in the interior, its kernel is smooth and is $O(\lambda ^{-M})$ for any $M$. Hence the terms $\theta _{i_1}G\theta_{i_2}G^{(N)}_\lambda $ that are supported in the interior have traces that are $O(\lambda ^{-M})$ for any $M$ and can be disregarded in the following. For the terms $\theta _{i_1}G\theta _{i_2}Q^N_{\lambda ,+}$ supported in the interior, $\theta _{i_1}G\theta _{i_2}$ is of order $-\infty $; they will be covered by Lemma 3.1 below, which gives: $$\Tr(\theta _{i_1}G\theta
_{i_2}Q^N_{\lambda ,+})=
\Tr(\theta _{i_1}G\theta
_{i_2})(-\lambda )^{-N}+O(\lambda ^{-N-1/2}).\tag3.2$$ For the terms supported in patches meeting the boundary we can use the results of [@GSc] in an accurate way.

Let us first recall some elements of the basic symbol calculus we use, namely the calculus of parameter-dependent $\psi $do’s introduced in [@GS1]. It will be used both in the $(n-1)$-dimensional setting, relevant for operators on $X'$, and in the $n$-dimensional setting, relevant for operators on $\widetilde X$.

For $m\in\Bbb R$, the usual pseudodifferential symbol space $S^m(\Bbb
R^{n'}\stimes\Bbb R^{n'})$ consists of the functions $s(x,\xi )\in
C^\infty (\Bbb R^{n'}\stimes\Bbb R^{n'})$ satisfying estimates$$|D_{x}^\beta D_{\xi }^\alpha s(x,\xi )|\le C_{\alpha ,\beta
}\ang{\xi }^{m-|\alpha |},$$ for all $\alpha ,\beta \in \Bbb N^{n'}$. (We denote $(1+|\xi
|^2)^{\frac12}=\ang\xi $, $\{0,1,2,\dots\}=\Bbb N$.) The parameter-dependent version we shall use 