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arXiv · 2304.10441

Sturm-Liouville Problems And Global Bounds By Small Control Sets And applications to quantum graphs

Michela Egidi·Delio Mugnolo·Albrecht Seelmann
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Abstract

We develop a Logvinenko--Sereda theory for one-dimensional vector-valued self-adjoint operators. We thus deliver upper bounds on $L^2$-norms of eigenfunctions -- and linear combinations thereof -- in terms of their $L^2$- and $W^{1,2}$-norms on small control sets that are merely measurable and suitably distributed along each interval. An essential step consists in proving a Bernstein-type estimate for Laplacians with rather general vertex conditions. Our results carry over to a large class of Schrödinger operators with magnetic potentials; corresponding results are unknown in higher dimension. We illustrate our findings by discussing the implications in the theory of quantum graphs.

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Michela Egidi, Delio Mugnolo, Albrecht Seelmann. 2023-04-20. Sturm-Liouville Problems And Global Bounds By Small Control Sets And applications to quantum graphs. https://doi.org/10.1016/j.jmaa.2024.128101

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