z_n) [R(z) - R^0(z)] \, dz\\
& = \sum_{j = 1}^{\infty} T_j(n)
\end{aligned} \label{eq:lamdiff}$$ where we put \[with $z_n = \lambda_n^0 = 2n + 1$\] $$T_j(n) = T_j(n; W) = \frac{1}{2\pi i} \operatorname{Trace}\int\limits_{{\partial (\mathcal{D}_n )}} (z - z_n) U_j(z) \, dz \label{eq:tjdef}$$ Proposition \[prop:normbds\] is used in , . *Trace* is a linear bounded functional of norm $1$, on the space ${\mathfrak{S}_{1}}$ of trace-class operators. It implies the following.

\[cor:tjnorm\] Under the assumptions of Proposition \[prop:normbds\], with $n \geq N_*$, we have $$\begin{aligned}
\vert T_j(n) \vert & \leq \left[ M(\alpha) \frac{\log en}{n^{2\alpha}}\right]^j, \quad j \geq 2 \label{eq:tjnorm}
\intertext{and}
\vert T_1(n) \vert & \leq  \frac{9}{4} M(\alpha) \frac{\log en}{n^{2\alpha}} \label{eq:tjnorm.b}\end{aligned}$$

With $\vert \operatorname{Trace}A \vert \leq \Vert A \Vert_1$ and $$\begin{aligned}
\vert z - z_n \vert \leq \frac{1}{\sqrt{2}},& \quad  z \in {\partial (\mathcal{D}_n )},\\
\text{length}(\mathcal{D}_n) = 4
\end{aligned} \label{eq:data2}$$ rough estimates of integrals with $j \geq 2$ and $j = 1$ based on and lead to and .

\[cor:lamest\] Under the assumptions of Proposition \[prop:normbds\], with $n \geq N_*$, $$\lambda_n = (2n + 1) + \sum_{j = 1}^q T_j(n) + r_q(n), \quad q \geq 1, \label{eq:lamest}$$ where $$\vert r_q(n) \vert \leq 2 \left( M(\alpha) \frac{\log en}{n^{2\alpha}} \right)^{q + 1} \label{eq:rqest}$$

The presentation of 