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, ${\mathrm{limsup}}_n\Vert P_n\Vert /\exp (\gamma n^{\sigma })=\infty$. 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 $\Vert P_n\Vert$ and of the resolvent $\Vert (z-L{)}^{-1}\Vert$ outside of the spectrum $\sigma (L)$; (b) partial fraction decompositions of special meromorphic functions $1/F$ where $F(w)={\prod }_{k=1}^{\infty }\left(1+\frac{w}{a_k}\right)$, $a_{k+1}\ge a_k>0$, $k\in \mathbb{N}$, and the generalization of the first resolvent identity.

55 pages