论文标题
$ \ mathbb {z}^2 $ on Dirac-type运算符的阈值频谱渐近学
Spectral Asymptotics at Thresholds for a Dirac-type Operator on $\mathbb{Z}^2$
论文作者
论文摘要
在本文中，我们通过描述与乘法运算符与标志性定义的微量级扰动相关联的频谱偏移函数的行为，从而在$ \ mathbb {z}^2 $上提供了dirac型运算符的光谱分析。我们证明它在单个阈值之外保持界限，并在无限的情况下获得其主要渐近术语。有趣的是，我们表明，主要渐近项中的常数编码平坦频段和整个非恒定带之间的相互作用。所使用的策略是将光谱偏移函数的降低到某些紧凑型操作员的特征值计数函数中，该函数可以作为环形伪差异操作员研究。
In this article, we provide the spectral analysis of a Dirac-type operator on $\mathbb{Z}^2$ by describing the behavior of the spectral shift function associated with a sign-definite trace-class perturbation by a multiplication operator. We prove that it remains bounded outside a single threshold and obtain its main asymptotic term in the unbounded case. Interestingly, we show that the constant in the main asymptotic term encodes the interaction between a flat band and whole non-constant bands. The strategy used is the reduction of the spectral shift function to the eigenvalue counting function of some compact operator which can be studied as a toroidal pseudo-differential operator.
