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A simple semiclassical H\'enon-Heiles model is constructed based on Dirac's time-dependent variational principle. We obtain an effective semiclassical Hamiltonian using a Hatree-type two-body trial wavefunction in the Jackiw-Kerman form.…
Quantum-classical correspondence for the average shape of eigenfunctions and the local spectral density of states are well-known facts. In this paper, the fluctuations that quantum mechanical wave functions present around the classical…
A fundamental spacetime scale in the universe leads to noncommutative spacetime and thence to a modified energy - momentum dispersion relation or equivalently to a modification of Lorentz symmetry as shown by the author and others. This…
We describe the quantum mechanical spreading of a Gaussian wave packet by means of the semiclassical WKB approximation of Berry and Balazs. We find that the time scale $\tau$ on which this approximation breaks down in a chaotic system is…
We consider the Riemannian random wave model of Gaussian linear combinations of Laplace eigenfunctions on a general compact Riemannian manifold. With probability one with respect to the Gaussian coefficients, we establish that, both for…
For an $N \times N$ random unitary matrix $U_N$, we consider the random field defined by counting the number of eigenvalues of $U_N$ in a mesoscopic arc of the unit circle, regularized at an $N$-dependent scale $\epsilon_N>0$. We prove that…
In this paper we obtain a Bernstein type inequality for the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality can be viewed as an extension to the matrix…
We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and…
In this work, we establish conditions ensuring convergence in distribution of a sequence admitting a Wiener-It\^o chaos representation to a nondegenerate Gaussian measure on a separable Hilbert space. Our first main result shows that,…
In this survey we collect some of the recent results on the "nodal geometry" of random eigenfunctions on Riemannian surfaces. We focus on the asymptotic behavior, for high energy levels, of the nodal length of Gaussian Laplace…
We perform a quantum probing of colliding plane-wave space-times. In particular, we consider the Khan-Penrose and the Ferrari-Ib\'a\~nez solutions, which admit a strong and a weak singularity after the two waves collide. While we find that,…
We present an elementary non-recursive formula for the multivariate moments of the Dirichlet distribution on the standard simplex, in terms of the pattern inventory of the moments' exponents. We obtain analog formulas for the multivariate…
Spatio-temporal deterministic chaos at small Taylor-Reynolds numbers $Re_{\lambda} \lesssim 40$ and distributed chaos at turbulent $Re_{\lambda} \gtrsim 40$ in passive scalar dynamics have been studied using results of direct numerical…
We show that the condition for the appearance of quantum chaos (Wigner-Dyson distribution of energy eigenvalues, gaussian-random energy eigenfunctions) in a dilute gas of many hard spheres is $\lambda \ll \ell$, where $\lambda$ is the…
States supported by chaotic open quantum systems fall into two categories: a majority showing instantaneous ballistic decay, and a set of quantum resonances of classically vanishing support in phase space. We present a theory describing…
The existence of both a minimum mass and a minimum density in nature, in the presence of a positive cosmological constant, is one of the most intriguing results in classical general relativity. These results follow rigorously from the…
In this paper, we investigate the fourth power moment of twisted Kloosterman sum and its relationship with Hurwitz class number. We derive an explicit formula expressing this moment in terms of weighted sums involving Hurwitz class numbers.…
We introduce a theory of probability in $\lambda$-rings designed to efficiently describe random variables valued in multisets of complex numbers, varieties over a field, or other similar enriched settings. A key role is played by the…
We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU(3) gauge theory and in full QCD as well as in quenched U(1) theory. As a measure of the fluctuation properties of the eigenvalues, we consider the nearest-neighbor…
The wave speed of a stochastic wave equation driven by Riesz noise on the unbounded multidimensional spatial domain is estimated based on discrete measurements. Central limit theorems for second-order variations of the observations in…