相关论文: Quantum Unique Ergodicity for maps on the torus

We study semi-classical limits of eigenfunctions of a quantized linear hyperbolic automorphism of the torus ("cat map"). For some values of Planck's constant, the spectrum of the quantized map has large degeneracies. Our first goal in this…

We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure…

We prove a quantum ergodic restriction theorem for the Cauchy data of a sequence of quantum ergodic eigenfunctions on a hypersurface $H$ of a Riemannian manifold $(M, g)$. The technique of proof is to use a Rellich type identity to relate…

In this paper we study the stochastic quantization problem on the two dimensional torus and establish ergodicity for the solutions. Furthermore, we prove a characterization of the $\Phi^4_2$ quantum field on the torus in terms of its…

We discuss limit distributions for hitting-time functions of certain exceptional families of asymptotically rare events for ergodic probability preserving transformations. The abstract core is an inducing argument. The latter applies, for…

Quantifying the ergotropy (a.k.a. available energy), namely the maximal amount of energy that can be extracted from a thermally isolated system, is a central problem in quantum thermodynamics. Notably, the same problem has been long studied…

We eliminate the possibility of "escape of mass" for Hecke-Maass forms of large eigenvalue for the modular group. Combined with the work of Lindenstrauss, this establishes the Quantum Unique Ergodicity conjecture of Rudnick and Sarnak for…

Given any compact hyperbolic surface $M$, and a closed geodesic on $M$, we construct of a sequence of quasimodes on $M$ whose microlocal lifts concentrate positive mass on the geodesic. Thus, the Quantum Unique Ergodicity (QUE) property…

We propose an extension of ergodic theory which focuses on the identification of ergodicity in terms of the uniqueness of the invariant measure. We first explain the concept for the doubling maps, which can be analyzed using Fourier…

There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the Eulerian numbers. This result may partially justify a frequent assumption…

A continuous-time quantum walk is modelled using a graph. In this short paper, we provide lower bounds on the size of a graph that would allow for some quantum phenomena to occur. Among other things, we show that, in the adjacency matrix…

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…

We investigate the asymptotic stability and ergodic properties of quantum trajectories under imperfect measurement, extending previous results established for the ideal case of perfect measurement. We establish a necessary and sufficient…

Given a smooth integral two-form and a smooth potential on the flat torus of dimension 2, we study the high energy properties of the corresponding magnetic Schr\"odinger operator. Under a geometric condition on the magnetic field, we show…

A map from a manifold to a Euclidean space is said to be k-regular if the image of any distinct k points are linearly independent. In this paper, we give some lower bounds of the dimension of the ambient Euclidean space for complex…

A finite group $G$ is called $C$-quasirandom (by Gowers) if all non-trivial irreducible complex representations of $G$ have dimension at least $C$. For any unit $\ell^{2}$ function on a finite group we associate the quantum probability…

We study the quantum mechanics of a generalized version of the baker's map. We show that the Ruelle resonances (which govern the approach to ergodicity of classical distributions on phase space) also appear in the quantum correlation…

We study joint quasimodes of the Laplacian and one Hecke operator on compact congruence surfaces, and give conditions on the orders of the quasimodes that guarantee positive entropy on almost every ergodic component of the corresponding…

We present here a canonical description for quantizing classical maps on a torus. We prove theorems analagous to classical theorems on mixing and ergodicity in terms of a quantum Koopman space $ L^2 (A_\hbar},\tau_\hbar) $ obtained as the…

In the case of a linear symplectic map A of the 2d-torus, semiclassical measures are A-invariant probability measures associated to sequences of high energy quantum states. Our main result is an explicit lower bound on the entropy of any…