相关论文: Gauss sums, superoscillations and the Talbot carpe…
We exhibit a family of autosimilar H\"older maps that satisfies a fractal version of the Van Der Corput Lemma, despite not being absolutely continuous. The result is a direct consequence of a recent work of Sahlsten and Steven…
We construct a continuous-time non-commutative random walk on $U(\mathfrak{gl}_N)$ with dilation maps $U(\mathfrak{gl}_N)\rightarrow L^2(U(N))^{\otimes\infty}$. This is an analog of a continuous-time non-commutative random walk on the group…
The central limit theorem has been found to apply to random vectors in complex Hilbert space. This amounts to sufficient reason to study the complex valued Gaussian, looking for relevance to quantum mechanics. Here we show that the…
The issue of separating Schr\"odinger-type quantum time evolution into a product of holonomic and dynamical parts in the non-adiabatic non-Abelian case is examined. We identify all special cases in which this kind of separation is possible,…
The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for $H^s$ initial data, $s>-1/2$, and for any $s_1<\min(3s+1,s+1)$, the difference of the nonlinear and linear evolutions is in $H^{s_1}$…
The paper describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis. For this purpose, the Egorov-type…
We consider time dependent harmonic oscillators and construct a parametrix to the corresponding Schr\"odinger equation using Gaussian wavepackets. This parametrix of Gaussian wavepackets is precise and tractable. Using this parametrix we…
We extend a previously successful discussion of the constrained Schr\"{o}dinger system through the Dirac--Bergmann algorithm to the case of the Dirac field. In order to follow the analogy, first we discuss the classical Dirac field as a…
An asymptotic expansion for the generalised quadratic Gauss sum $$S_N(x,\theta)=\sum_{j=1}^{N} \exp (\pi ixj^2+2\pi ij\theta),$$ where $x$, $\theta$ are real and $N$ is a positive integer, is obtained as $x\rightarrow 0$ and…
For almost 75 years, the general solution for the Schr\"odinger equation was assumed to be generated by an exponential or a time-ordered exponential known as the Dyson series. We study the unitarity of a solution in the case of a singular…
. We study the evolution of the distribution of eigenvalues of a $N\times N$ matrix subject to a random perturbation drawn from (i) a generalized Gaussian ensemble (ii) a non-Gaussian ensemble with a measure variable under the change of…
In this paper, we concentrate on the superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional linear time-dependent fourth-order equations. The adjustable numerical viscosity of the…
We study the spatio-temporal two-point correlation function of passively advected scalar fields in the inertial-convective range in three dimensions by means of numerical simulations. We show that at small time delays $t$ the correlations…
We consider the relativistic Schr\"odinger equation with a time dependent vector and scalar potential on a bounded cylindrical domain. Using a Geometric Optics Ansatz we establish a logarithmic stability estimate for the recovery of the…
We consider the relativistic Schr\"odinger (Gordon-Klein) equation with a time dependent vector and a scalar potential on a bounded cylindrical domain. Using a standard Geometric Optics Ansatz, we establish a logarithmic stability estimate…
We explore the problem of time in quantum gravity in a point-particle analogue model of scale-invariant gravity. If quantized after reduction to true degrees of freedom, it leads to a time-independent Schr\"odinger equation. As with the…
This article deals with a quantum-mechanical system which generalizes the ordinary isotropic harmonic oscillator system. We give the coefficients connecting the polar and Cartesian bases for D=2 and the coefficients connecting the Cartesian…
In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schr\"odinger equation for rational time dependent KdV potentials. In particular, we compute the fundamental matrices of the linear systems…
We show how the Dirac equation in three space-dimensions emerges from the large-scale dynamics of the minimal nontrivial quantum cellular automaton satisfying unitariety, locality, homogeneity, and discrete isotropy, without using the…
We study the evolution of the one dimensional periodic cubic Schr\"odinger equation (NLS) with bounded variation data. For the linear evolution, it is known that for irrational times the solution is a continuous, nowhere differentiable…