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 $V(x) = x^{2a} + i x^b$, $x \in \mathbb R$, $a, b \in \mathbb N$, we show that the system of spectral projections $\{P_n\}_n$ 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 $a - 1 < b < 2a$. Moreover, for $\sigma = [b - (a - 1)]/(1 + a)$ and $\gamma > 0$ small enough, $\limsup_n \|P_n\|/ \exp(\gamma n^\sigma) = \infty$. 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 $\|(z - L)^{-1}\|$ outside of the spectrum $\sigma(L)$; (b) partial fraction decompositions of special meromorphic functions $1/F$ where $F(w) = \prod_{k=1}^\infty \left( 1 + \frac{w}{a_k} \right)$, $a_{k+1} \geq a_k>0$, $k \in \mathbb N$, and the generalization of the first resolvent identity.