Spectral projections of an anharmonic oscillator with complex polynomial potential

TL;DR

This paper investigates the spectral properties of anharmonic oscillators with complex polynomial potentials, showing that their spectral projections do not form a basis under certain conditions and analyzing the growth of projection norms.

Contribution

It provides new insights into the spectral projection behavior of complex anharmonic oscillators and establishes links between projection norms and resolvent growth, with novel decompositions of meromorphic functions.

Findings

Spectral projections do not form a basis if a - 1 < b < 2a.

Projection norms grow faster than any exponential for certain parameters.

Established relationships between projection norms and resolvent behavior.

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.