[Paper Review] Spectral projections of an anharmonic oscillator with complex polynomial potential

Boris Mityagin, Petr Siegl|arXiv (Cornell University)|Jan 14, 2026

Holomorphic and Operator Theory0 citations

TL;DR

The paper shows that for a broad class of polynomial potentials with complex perturbations, the system of spectral projections of the anharmonic operator does not form a (Riesz) basis in L2(R), and spectral projection norms grow super-exponentially under certain conditions; it also develops a resolvent-based framework and a novel partial fraction identity to relate projection norms to resolvent growth.

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 $σ= [b - (a - 1)]/(1 + a)$ and $γ> 0$ 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 $\|(z - L)^{-1}\|$ outside of the spectrum $σ(L)$; (b) partial fraction decompositions of special meromorphic functions $1/F$ where $F(w) = \prod_{k=1}^\infty \left( 1 + \frac{w}{a_k} ight)$, $a_{k+1} \geq a_k>0$, $k \in \mathbb N$, and the generalization of the first resolvent identity.

Motivation & Objective

• Investigate whether spectral projections of L = -d^2/dx^2 + V(x) with complex polynomial V form a (Riesz) basis in L^2(R).
• Characterize how the growth of projection norms ||Pn|| relates to the resolvent norm ||(z-L)^{-1}|| outside the spectrum.
• Develop a regularized resolvent expansion using infinite-product (partial fraction) decompositions to control resolvent growth.
• Establish explicit conditions on the polynomial potential V (specifically a-1 < b < 2a) under which the basis property fails and norms grow exponentially.
• Extend the framework to abstract m-accretive operators and to imaginary/conjugated oscillators as applications.

Proposed method

• Analyze L = -d^2/dx^2 + x^{2a} + i x^b with a,b ∈ N and a-1 < b < 2a.
• Use spectral projection theory to relate Pn norms to resolvent growth via sectorial numerical range estimates.
• Introduce a regularization via a gauge function F from infinite products F(w)=∏(1+w/ak) and a resolvent identity Bz(T)=(1/F(z))(z-T)^{-1} + Σ (1/(z+an)F′(−an))(an+T)^{-1}.
• Derive partial fraction decompositions of 1/F for ρ<1/2 to enable convergent series representations of resolvent terms.
• Prove bounds on ||(z-L)^{-1}|| outside the spectrum and deduce non-basis properties of the projection system.
• Generalize key resolvent identities to abstract operators T and apply to imaginary and conjugated oscillators.

Experimental results

Research questions

• RQ1 Does the system of spectral projections {Pn} of L fail to form a Riesz basis in L^2(R) for the considered complex polynomial potentials?
• RQ2 How does the growth of the projection norms ||Pn|| relate to the growth of the resolvent norms ||(z-L)^{-1}|| outside the spectrum?
• RQ3 Can a regularized resolvent expansion via an infinite-product gauge function control or reveal the resolvent’s growth?
• RQ4 Under what precise relations between a and b (specifically a-1 < b < 2a) does the exponential growth of projections occur, and what is the sharp rate?
• RQ5 Do the developed methods extend to abstract m-accretive operators and to other oscillator variants (imaginary, odd/even, conjugated)?

Key findings

• For a broad class of polynomial potentials V with V(x)=x^{2a}+i x^b, the spectral projections of L do not form a basis in L^2(R) when a-1 < b < 2a.
• There is an explicit exponential growth rate bound for the projection norms, with limsup_n ||Pn||/exp(γ n^σ) = ∞ for σ=(b−(a−1))/(1+a) and small γ>0.
• A key identity Bz(T) = (1/F(z))(z−T)^{-1} + Σ (1/(z+an)F′(−an))(an+T)^{-1} enables propagating exponential-projection bounds to resolvent growth.
• Partial fraction decompositions of the auxiliary function 1/F, with F defined as an infinite product, yield convergent representations that regularize resolvent expansions and derive lower/upper resolvent bounds.
• The approach applies to abstract m-accretive operators with compact resolvent and to several oscillator variants (imaginary even/odd, conjugated real), yielding corresponding resolvent growth results.
• The paper connects spectral-projection growth to resolvent growth via detailed asymptotic eigenvalue behavior and proves completeness of the spectral projections despite lack of a basis.