The Spectral Shift Function and The Friedel Sum Rule

Friedel Sum Rule as a Trace Formula

We examine the Friedel sum rule which states that the "excess charge" due to a single impurity potential in a metal is equal to a sum of phase shifts for scatterings of electrons by the impurity. For finite volume, the ``excess charge" is given by the difference between total numbers of levels in the Fermi sea with and without the impurity potential. However, a sequence of the "excess charge" for finite volume is not necessarily bounded in the infinite volume limit, as was pointed out by Kirsch. In order to circumvent this difficulty, we define "excess charge" directly for the infinite volume.…

Friedel oscillations in quantum degenerate collisional plasma. Screening of point charge

Research of influence of collisions on Friedel oscillations in quantum degenerate collisional plasma (T=0) is carried out for the first time. It is shown that presence of collisions in plasma leads to exponential decreasing of amplitude and phase shift of the Friedel oscillations. In linear approximation the phase shift is equal to the half of quantity inverse to product of Fermi's wave number by free length path of electrons. The correct expression for longitudinal dielectric permeability of the quantum collisional plasma found by the authors (see arxiv:1001.3937 [math-ph] 22 Jan 2010) is use…

Coulomb screening in graphene with topological defects

We analyze the screening of an external Coulomb charge in gapless graphene cone, which is taken as a prototype of a topological defect. In the subcritical regime, the induced charge is calculated using both the Green's function and the Friedel sum rule. The dependence of the polarization charge on the Coulomb strength obtained from the Green's function clearly shows the effect of the conical defect and indicates that the critical charge itself depends on the sample topology. Similar analysis using the Friedel sum rule indicates that the two results agree for low values of the Coulomb charge bu…

微扰参数化量子演化导数的“真”移位规则

"Proper" Shift Rules for Derivatives of Perturbed-Parametric Quantum Evolutions

本文针对含微扰项的参数化量子演化期望值导数估计问题，提出仅需移位参数的真移位规则，实现精确无偏估计且最坏情况方差与Banchi-Crooks方法相同，并通过傅里叶分析刻画其理论性质。