Uniform Polynomial Decay and Approximation in Control of a Family of Abstract Thermoelastic Models

2021 ◽  
Author(s):  
S. Nafiri
Keyword(s):  
Polynomial Decay

scholarly journals Fast and slow dynamics for classical and quantum walks on mean-field small world networks

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Andre M. C. Souza ◽  
Roberto F. S. Andrade
Keyword(s):  
Transition Probabilities ◽  
Linear Chain ◽  
Mean Field ◽  
Small World ◽  
Quantum Walks ◽  
Polynomial Decay ◽  
Small World Networks ◽  
Quantum Random Walks ◽  
Local Maxima ◽  
Mathematical Properties

AbstractThis work investigates the dynamical properties of classical and quantum random walks on mean-field small-world (MFSW) networks in the continuous time version. The adopted formalism profits from the large number of exact mathematical properties of their adjacency and Laplacian matrices. Exact expressions for both transition probabilities in terms of Bessel functions are derived. Results are compared to numerical results obtained by working directly the Hamiltonian of the model. For the classical evolution, any infinitesimal amount of disorder causes an exponential decay to the asymptotic equilibrium state, in contrast to the polynomial behavior for the homogeneous case. The typical quantum oscillatory evolution has been characterized by local maxima. It indicates polynomial decay to equilibrium for any degree of disorder. The main finding of the work is the identification of a faster classical spreading as compared to the quantum counterpart. It stays in opposition to the well known diffusive and ballistic for, respectively, the classical and quantum spreading in the linear chain.

scholarly journals The Distribution of Phase Shifts for Semiclassical Potentials with Polynomial Decay

2018 ◽  
Vol 2020 (19) ◽  
pp. 6294-6346
Author(s):  
Jesse Gell-Redman ◽  
Andrew Hassell
Keyword(s):  
Asymptotic Formula ◽  
Unit Circle ◽  
Hamiltonian Dynamics ◽  
Semiclassical Limit ◽  
Scattering Matrix ◽  
Phase Shifts ◽  
Polynomial Decay ◽  
Homogeneous Distribution ◽  
Fixed Energy ◽  
Atomic Measure

Abstract This is the 3rd paper in a series [6, 9] analyzing the asymptotic distribution of the phase shifts in the semiclassical limit. We analyze the distribution of phase shifts, or equivalently, eigenvalues of the scattering matrix $S_h$, at some fixed energy $E$, for semiclassical Schrödinger operators on $\mathbb{R}^d$ that are perturbations of the free Hamiltonian $h^2 \Delta $ on $L^2(\mathbb{R}^d)$ by a potential $V$ with polynomial decay. Our assumption is that $V(x) \sim |x|^{-\alpha } v(\hat x)$ as $x \to \infty $, $\hat x = x/|x|$, for some $\alpha> d$, with corresponding derivative estimates. In the semiclassical limit $h \to 0$, we show that the atomic measure on the unit circle defined by these eigenvalues, after suitable scaling in $h$, tends to a measure $\mu $ on $\mathbb{S}^1$. Moreover, $\mu $ is the pushforward from $\mathbb{R}$ to $\mathbb{R} / 2 \pi \mathbb{Z} = \mathbb{S}^1$ of a homogeneous distribution. As a corollary we obtain an asymptotic formula for the accumulation of phase shifts in a sector of $\mathbb{S}^1$. The proof relies on an extension of results in [14] on the classical Hamiltonian dynamics and semiclassical Poisson operator to the larger class of potentials under consideration here.

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