Spectral projections of an anharmonic oscillator with complex polynomial potential

Abstract: For a broad class of polynomial potentials $V$, with an important and instructive representative being $V(x) = x^{2a} + i x<sup>b$, $x \in \mathbb R$, $a, b \in \mathbb N$, we show that the system of spectral projections ${P_n}<em>n$ of an anharmonic operator $L = - (\mathrm{d}/ \mathrm{d}x)² + V(x)$ does not generate a (Riesz) basis in $L^2(\mathbb R)$ if $a - 1 &lt; b &lt; 2a$. Moreover, for $σ= [b - (a - 1)]/(1 + a)$ and $γ&gt; 0$ small enough, $\limsup_n |P_n|/ \exp(γn^σ) = \infty$. Proofs are based on two groups of results which are of great interest on their own: (a) relationship between behavior (growth) of the norms of projections $|P_n|$ and of the resolvent $|(z - L)<sup>{-1}|$ outside of the spectrum $σ(L)$; (b) partial fraction decompositions of special meromorphic functions $1/F$ where $$F(w) = \prod</em>{k=1}^\infty \left( 1 + \frac{w}{a_k} \right)$$, $a_{k+1} \geq a_k&gt;0$, $k \in \mathbb N$, and the generalization of the first resolvent identity.