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Spectral projections of an anharmonic oscillator with complex polynomial potential

Boris Mityagin, Petr Siegl

TL;DR

We analyze the non-selfadjoint Schrödinger-type operator \(L = -\dfrac{d^2}{dx^2} + V(x)\) with the complex polynomial potential \(V(x) = x^{2a} + i x^b\) and establish that for the regime \(a-1 < b < 2a\) the spectral projections \(\{P_n\}\) fail to form a (Riesz) basis in \(L^2(\mathbb{R})\). A novel resolvent identity combined with a gauge function \(F\) built from an infinite product enables a regularized resolvent expansion and a partial fraction decomposition of \(1/F\), linking bounds on \|P_n\| to bounds on the resolvent \|(z-L)^{-1}\|. The key finding is that \limsup_{n\to\infty} \|P_n\|/\exp(\gamma n^{\sigma}) = \infty\ for small \gamma>0\) with \(\sigma = \frac{b-(a-1)}{1+a} \in (0,1)\), proving absence of a basis. The framework further yields resolvent-growth bounds and basis-implications for related models, including imaginary even/odd oscillators and conjugated real oscillators, illustrating how exponential-type projection growth drives non-basis behavior in a broad class of non-selfadjoint perturbations.

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 $F(w) = \prod_{k=1}^\infty \left( 1 + \frac{w}{a_k} \right)$, $a_{k+1} \geq a_k>0$, $k \in \mathbb N$, and the generalization of the first resolvent identity.

Spectral projections of an anharmonic oscillator with complex polynomial potential

TL;DR

We analyze the non-selfadjoint Schrödinger-type operator \(L = -\dfrac{d^2}{dx^2} + V(x)\) with the complex polynomial potential \(V(x) = x^{2a} + i x^b\) and establish that for the regime the spectral projections fail to form a (Riesz) basis in \(L^2(\mathbb{R})\). A novel resolvent identity combined with a gauge function built from an infinite product enables a regularized resolvent expansion and a partial fraction decomposition of , linking bounds on \|P_n\| to bounds on the resolvent \|(z-L)^{-1}\|. The key finding is that \limsup_{n\to\infty} \|P_n\|/\exp(\gamma n^{\sigma}) = \infty\ for small \gamma>0\) with \(\sigma = \frac{b-(a-1)}{1+a} \in (0,1)\), proving absence of a basis. The framework further yields resolvent-growth bounds and basis-implications for related models, including imaginary even/odd oscillators and conjugated real oscillators, illustrating how exponential-type projection growth drives non-basis behavior in a broad class of non-selfadjoint perturbations.

Abstract

For a broad class of polynomial potentials , with an important and instructive representative being , , , we show that the system of spectral projections of an anharmonic operator does not generate a (Riesz) basis in if . Moreover, for and small enough, . 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 and of the resolvent outside of the spectrum ; (b) partial fraction decompositions of special meromorphic functions where , , , and the generalization of the first resolvent identity.
Paper Structure (43 sections, 22 theorems, 372 equations)

This paper contains 43 sections, 22 theorems, 372 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some facts from operator theory
  4. One-dimensional Schrödinger operators with complex potentials
  5. Infinite products and partial fraction decompositions
  6. Spectral projections of complex perturbations of anharmonic oscillators
  7. Spectral properties of $L$
  8. From spectral projections to the resolvent
  9. Regularization of the resolvent expansion
  10. Resolvent estimate for $L$ and its corollaries
  11. Absence of basis property
  12. Faster resolvent growth
  13. From projections to the resolvent for an abstract $T$
  14. Imaginary even anharmonic oscillators
  15. Imaginary odd anharmonic oscillators
  16. ...and 28 more sections

Key Result

Proposition 2.1

Let $T$ be a densely defined closed operator in a separable Hilbert space $\mathcal{H}$. Assume that $\rho(T) \neq \emptyset$ and for some $z_0 \in \rho(T)$ with $q \in (0,\infty)$. Let $\gamma_1, \dots, \gamma_j$ be non-overlapping differentiable arcs having a limit direction at infinity and suppose that no adjacent pair of arcs forms an angle as great as $\pi/q$ at infinity. Suppose that, for so

Theorems & Definitions (51)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Proposition 3.1
  • proof : Sketch of proof
  • Proposition 3.2: Shin-2010-6
  • Theorem 3.3
  • ...and 41 more