arXiv subjects

Jeremy Clark

Publications and source records attributed to Jeremy Clark.

40 records · Page 3Linked to original sources

arXiv2011

Bounds for the state-modulated resolvent of a linear Boltzmann generator

We study a generalized resolvent for the generator of a Markovian semigroup. The Markovian generator appears in a linear Boltzmann equation modeling a one-dimensional test particle in a periodic potential and colliding elastically with particles from an ideal background gas. We obtain bounds for the state-modulated resolvent which are relevant in the regime where the mass ratio between the test particle and a particle from the gas is large. These bounds relate to the typical amount of time that the particle spends in different regions of phase space before arriving to a region around the origin.

math-ph↗

arXiv2009

The reduced effect of a single scattering with a low-mass particle via a point interaction

In this article, we study a second-order expansion for the effect induced on a large quantum particle which undergoes a single scattering with a low-mass particle via a repulsive point interaction. We give an approximation with third-order error in $λ$ to the map $G\to \Tr_{2}[(I\otimes ρ)S_λ^{*}(G\otimes I)S_λ]$, where $G\in \Bi(L^{2}(\R^{n}))$ is a heavy-particle observable, $ρ\in \Bi_{1}(\R^{n})$ is the density matrix corresponding to the state of the light particle, $λ=\frac{m}{M}$ is the mass ratio of the light particle to the heavy particle, $S_λ\in \Bi(L^{2}(\R^{n})\otimes L^{2}(\R^{n}))$ is the scattering matrix between the two particles due to a repulsive point interaction, and the trace is over the light-particle Hilbert space. The third-order error is bounded in operator norm for dimensions one and three using a weighted operator norm on $G$.

math-ph↗

arXiv2008

An infinite-temperature limit for a quantum scattering process

We study a quantum dynamical semigroup driven by a Lindblad generator with a deterministic Schrödinger part and a noisy Poission-timed scattering part. The dynamics describes the evolution of a test particle in $\R^{n}$, $n=1,2,3$, immersed in a gas, and the noisy scattering part is defined by the reduced effect of an individual interaction, where the interaction between the test particle and a single gas particle is via a repulsive point potential. In the limit that the mass ratio $λ=\frac{m}{M}$ tends to zero and the collisions become more frequent as $\frac{1}λ$, we show that our dynamics $Φ_{t,λ}$ approaches a limiting dynamics $Φ_{t,λ}^{\diamond}$ with second order error. Working in the Heisenberg representation, for $G\in \Bi(L^{2}(\R^{n}))$ $n=1,3$ we bound the difference between $Φ_{t,λ}(G)$ and $Φ_{t,λ}^{\diamond}(G)$ in operator norm proportional to $λ^{2}$.

math-ph↗

arXiv2008

Decoherence rates for Galilean covariant dynamics

We introduce a measure of decoherence for a class of density operators. For Gaussian density operators in dimension one it coincides with an index used by Morikawa (1990). Spatial decoherence rates are derived for three large classes of the Galilean covariant quantum semigroups introduced by Holevo. We also characterize the relaxation to a Gaussian state for these dynamics and give a theorem for the convergence of the Wigner function to the probability distribution of the classical analog of the process.

math-ph↗