Two-scale estimates for limit-periodic Schrödinger operators
Abstract
We prove strong ballistic transport for one-dimensional continuum Schrödinger operators by comparing the spectral geometry at two consecutive periods. Enlarging a period decomposes each fiber into the eigenspaces of translation by the smaller cell. Away from the larger-period symmetry momenta, a simple ordered band remains in one translation sector and agrees with one translated smaller-period band. This permits us to combine large-period ordering and non-overlap with a smaller-period derivative bound. We obtain lower bounds for folded band widths and confine all near-collisions to a common neighborhood of the symmetry momenta, with the exponential cost at the smaller period. For nested periods pn with pn+1/pn∈ℕ and pn+1≥2pn, approximation errors o(pn+1−2e−ηpn), where η>7√(2R) and |V|∞≤ R, give a nonzero strong asymptotic velocity on a weighted H2 class containing the Schwartz space. Every position moment of order 0<r≤4 has two-sided ballistic growth. A period-uniform form estimate, ordered-gap summation, and a linear dynamical comparison complete the argument. No upper bound on successive period ratios is assumed, and the sufficient approximation exponents are not claimed to be optimal.
Related
- Polynomial Spectral Gap Estimates for Finitely Differentiable Quasi-Periodic Schrödinger Operators
- Bilinear Strichartz Estimates and Linear Morawetz Estimates for Schrödinger Operators with Potentials
- Band edges of periodic Schrödinger operators are generically isolated and nondegenerate
- Irreducibility of the Bloch variety for periodic Schrödinger operators in arbitrary dimension
- Local Smoothing Estimates of Schrödinger Operators on Waveguides Using Refined Decoupling
BibTeX
@misc{solis2026twoscale,
title = {{Two-scale estimates for limit-periodic Schrödinger operators}},
author = {Moises Gomez Solis},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01303},
eprint = {2611.01303},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01303},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two-scale estimates for limit-periodic Schrödinger operators
AU - Moises Gomez Solis
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01303
UR - https://arxiv.org/abs/2611.01303
AB - We prove strong ballistic transport for one-dimensional continuum Schrödinger operators by comparing the spectral geometry at two consecutive periods. Enlarging a period decomposes each fiber into the eigenspaces of translation by the smaller cell. Away from the larger-period symmetry momenta, a simple ordered band remains in one translation sector and agrees with one translated smaller-period band. This permits us to combine large-period ordering and non-overlap with a smaller-period derivative bound. We obtain lower bounds for folded band widths and confine all near-collisions to a common neighborhood of the symmetry momenta, with the exponential cost at the smaller period. For nested periods p_n with p_n+1/p_n∈ℕ and p_n+1≥2p_n, approximation errors o(p_n+1^−2e^−ηp_n), where η>7√(2R) and |V|_∞≤ R, give a nonzero strong asymptotic velocity on a weighted H^2 class containing the Schwartz space. Every position moment of order 0<r≤4 has two-sided ballistic growth. A period-uniform form estimate, ordered-gap summation, and a linear dynamical comparison complete the argument. No upper bound on successive period ratios is assumed, and the sufficient approximation exponents are not claimed to be optimal.
ER -