by spectrally wild ones. A family of eigenvectors of spectrally wild operator, although can be complete and minimal in a space, cannot constitute a Schauder basis. E.g., the eigenvectors of $\mathcal{L}_{\alpha}$, where $\Re (c)>0$, are dense in $L_2\left(\mathbb{R}\right)$ if either $\alpha\geq 1$, or $0<\alpha<1$ and $|\arg c|<\alpha\pi/2$, see [@Davies3]. We recall that a sequence $\{\phi_n\}_{n=1}^{\infty}$ of a Banach space $X$ forms a Schauder basis of $X$ provided each $x\in X$ has a unique norm-convergent expansion $$x=\sum\limits_{n=1}^{\infty} c_n \phi_n.$$

In 2004 E.B. Davies and A. B. J. Kuijlaars proved that spectral projections $P_n$ of the operator $\mathcal{L}_2$, where $c=e^{i\theta}$, $0<|\theta| < \pi,$ grow exponentially [@Davies4]: $$\lim\limits_{n\to\infty} \frac{1}{n}\ln\left\|P_n\right\|=2 \Re \left\{f\left(r(\theta)e^{\frac{i\theta}{4}} \right)\right\},$$ where $f(z)= \ln \left(z+  g(z)\right) - z g(z)$, $g(z)=(z^2-1)^{1/2}$, $r(\theta)= \left(2\cos \left(\theta/2\right) \right)^{-1/2}$.

These studies were continued by R. Henry, who determined exponential growth rates of spectral projections of the so-called complex Airy operator $\mathcal{L}_1$, where $c=e^{i\theta}$, $0<|\theta| <\frac{3\pi}{4}$, and anharmonic oscillators $\mathcal{L}_{2k},$ $k\in\mathbb{N},$ where $c=e^{i\theta}$, $0<|\theta| <\frac{(k+1)\pi}{2k},$ see [@Henry1], [@Henry2]. Moreover, in [@Henry3] Henry studied spectral projections $P_n$ of the complex cubic oscillator $ \mathcal{C}_{\beta}=-\frac{d^2}{d x^2}+ i x^3 +i\beta x,\: \beta\geq 0$ with domain $H^2\left(\mathbb{R}\right)\cap L_2\left(\mathbb{R};x^6 dx\right)\subset L_2\left(\mathbb{R}\right)$ and showed that for all $\beta\geq 0,$ $$\lim\limits_{n\to\infty} \frac{1}{n}\ln\left\|P_n\right\|=\frac{\pi}{\sqrt{3}}.$$

Recently, B. Mityagin 