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

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.

References in corpus (14)

• Quantum wires with magnetic fluxes
• Surgery principles for the spectral analysis of quantum graphs
• Localization of eigenfunctions via an effective potential
• Edge connectivity and the spectral gap of combinatorial and quantum graphs
• Sharp estimates and homogenization of the control cost of the heat equation on large domains
• An abstract Logvinenko-Sereda type theorem for spectral subspaces
• Localization effects in a periodic quantum graph with magnetic field and spin-orbit interaction
• The effective potential of an $M$-matrix
• Scale-free unique continuation estimates and Logvinenko-Sereda Theorems on the torus
• The Logvinenko-Sereda Theorem for the Fourier-Bessel transform
• Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density
• Control problem for quadratic parabolic differential equations with sparse sensor sets of finite volume or anisotropically decaying density
• $L^p$ norms and support of eigenfunctions on graphs
• Always detectable eigenfunctions on metric graphs