arXiv subjects

Bernhard Pfirsch

Publications and source records attributed to Bernhard Pfirsch.

3 recordsLinked to original sources

arXiv2021

The almost periodic gauge transform -- An abstract scheme with applications to Dirac Operators

One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schr\"odinger operators is the method of gauge transform. In this paper, we extend this method to an abstract setting, thus allowing for greater flexibility in its applications that include, among others, matrix-valued operators. In particular, we obtain asymptotic expansions for the density of states of certain almost periodic systems of elliptic operators, including systems of Dirac type. We also prove that a range of periodic systems including the two-dimensional Dirac operators satisfy the Bethe--Sommerfeld property, that the spectrum contains a semi-axis -- or indeed two semi-axes in the case of operators that are not semi-bounded.

math-ph↗

arXiv2018

A Szegő limit theorem for translation-invariant operators on polygons

We prove Szegő-type trace asymptotics for translation-invariant operators on polygons. More precisely, consider a Fourier multiplier $A=\mathcal{F}^\ast σ\mathcal{F}$ on $\mathsf{L}^2(\mathbb{R}^2)$ with a sufficiently decaying, smooth symbol $σ:\mathbb{C}\to\mathbb{C}$. Let $P\subset \mathbb{R}^2$ be the interior of a polygon and, for $L\geq 1$, define its scaled version $P_L:=L\cdot P$. Then we study the spectral asymptotics for the operator $A_{P_L}=χ_{P_L}Aχ_{P_L}$, the spatial restriction of $A$ onto $P_L$: for entire functions $h$ with $h(0)=0$ we provide a complete asymptotic expansion of $\operatorname{tr}h(A_{P_L})$ as $L\to\infty$. These trace asymptotics consist of three terms that reflect the geometry of the polygon. If $P$ is replaced by a domain with smooth boundary, a complete asymptotic expansion of the trace has been known for more than 30 years. However, for polygons the formula for the constant order term in the asymptotics is new. In particular, we show that each corner of the polygon produces an extra contribution; as a consequence, the constant order term exhibits an anomaly similar to the heat trace asymptotics for the Dirichlet Laplacian.

math.SP↗

arXiv2016

Formulas of Szegő type for the periodic Schrödinger operator

We prove asymptotic formulas of Szegő type for the periodic Schrödinger operator $H=-\frac{d^2}{dx^2}+V$ in dimension one. Admitting fairly general functions $h$ with $h(0)=0$, we study the trace of the operator $h(χ_{(-α,α)}χ_{(-\infty,μ)}(H)χ_{(-α,α)})$ and link its subleading behaviour as $α\to\infty$ to the position of the spectral parameter $μ$ relative to the spectrum of $H$.

math.SP↗