$C_0$-groups from Theorem \[construction\] and to characterize the asymptotic properties of the resolvent on a complex plane. Theorem \[construction\] together with the XYZ Theorem show that Theorem 1.1 from [@Zwart] dealing with the case of simple eigenvalues $\{\lambda_n\}_{n=1}^{\infty}$ in the XYZ Theorem, satisfying $$\inf\limits_{n\neq m} |\lambda_{n}-\lambda_{m}|>0,$$ is sharp, see also Example 1.3 in [@Zwart]. In the present paper we will demonstrate that Theorem \[construction\] means that the XYZ Theorem is also sharp, see Section 2.

Operators with simple eigenvalues $\{\lambda_n\}_{n=1}^{\infty}$ not satisfying the condition 2 of the XYZ Theorem and non-basis family of eigenvectors are considered in recent applications. In [@Almog] the author study the stability of the normal state of superconductors in the presence of electric currents in the large domain limit using the time-dependent Ginzburg-Landau model. The study involves spectral analysis of the operator $\mathcal{L}: D(\mathcal{L}) \mapsto L_2 \left(\mathbb{R},\mathbb{C}\right)$, defined by $$\mathcal{L}  = - \frac{d^2}{d x^2} +i x ,$$ where $D(\mathcal{L})=\left\{\psi\in  L_2 \left(\mathbb{R_{+}},\mathbb{C}\right):\: x\psi \in L_2 \left(\mathbb{R_{+}},\mathbb{C}\right), \psi\in H_{0}^{2} \left(\mathbb{R_{+}},\mathbb{C}\right)\right\}.$ Let $\{\mu_n\}_{n=1}^{\infty} \subset \mathbb{R}$ denotes the non-increasing sequence of zeroes of $Ai(z)$, Airy function. Then $\{\lambda_n\}_{n=1}^{\infty}$, where $\lambda_n=e^{-\frac{2\pi}{3} i} \mu_n,\:n\in\mathbb{N},$ is a sequence of eigenvalues of $\mathcal{L}$ [@Almog]. Since $\lim\limits_{n\rightarrow \infty} \mu_n=-\infty$ and $\lim\limits_{n\rightarrow \infty} |\mu_{n+1}-\mu_n|=0$ (see [@Vallee]), 