Related papers: On quantum ergodicity for linear maps of the torus

We previously introduced a family of symplectic maps of the torus whose quantization exhibits scarring on invariant co-isotropic submanifolds. The purpose of this note is to show that in contrast to other examples, where failure of Quantum…

We prove quantum unique ergodicity for a subspace of the continuous spectrum spanned by the degenerate Eisenstein Series on GL(n).

We state a theorem relating the ergodicity of the action of a given subgroup of the mapping class group of a surface on the character variety, to the asymptotic of its invariant subspaces through the Witten-Reshetikhin-Turaev…

We show that on a suitable time scale, logarithmic in $\hbar$, the coherent states on the two-torus, evolved under a quantized perturbed hyperbolic toral automorphism, equidistribute on the torus. We then use this result to obtain control…

We analyze the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed, deterministic phase. The density of roots is shown to possess a singular component only in the case for…

We prove that if $E$ is a compact subset of the unit disk ${\mathbb D}$ in the complex plane, if $E$ contains a sequence of distinct points $a_n\not= 0$ for $n\geq 1$ such that $\lim_{n\to\infty} a_n=0$ and for all $n$ we have $ |a_{n+1}|…

We prove a quantitative statement of the quantum ergodicity for Hecke--Maass cusp forms on the modular surface. As an application of our result, along a density $1$ subsequence of even Hecke--Maass cusp forms, we obtain a sharp lower bound…

Quantum ergodicity, which expresses the semiclassical convergence of almost all expectation values of observables in eigenstates of the quantum Hamiltonian to the corresponding classical microcanonical average, is proven for…

In this paper, we investigate capacity preserving transformations and their ergodicity. We show that for any measurable transformation $\theta$ there always exists a $\theta$-invariant capacity. We investigate some limit properties under…

For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to…

We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers $L^N$ of a positive holomorphic Hermitian line bundle $L$ over a compact complex manifold $M$. Our first result concerns `random' sequences…

In the customary random matrix model for transport in quantum dots with $M$ internal degrees of freedom coupled to a chaotic environment via $N\ll M$ channels, the density $\rho$ of transmission eigenvalues is computed from a specific…

This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results…

We prove the arithmetic quantum unique ergodicity (AQUE) conjecture for non-degenerate sequences of Hecke eigenfunctions on quotients $\Gamma \backslash G/K$, where $G\simeq\mathrm{PGL}_{d}(\mathbb{R})$, $K$ is a maximal compact subgroup of…

For any accessible partially hyperbolic homogeneous flow, we show that all smooth time changes are K and hence mixing of all orders. We also establish stable ergodicity for time-one map of these time changes.

When a granular system is tapped, its volume changes. Here, using a well-defined macroscopic protocol, we prepare an ensemble of granular systems and track the statistics of volume changes as a function of the number of taps. This is in…

The power-law random banded matrices and the ultrametric random matrices are investigated numerically in the regime where eigenstates are extended but all integer matrix moments remain finite in the limit of large matrix dimensions. Though…

In a classically chaotic system that is ergodic, any trajectory will be arbitrarily close to any point of the available phase space after a long time, filling it uniformly. Using Born's rules to connect quantum states with probabilities,…

We study the limiting distributions of expanding translates of a compact segment of a smooth curve under a diagonal subgroup of $G=\mathrm{SO}(n_1,1)\times\cdots\times\mathrm{SO}(n_k,1)$, where $G$ acts on a finite volume homogeneous space…

We investigate the harmonic map heat flow from a compact Riemannian manifold \( M \) into the moduli space \( \mathcal{M}_1 \) of unit-area flat tori, which carries a natural hyperbolic structure as the quotient \( \mathrm{SL}(2,\mathbb{Z})…