\frac{4}{3} \cdot 2 + \frac{2}{3} < 10, \text{ and} \label{eq:alph1qtr1}\\
\mu \leq \frac{2}{\sqrt{n}} (5 + \log n) \label{eq:alph1qtr2}\end{gathered}$$ This case is important in analysis of the harmonic oscillator and its perturbations . The estimates will be used later as well.

\[rem:sbds\] Let $\displaystyle s \equiv \sum_{\substack{ j = 0 \\ j \neq n }}^{\infty} \frac{1}{(1 + j)^{\beta}} \cdot \frac{1}{\vert n - j \vert}$. Then

$$\begin{aligned}
s & \leq \frac{M(\beta)}{n^{\beta}} \log en, && \text{ if } 0 < \beta \leq 1, \label{eq:sbetasmall}\\
s & \leq \frac{M}{n}, && \text{ if } \beta > 1.  \label{eq:sbetalarge}\end{aligned}$$

\[eq:sconds\]

The case $\beta = 2 \alpha < 1$ is done in the proof of Lemma \[lem:hs\]. Other cases could be explained in the same way; we omit details.

By the operator $K$ is bounded if $z \in H_n \setminus \mathcal{D}_n$ and by , its norm $$\Vert K \Vert \leq \sqrt{2}. \label{eq:knorm}$$ Therefore, for $U_j \in $ if $j \geq 2$ $$\Vert U_j \Vert_1 \leq 2 \Vert K W K \Vert_2^j \leq 2 \nu^j \leq 2 \left[ M(\alpha) \frac{\log e n}{n^{2\alpha}} \right]^j. \label{eq:ujest}$$ But we can claim that $U_1$ is a trace-class operator as well, and $$\Vert U_1 \Vert_1 = \Vert K(KWK) K \Vert_1 \leq \Vert K 