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Svetlana Jitomirskaya

Publications and source records attributed to Svetlana Jitomirskaya.

At least 19 recordsLinked to original sources
arXiv2026

(Sharp arithmetic) localization beyond transversality

We prove sharp arithmetic localization for one-dimensional quasiperiodic potentials with jump discontinuities, beyond uniform transversality. Our results include every non-constant $C^1$ monotone sampling function, even with arbitrarily large flat regions, and also certain non-monotone potentials. A key new mechanism is a symbolic unique-continuation principle: first-return words to the region where $f'>0$ have no common Dirichlet resonance, forcing eigenfunction mass to transfer away from flat or negatively sloped regions and recovering quantitative monotonicity of finite-volume eigenvalue branches.

math.SP↗
arXiv2026

Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum

We obtain a representation of the critical almost Mathieu family as a Jacobi matrix that has a singularity. This allows us to prove that the Hausdorff dimension of its spectrum is not larger than 1/2 for all irrational frequencies, solving a long-standing problem. Other corollaries include two very short proofs of zero measure of the spectrum (e.g. Problem 5 in B. Simon's list of the 21'st century problems). We also obtain continuity of the measure of the spectrum for general singular Jacobi matrices, and prove a similar Hausdorff dimension result for the quantum graph graphene.

math.SP↗
arXiv2026

The Robust Ten Martini Problem

We solve the robust ten martini problem: Cantor spectrum with no arithmetic restriction on the irrational frequency for Type I analytic one-frequency quasiperiodic Schrodinger operators. We introduce Type I through a new acceleration-type quantity with stability under analytic perturbations. This yields a global open class of analytic potentials whose associated operators have Cantor spectrum for every irrational frequency.

math-ph↗
arXiv2026

Capacity and measure approximations for Schrödinger operators

We prove that logarithmic capacity convergence for phase-union spectra of quasi-periodic Schrödinger operators in the zero Lyapunov exponent regime is robust, requiring only continuity of the potential. Let $S^+(p/q)$ denote the union, over the phase, of the spectra at rational frequency $p/q$. We show that if the Lyapunov exponent vanishes on the spectrum $Σ(α)$ at an irrational frequency $α$, then for every sequence $p_n/q_n\toα$, the logarithmic capacities $Cap(S^+(p_n/q_n))\longrightarrow Cap(Σ(α)).$ We also prove convergence of the corresponding harmonic measures. As a consequence, the equilibrium measures of $S^+(p_n/q_n)$ converge in the weak$^*$ topology to the density of states measure of the quasi-periodic Schrödinger operator. We extend these results to multi-frequency Schrödinger operators and prove analogous convergence theorems, for logarithmic capacity, harmonic measure, and equilibrium measure, for ergodic Schrödinger operators in a general setting where the almost sure spectrum is approximated in the Hausdorff metric by union spectra of periodic operators. This abstract formulation applies, in particular, to uniformly almost periodic potentials along sequences of almost periods. We also provide counterexamples when the limiting frequency is rational.

math.SP↗
arXiv2026

Intrinsic Symplectic Structure and Sharp Arithmetic Universality

We show that formal eigenvalue equations of analytic one-frequency Schröd-inger operators admit intrinsic analytic $Sp(2k,\C)$ structures, where $k=k(E)$ is the T-acceleration in global theory. For trigonometric potentials those structures govern the center dynamics of partially hyperbolic dual cocycles; for general analytic potentials they persist, without loss of analyticity, as an intrinsic object even when the dual operator has infinite range and no cocycles exist. For $k=1$, we also introduce the concept of projectively real cocycles: complex symplectic systems whose projective action is algebraically conjugate, up to a scalar phase, to that of a real $\SL(2,\R)$ cocycle. This allows us to define a rotation pair and establish a rotation--IDS correspondence in the general analytic setting, where standard dynamical methods fail. Using these tools, we solve two spectral arithmetic conjectures: universality of the sharp arithmetic transition in frequency (AAJ) and of the absolute continuity of the integrated density of states for all frequencies, throughout the class of non-critical Type I operators, an open and conjecturally dense set. We also prove universality of sharp $1/2$-Hölder continuity of the integrated density of states for Type I operators with Diophantine frequencies, establishing part of You's conjecture. These results also provide the first duality-based spectral framework for general analytic potentials, overcoming the symmetry and finite-range restrictions present in previous work.

