关于微量缺陷公式中的对数分量
On the logarithm component in trace defect formulas
摘要：
在闭流形上经典伪微分算子的预解迹$\Tr(A(P-λ)^{-1})$的渐近展开式中，$(-λ)^{-1}$的系数$C_0(A,P)$是特别有趣的，因为它是第一个包含来自$A$的非局部元素的系数;另一方面，如果$A=I$和$P=D^*D$，它给出了$D$的部分指标。当$P$可逆时，$C_0(A,P)$也等于0处的ζ函数值。$C_0(A,P)$是模局部项的迹，因为$C_0(A,P)-C_0(A,P')$和$C_0([A,A'],P)$是局部的。通过使用复杂的权力$P^s$（或类似的全纯家庭的顺序$s$），Okikiolu，Kontsevich和Vishik，梅尔罗斯和尼斯托尔显示公式为这些微量缺陷的残留运营商定义的$A$，$A'$，$\log P$和$\log P'$。本文有两个目的：一是说明如何从预解式中以一种简单的方式获得微量亏损公式，而不像原来的证明那样使用$P$的复数幂。本文还给出了Scott最近关于$C_0(I,P)$的一个留数公式的一个简单的直接证明.另一个目的是建立跟踪缺陷剩余公式的经营者流形上的边界，复杂的权力是不容易获得的，我们这样做只使用预解式。我们还将Scott公式推广到边界问题。
英文摘要：
In asymptotic expansions of resolvent traces $\Tr(A(P-λ)^{-1})$ for classical pseudodifferential operators on closed manifolds, the coefficient $C_0(A,P)$ of $(-λ)^{-1}$ is of special interest, since it is the first coefficient containing nonlocal elements from $A$; on the other hand if $A=I$ and $P=D^*D$ it gives part of the index of $D$. $C_0(A,P)$ also equals the zeta function value at 0 when $P$ is invertible. $C_0(A,P)$ is a trace modulo local terms, since $C_0(A,P)-C_0(A,P')$ and $C_0([A,A'],P)$ are local. By use of complex powers $P^s$ (or similar holomorphic families of order $s$), Okikiolu, Kontsevich and Vishik, Melrose and Nistor showed formulas for these trace defects in terms of residues of operators defined from $A$, $A'$, $\log P$ and $\log P'$. The present paper has two purposes: One is to show how the trace defect formulas can be obtained from the resolvents in a simple way without use of the complex powers of $P$ as in the original proofs. We here also give a simple direct proof of a recent residue formula of Scott for $C_0(I,P)$. The other purpose is to establish trace defect residue formulas for operators on manifolds with boundary, where complex powers are not easily accessible; we do this using only resolvents. We also generalize Scott's formula to boundary problems.
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