Quantitative local inversion and a minimax depth barrier for canonical systems from fixed-height Weyl-Schur seam data
Sharan Thota
TL;DR
This work analyzes fixed-height seam sampling for trace-normed $2\times2$ canonical systems with a free tail, focusing on how spectral data encoded by the Weyl function $m_{H,\Λ}$ and its Cayley transform $v$ respond to perturbations. By linearizing around the free Hamiltonian and exploiting Paley–Wiener/de Branges structures, the authors derive explicit stability bounds, depth-barrier phenomena, and optimal sampling designs, particularly in block-constant models. They prove that fixed-height data induces an exponential minimax barrier for recovering high-frequency spectral features and establish a precise input–output framework via kernel frames and time–frequency limiting operators, enabling quantitative local inversion and constructive reconstruction. The results connect de Branges’ correspondence, Poisson smoothing, and operator-theoretic descriptions of Dirac-type systems, offering intrinsic limits and practical guidance for robust spectral estimation in one-dimensional canonical systems. The findings have potential implications for inverse spectral problems and related areas where fixed-height sampling and depth attenuation govern the conditioning of spectral deconvolution.
Abstract
Fix $Λ>0$ and consider trace-normed $2\times 2$ canonical systems $\begin{pmatrix}0&-1\\1&0\end{pmatrix}Y'(s)=zH(s)Y(s)$ on $[0,Λ]$, extended by the free tail $H(s)=\frac12 I$ for $s\geΛ$. The Weyl-Titchmarsh coefficient $m_{H,Λ}$ and its Cayley transform $v$ encode the spectral measure. We study the finite sampling operator (the seam map) that sends $H$ (or a finite-dimensional parameter $θ$) to the values $v(z_k)$ at points $z_k=x_k+iη$ with $η>0$, $k=1,\dots,M$. For piecewise-constant (block) Hamiltonians, with block length $\ell=Λ/N$, we analyze stability and optimal sampling. We prove three types of results. (1) The free tail forces the transfer matrix at $Λ$ to have entire entries of exponential type at most $Λ$; consequently $m_{H,Λ}$ and $v$ extend meromorphically to $\mathbb{C}$, showing that the free-tail class is a strict spectral subclass. (2) Fixed-height data is intrinsically smoothed: $\operatorname{Im} m(\cdot+iη)$ is the Poisson extension of the spectral measure, yielding an exponential minimax obstruction to recovering oscillatory spectral features from finitely many noisy samples. (3) Linearization at the free Hamiltonian identifies seam sampling with an evaluation operator in a Paley-Wiener/de Branges space, so stability is governed by kernel frame bounds; shifted equispaced designs give optimal tight frames and explicit conditioning estimates. Finally, a perturbation formula expresses $Dm$ as an $L^2([0,Λ];\mathbb{R}^2,H)$ pairing with $ΔH$, linking depth attenuation to decay of Weyl solutions.