math.SP↗
arXiv2025

Lower bounds on concentration through Borel transforms and quantitative singularity of spectral measures near the arithmetic transition

We develop tools to study arithmetically induced singular continuous spectrum in the neighborhood of the arithmetic transition in the hyperbolic regime. This leads to first transition-capturing upper bounds on packing and multifractal dimensions of spectral measures. We achieve it through the proof of partial localization of generalized eigenfunctions, another first result of its kind in the singular continuous regime. The proof is based also on a general criterion for lower bounds on concentrations of Borel measures as a corollary of boundary behavior of their Borel-type transforms, that may be of wider use and independent interest.

math.SP↗
arXiv2024

Sharp palindromic criterion for semi-uniform dynamical localization

We develop a sharp palindromic argument for general 1D operators, that proves absence of semi-uniform localization in the regime of exponential symmetry-based resonances. This provides the first examples of operators with dynamical localization but no SULE/SUDL, as well as with nearly uniform distribution of centers of localization in absence of SULE. For the almost Mathieu operators, this also leads to a sharp arithmetic criterion for semi-uniformity of dynamical localization in the Diophantine case.

math-ph↗
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↗
arXiv2023

Multiplicative Jensen's formula and quantitative global theory of one-frequency Schrödinger operators

We introduce the concept of dual Lyapunov exponents, leading to a multiplicative version of the classical Jensen's formula for one-frequency analytic Schrödinger cocycles. This formula, in particular, gives a new proof and a quantitative version of the fundamentals of Avila's global theory \cite{avila}, fully explaining the behavior of complexified Lyapunov exponent through the dynamics of the dual cocycle. In particular, concepts of (sub/super) critical regimes and acceleration are all explained (in a quantitative way) through the duality approach. This leads to a number of powerful spectral and physics applications

math.DS↗
arXiv2023

Kotani theory, Puig's argument, and stability of The Ten Martini Problem

We solve the ten martini problem (Cantor spectrum with no condition on irrational frequencies, previously only established for the almost Mathieu) for a large class of one-frequency quasiperiodic operators, including nonperturbative analytic neighborhoods of several popular explicit families. The proof is based on the structural analysis of dual cocycles as introduced in [35]. As a part of the proof, we develop several general ingredients of independent interest: Kotani theory, for a class of finite-range operators over general minimal underlying dynamics, making the first step towards and providing a partial solution of the Kotani-Simon problem, simplicity of point spectrum for the same class, and the all-frequency version of Puig's argument.

math.SP↗
arXiv2022

Growth of the Wang-Casati-Prosen counter in an integrable billiard

This work is motivated by an article by Wang, Casati, and Prosen [Phys. Rev. E vol. 89, 042918 (2014)] devoted to a study of ergodicity in two-dimensional irrational right-triangular billiards. Numerical results presented there suggest that these billiards are generally not ergodic. However, they become ergodic when the billiard angle is equal to $π/2$ times a Liouvillian irrational, a Liouvillian irrational, a class of irrational numbers which are well approximated by rationals. In particular, Wang et al. study a special integer counter that reflects the irrational contribution to the velocity orientation; they conjecture that this counter is localized in the generic case, but grows in the Liouvillian case. We propose a generalization of the Wang-Casati-Prosen counter: this generalization allows to include rational billiards into consideration. We show that in the case of a $45^{\circ} \!\! : \! 45^{\circ} \!\! : \! 90^{\circ}$ billiard, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al.

nlin.CD↗
arXiv2020

Honeycomb structures in magnetic fields

We consider reduced-dimensionality models of honeycomb lattices in magnetic fields and report results about the spectrum, the density of states, self-similarity, and metal/insulator transitions under disorder. We perform a spectral analysis by which we discover a fractal Cantor spectrum for irrational magnetic flux through a honeycomb, prove the existence of zero energy Dirac cones for each rational flux, obtain an explicit expansion of the density of states near the conical points, and show the existence of mobility edges under Anderson-type disorder. Our results give a precise description of de Haas-van Alphen and Quantum Hall effects, and provide quantitative estimates on transport properties. In particular, our findings explain experimentally observed asymmetry phenomena by going beyond the perfect cone approximation.

math-ph↗