by the ${\mathfrak{S}_{1}}$-norm and} 
\Vert R - R^0 \Vert_1 &\leq 7 M(\alpha) \frac{\log en}{n^{2\alpha}}\,\label{eq:rdiffnorm}\\
\intertext{and}
{\left\Vert \sum_{j = m}^{\infty} U_j \right\Vert}_1 & \leq 4 \left(  M(\alpha) \frac{\log en}{n^{2\alpha}} \right)^m,\quad m \geq 2. \label{eq:ujsumnorm}\end{aligned}$$

Inequality is identical with (proven) line . come if we combine , , and . Therefore, for $m \geq 2$, by and , $$\begin{aligned}
{\left\Vert \sum_{j = m}^{\infty} U_j \right\Vert}_1 & \leq 2 \sum_{j = m}^{\infty} \left(  M(\alpha) \frac{\log en}{n^{2\alpha}} \right)^j \leq \\
& \leq 4 \left(  M(\alpha) \frac{\log en}{n^{2\alpha}} \right)^m.
\end{aligned} \label{eq:umanynorm}$$ Then by $$\begin{aligned}
\Vert R - R^0 \Vert_1 & \leq \Vert U_1 \Vert_1 + {\left\Vert \sum_{j = 2}^{\infty} U_j \right\Vert}_1 \leq \\
& \leq   M(\alpha) \frac{\log en}{n^{2\alpha}}  \cdot \left(5 + 4 M(\alpha) \frac{\log en}{n^{2\alpha}} \right)\\ 
& \leq  7M(\alpha) \frac{\log en}{n^{2\alpha}}.
\end{aligned} \label{eq:rdiffnormest}$$

Deviations of eigenvalues of the harmonic oscillator operator and its perturbations {#sec:evalperturb}
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Although the constructions and methods of this section are general and applicable to many operators with discrete spectrum and their perturbations, we’ll focus later in this section on the case of Harmonic Oscillator operator and its functional representation $$L^0 y = - y^{\prime \prime} + x^2 y \label{eq:ell0reit}$$ in $L^2({\mathbb{R}})$.

The Riesz-Neumann Series , — as soon as its 