natural sciences, see, e.g., [@Bagarello], [@Davies1], [@Davies2], [@Davies3], [@Henry1], [@Henry2], [@Henry3] [@Mityagin], [@Sklyar3], [@Xu], [@Zwart] and the references therein. This is primarily caused by the recent progress in theoretical physics of non-Hermitian systems [@Bagarello] on the one hand, and, on the other, by the fact that many mathematical models of dynamical processes lead to the study of linear evolution equations $$\label{Cauchy problem}
    \left \{
\begin{array}{l}
\dot{x}(t)=Ax(t),\quad t \geq 0,\\
x(0)=x_0\in H,\\
\end{array}
\right .$$ in Hilbert spaces $H$ with unbounded NSA operator $A$.

In last years NSA Schrödinger operators are studied very intensively, see [@Davies3], [@Davies4], [@Henry1], [@Henry2], [@Henry3], [@Mityagin] and, especially, [@Bagarello], [@Davies1], [@Davies2]. In 2000 E.B. Davies [@Davies3] studied NSA anharmonic oscillators $$\label{L alpha}
    \mathcal{L}_{\alpha} = -\frac{d^2}{d x^2}+c |x|^{\alpha},$$ defined on $L_2\left(\mathbb{R}\right)$ as the closure of the associated quadratic form defined on $C_{0}^{\infty}\left(\mathbb{R}\right)$, where $\alpha>0,$ $c\in\mathbb{C}\setminus \mathbb{R},$ $|\arg c|<C(\alpha)$. He proved that for all $\alpha>0$ the spectrum of $\mathcal{L}_{\alpha}$ consists of discrete simple eigenvalues and, if we denote them in nondecreasing modulus order by $\lambda_n,$ $|\lambda_n|\to \infty$, and consider corresponding one-dimensional spectral projections $P_n$, then the norms $\left\|P_n\right\|$ grow more rapidly than any polynomial of $n$ as $n\to\infty$, see [@Davies2], [@Davies3]. Davies called operators with such spectral behavior 