic quasiperiodic $XY$ spin chains

We 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 $XY$ spin chains, we obtain lower bounds on their Lieb--Robinson velocities in terms of group velocities of their effective Hamiltonians: $\mathfrak v\ge\mathrm{ess\,sup}_{[0,1]} \frac{2}π\frac{dE}{dN},$ where $E$ is considered as a function of the integrated density of states.

math-ph↗