[论文解读] Curved Ingham inequalities and observability of the toroidal Schr{ö}dinger equation

Bernhard H. Haak, Philippe Jaming+2 more|arXiv (Cornell University)|Mar 16, 2026

Mathematical Analysis and Transform Methods被引用 0

一句话总结

本文通过建立 curved Ingham 型不等式并分别分析低频成分与高频成分，证明了从曲线时空轨迹得到 toroidal 薛定谔方程的可观测性。

ABSTRACT

We prove that solutions of the toroidal Schr{ö}dinger equation can be observed from suitably curved space-time trajectories, thus of zero Lebesgue measure. To do so, we establish new upper and lower bounds for certain trigonometric sums along curves, in the spirit of the celebrated Ingham inequality. In a second part, we establish observability properties over arbitrarily short curves of the low-and high-frequency components separately. For the low-frequency component, we establish strong restrictions on the zero sets of the trigonometric sums under consideration.

研究动机与目标

• Motivate the study of observability for dispersive equations on the torus from sets of zero measure.
• Develop curved Ingham-type inequalities that handle general curves via curvature assumptions.
• Separate analysis of low- and high-frequency components to obtain observability with minimal curve length.
• Provide conditions under which traces along curved trajectories control initial data.
• Extend the framework to bounded potentials and fractional Schrödinger equations on the torus.

提出的方法

• Construct curved Ingham inequalities by examining the trace of solutions along curves and proving the associated trigonometric sums form a Riesz system in L^2([0,T], dσ).
• Apply stationary phase and Van der Corput-type estimates to bound off-diagonal terms I_{n,m}(T).
• Partition the index set into good/bad regions to exploit dispersion and curvature.
• Obtain upper bounds for ∫_0^T |∑ c_n e^{2πi(n p(t)+|n|^s t)}|^2 dt and lower bounds under a minimal time T(p).
• Prove high-frequency curved Ingham inequalities showing equivalence between ℓ^2-norms of coefficients and L^2-curve traces for large N; derive corollaries for curved settings.
• Extend results to Schrödinger equations with bounded potentials and to fractional cases on the torus.

实验结果

研究问题

• RQ1 Can observability for toroidal Schrödinger equations be achieved from sets of zero Lebesgue measure by restricting to curved space–time trajectories?
• RQ2 What curvature and frequency conditions ensure curved Ingham-type inequalities hold for fractional Schrödinger equations on the torus?
• RQ3 How do high-frequency and low-frequency components behave differently under curved observability, and what minimal time is needed?
• RQ4 How do bounded potentials affect trace observability along curves and the control of initial data?
• RQ5 Under what conditions is the system of exponentials along curved trajectories a Riesz basis in L^2(μ) on curves?

主要发现

• Curved Ingham inequalities are established for fractional toroidal Schrödinger equations under Assumption (H_α) for p and s>3/2, providing upper bounds for curve traces.
• There exist T(p) and C(p) such that for T>T(p) a lower bound holds, showing observability from curved trajectories for large enough time.
• For high frequencies (large n), the trace along a smooth curve with nonzero curvature yields a polynomial-Fourier decay-based lower bound, enabling a curved Ingham-type estimate with N-dependent thresholds.
• For low frequencies, vanishing lemmas imply rigidity: a nontrivial finite-frequency sum cannot vanish along a general curve unless the curve is holomorphic/entire in a certain sense.
• The results extend to bounded potentials and to the fractional Schrödinger equation, yielding L^2 traces along curves and potential control of the initial data for small potentials and large times.
• Corollaries provide a High-Frequency Curved Ingham Inequality and a result on effective observability from curved trajectories when N is sufficiently large.