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Journal of Spectral Theory


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Volume 10, Issue 3, 2020, pp. 843–885
DOI: 10.4171/JST/314

Published online: 2020-08-22

Unique continuation and lifting of spectral band edges of Schrödinger operators on unbounded domains

Ivica Nakić[1], Matthias Täufer[2], Martin Tautenhahn[3] and Ivan Veselić[4]

(1) University of Zagreb, Croatia
(2) Queen Mary University of London, UK
(3) Technische Universität Chemnitz, Germany
(4) Technische Universität Dortmund, Germany

We prove and apply two theorems: first, a quantitative, scale-free unique continuation estimate for functions in a spectral subspace of a Schrödinger operator on a bounded or unbounded domain; second, a perturbation and lifting estimate for edges of the essential spectrum of a self-adjoint operator under a semi-definite perturbation. These two results are combined to obtain lower and upper Lipschitz bounds on the function parametrizing locally a chosen edge of the essential spectrum of a Schrödinger operator in dependence of a coupling constant. Analogous estimates for eigenvalues, possibly in gaps of the essential spectrum, are exhibited as well.

Keywords: Unique continuation, uncertainty principle, semi-definite perturbation, spectral edges, lifting estimates, spectral engineering

Nakić Ivica, Täufer Matthias, Tautenhahn Martin, Veselić Ivan: Unique continuation and lifting of spectral band edges of Schrödinger operators on unbounded domains. J. Spectr. Theory 10 (2020), 843-885. doi: 10.4171/JST/314