[Paper Review] Spectral Inequalities For Anisotropic Shubin Operators

Jérémy Martin|arXiv (Cornell University)|May 24, 2022

Stability and Controllability of Differential Equations4 citations

TL;DR

This paper establishes new spectral inequalities for finite combinations of eigenfunctions of anisotropic Shubin operators, using uncertainty principles in Gelfand-Shilov spaces and Bernstein-type estimates. It proves null-controllability in any positive time for evolution equations associated with these operators from any control set with positive Lebesgue measure, excluding the harmonic oscillator case.

ABSTRACT

In this paper, new spectral inequalities for finite combinations of eigenfunctions of anisotropic Shubin operators are presented. Given a subset $ω$ and an energy level, we provide an explicit control of the ratio of the L 2 (R d)-norm over the L 2 ($ω$)-norm with respect to the energy level. The proofs are based on recent uncertainty principles holding in Gelfand-Shilov spaces and Bernstein type estimates deduced from quantitative smoothing effects proved by Paul Alphonse. These spectral inequalities allow to derive the null-controllability in any positive time from any control subset with positive Lebesgue measure of evolution equations associated to anisotropic Shubin operators, except for the harmonic oscillator.

Motivation & Objective

• To establish quantitative spectral inequalities for finite-dimensional spectral projections of anisotropic Shubin operators.
• To extend known results on spectral inequalities beyond the harmonic oscillator case to general anisotropic Shubin operators with $k,m \geq 1$.
• To derive null-controllability results for evolution equations governed by these operators using observability and spectral estimates.
• To provide explicit, quantitative bounds on the spectral constants depending on the geometry of the control set and operator parameters.
• To generalize existing results on thick and weakly thick sets to the anisotropic setting using Gelfand-Shilov regularity and refined uncertainty principles.

Proposed method

• Leverages recent uncertainty principles in Gelfand-Shilov spaces to control the concentration of eigenfunctions.
• Applies new Bernstein-type estimates derived from Paul Alphonse's quantitative smoothing effects for Shubin operators.
• Uses the notion of $\delta$-weakly thick sets with respect to a density $\rho(x) = R\langle x\rangle^\delta$ to characterize control sets.
• Applies the Hilbert Uniqueness Method to link observability of the adjoint system to null-controllability of the forward system.
• Employs Theorem 6.4 (Beauchard, Egidi & Pravda-Starov) to derive observability estimates from spectral and dissipation inequalities.
• Combines spectral projection estimates with $L^2$-norm control on measurable subsets to establish the main inequality $\|f\|_{L^2} \leq C_{k,m,\lambda}(\omega)\|f\|_{L^2(\omega)}$.

Experimental results

Research questions

• RQ1 What are the sharp spectral inequalities for finite combinations of eigenfunctions of anisotropic Shubin operators?
• RQ2 How do spectral constants depend on the geometry of the control set $\omega$ and the parameters $k,m$?
• RQ3 Can null-controllability be established for evolution equations associated with anisotropic Shubin operators from sets of positive Lebesgue measure?
• RQ4 What is the role of Gelfand-Shilov regularity and uncertainty principles in deriving such spectral estimates?
• RQ5 How do the results extend beyond the harmonic oscillator case ($k=m=1$) to general $k,m \geq 1$?

Key findings

• The paper establishes spectral inequalities of the form $\|f\|_{L^2} \leq C_{k,m,\lambda}(\omega)\|f\|_{L^2(\omega)}$ for $f$ in finite spectral projections of anisotropic Shubin operators.
• For $\delta$-weakly thick sets with respect to $\rho(x) = R\langle x\rangle^\delta$, the spectral constant satisfies $C_{k,m,\lambda}(\omega) \leq C(8^\nu e^\nu A)^{\lambda^{\delta}}$ for some $\nu, \mu > 0$, with explicit dependence on $k,m,\lambda$.
• Null-controllability in any positive time is proven for evolution equations associated with $H_{k,m}$, except for the harmonic oscillator ($k=m=1$), from any control set $\omega$ with positive Lebesgue measure.
• The results generalize prior work on the harmonic oscillator by providing explicit, quantitative bounds on the spectral constants under weaker geometric assumptions.
• The key technical innovation lies in combining Gelfand-Shilov uncertainty principles with refined Bernstein-type estimates from quantitative smoothing effects.
• The proof relies on the equivalence between observability and null-controllability via the Hilbert Uniqueness Method and the application of Theorem 6.4 to derive exponential observability estimates.