et al. considered NSA perturbations of selfadjoint Schrödinger operators with single-well potentials and demonstrated that norms of spectral projections $P_n$ of these operators can grow at intermediate levels, from arbitrary slowly to exponentially fast [@Mityagin]. In particular, natural classes of operators with projections obeying $$\lim\limits_{n\to\infty} \frac{1}{n^{\gamma}}\ln\left\|P_n\right\|=C,$$ where $C\in (0,\infty)$ and $\gamma \in (0,1),$ were found.

On the other hand, in “good” situation, i.e. when the operator $A$ has a Riesz basis of $A$-invariant subspaces, the system (\[Cauchy problem\]) can be split into countable family of subsystems (each subsystem lives in a corresponding $A$-invariant subspace) and we can make conclusions on the behavior of (\[Cauchy problem\]) on the basis of the study of its subsystems, see, e.g., [@Miloslavskii], [@Rabah1], [@Rabah2], [@Sklyar1], [@Sklyar2], [@Xu], [@Zwart] and the references therein. That is why Riesz bases are convenient tools of infinite-dimensional linear systems theory and the following question is important.

\[q\] Which conditions are sufficient to guarantee that $$\label{question}
    A\:\textit{has a Riesz basis of eigenvectors ($A$-invariant subspaces)?}$$

For equivalent definitions and stability properties of Schauder bases of subspaces (Schauder decompositions) and Riesz bases of subspaces we refer to [@Marchenko1], [@Marchenko2], [@Marchenko3] and the references therein.

A number of recent papers are devoted to Question 