强耦合极化子能量-动量关系的最优抛物线上界

强耦合下极化子有效质量的Landau-Pekar公式的证明

Proof of the Landau-Pekar Formula for the effective Mass of the Polaron at strong coupling

该研究针对强耦合区Fröhlich极化子，证明了有效质量的渐近严格下界，结合已有的上界结果，证实了1948年提出的Landau-Pekar公式的有效性，解决了Spohn于1987年提出的相关猜想。

Bounds on the Minimal Energy of Translation Invariant $N$-Polaron Systems

For systems of $N$ charged fermions (e.g. electrons) interacting with longitudinal optical quantized lattice vibrations of a polar crystal we derive upper and lower bounds on the minimal energy within the model of H. Fröhlich. The only parameters of this model, after removing the ultraviolet cutoff, are the constants $U>0$ and $α>0$ measuring the electron-electron and the electron-phonon coupling strengths. They are constrained by the condition $\sqrt{2}α<U$, which follows from the dependence of $U$ and $α$ on electrical properties of the crystal. We show that the large $N$ asymptotic behavior…

Universal estimates for parabolic equations and applications for non-linear and non-local problems

We obtain some "universal" estimates for $L_2$-norm of the solution of a parabolic equation via a weighted version of $H^{-1}$-norm of the free term. More precisely, we found the limit upper estimate that can be achieved by transformation of the equation by adding a constant to the zero order coefficient. The inverse matrix of the higher order coefficients of the parabolic equation is included into the weight for the $H^{-1}$-norm. The constant in the estimate obtained is independent from the choice of the dimension, domain, and the coefficients of the parabolic equation, it is why it can be c…

The Definition and Measurement of the Topological Entropy per Unit Volume in Parabolic PDE's

We define the topological entropy per unit volume in parabolic PDE's such as the complex Ginzburg-Landau equation, and show that it exists, and is bounded by the upper Hausdorff dimension times the maximal expansion rate. We then give a constructive implementation of a bound on the inertial range of such equations. Using this bound, we are able to propose a finite sampling algorithm which allows (in principle) to measure this entropy from experimental data.

Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model

We consider the large-time behavior of the solution $u\colon [0,\infty)\times\Z\to[0,\infty)$ to the parabolic Anderson problem $\partial_t u=κΔu+ξu$ with initial data $u(0,\cdot)=1$ and non-positive finite i.i.d. potentials $(ξ(z))_{z\in\Z}$. Unlike in dimensions $d\ge2$, the almost-sure decay rate of $u(t,0)$ as $t\to\infty$ is not determined solely by the upper tails of $ξ(0)$; too heavy lower tails of $ξ(0)$ accelerate the decay. The interpretation is that sites $x$ with large negative $ξ(x)$ hamper the mass flow and hence screen off the influence of more favorable regions of the potential…