预解可比扰动的高阶谱位移函数
Higher-order spectral shift function for resolvent comparable perturbations
摘要：
给定一对自伴算子$H$和$V$，使得$V$有界且$(H+V-i)^{-1}-(H-i)^{-1}$属于可分Hilbert空间上算子的Schatten-von Neumann理想$\mathcal{S}^n$，$n\ge 2$，我们建立了包含几个主要测试函数类的广泛的函数集合$f$的高阶迹公式，并且还建立了相应的局部可积实-值谱移位函数确定唯一的低次多项式被加数。我们的结果推广了\cite{PSS13}的结果Schatten-von Neumman扰动$V$和结算较早的尝试，包括一般的扰动与Schatten-von Neumman差异的预解式，这导致了更复杂的跟踪公式更严格的功能集$f$和类似物的光谱移位功能缺乏真正的价值和/或预期程度的独特性。我们的证明建立在本文推导出的变量的一般变化的方法，并显着细化出现在\cite{vNS21,PSS15,S17}的几个参数在一次。
英文摘要：
Given a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $(H+V-i)^{-1}-(H-i)^{-1}$ belongs to the Schatten-von Neumann ideal $\mathcal{S}^n$, $n\ge 2$, of operators on a separable Hilbert space, we establish higher order trace formulas for a broad set of functions $f$ containing several major classes of test functions and also establish existence of the respective locally integrable real-valued spectral shift functions determined uniquely up to a low degree polynomial summand. Our result generalizes the result of \cite{PSS13} for Schatten-von Neumman perturbations $V$ and settles earlier attempts to encompass general perturbations with Schatten-von Neumman difference of resolvents, which led to more complicated trace formulas for more restrictive sets of functions $f$ and to analogs of spectral shift functions lacking real-valuedness and/or expected degree of uniqueness. Our proof builds on a general change of variables method derived in this paper and significantly refining those appearing in \cite{vNS21,PSS15,S17} with respect to several parameters at once.
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