具有复多项式势的非谐振子的谱投影
Spectral projections of an anharmonic oscillator with complex polynomial potential
AI总结：
研究具有复多项式势的非谐振子的谱投影性质，揭示其在特定条件下不构成Riesz基，并证明投影范数的增长行为。
摘要：
对于广义多项式势$V$，其中重要的且具有指导性的代表是$V(x) = x^{2a} + i x^b$，$x \in \mathbb R$，$a, b \in \mathbb N$，我们证明了非谐振子算子$L = - (\mathrm{d}/ \mathrm{d}x)^2 + V(x)$的谱投影系统$\{P_n\}_n$在$a - 1 < b < 2a$时不会在$L^2(\mathbb R)$中生成(Riesz)基。此外，对于$σ= [b - (a - 1)]/(1 + a)$和$γ> 0$足够小，$\limsup_n \|P_n\|/ \exp(γn^σ) = \infty$。证明基于两个小组的结果，这些结果本身具有极大的兴趣：(a) 投影$\|P_n\|$和谱外解析子$\|(z - L)^{-1}\|$的性质(增长)之间的关系；(b) 特殊的解析函数$1/F$的部分分式分解，其中$F(w) = \prod_{k=1}^\infty \left( 1 + \frac{w}{a_k} \right)$，$a_{k+1} \geq a_k>0$，$k \in \mathbb N$，以及第一解析子恒等式的推广。
英文摘要：
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 $F(w) = \prod_{k=1}^\infty \left( 1 + \frac{w}{a_k} \right)$, $a_{k+1} \geq a_k>0$, $k \in \mathbb N$, and the generalization of the first resolvent identity.
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