论文标题
分辨出可比扰动的高阶光谱函数
Higher-order spectral shift function for resolvent comparable perturbations
论文作者
论文摘要
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功能$ f $包含几个主要类别的测试功能，还建立了各个局部可实现的实价频谱变化函数的存在，直到唯一地确定为低度多项式求和。我们的结果概括了Schatten-von neumman扰动$ v $的结果\ cite {pss13}的结果，并在早期尝试与schatten-von neumman差异的一般扰动进行了解决，这导致了更为复杂的痕量公式和更为复杂的痕量范围和更为复杂的函数函数$ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f》;独特性。我们的证明基于本文中得出的变量方法的一般变化，并显着完善了\ cite {vns21，pss15，s17}中出现的变量方法。
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.
