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

Abstract

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.

On a Minimax Principle in Spectral Gaps

“…To this end, the result by Griesemer, Lewis, and Siedentop in [11] is adapted to a perturbative setting. In the particular case of bounded additive perturbations, this has already been done by the present author in the appendix to [27] with hypotheses that can, under reasonable assumptions, be verified explicitly by means of the Davis-Kahan sin 2 theorem from [3] or variants thereof. The latter has been successfully applied in [27] to study lower bounds on the movement of eigenvalues in gaps of the essential spectrum and of edges of the essential spectrum.…”

“…In the particular case of bounded additive perturbations, this has already been done by the present author in the appendix to [27] with hypotheses that can, under reasonable assumptions, be verified explicitly by means of the Davis-Kahan sin 2 theorem from [3] or variants thereof. The latter has been successfully applied in [27] to study lower bounds on the movement of eigenvalues in gaps of the essential spectrum and of edges of the essential spectrum. In the current work, the considerations from [27,Appendix A] are extended and supplemented to cover also certain unbounded perturbations, in particular ones that are off-diagonal with respect to the spectral gap under consideration.…”

“…Four results in this direction are presented here, each addressing a different situation, which are not contained in the previously known results in the sense that their hypotheses do not seem to imply (1.1), its operator analogue, or (1.2) directly. We first treat the case of operator perturbations and start with the direct extension of [27,Theorem A.2] to infinitesimal perturbations. Recall that an operator V with Dom(V…”

“…Theorem 1.2 is proved in Sect. 4, which is based on the author's appendix to [27] and extends the corresponding considerations to certain unbounded perturbations. Theorems 1.3-1.5 are proved in Sect.…”

“…We first consider the situation of indefinite or semidefinite bounded perturbations, which has partially been discussed in a slightly different form in [27]. For a bounded self-adjoint operator V we define bounded nonnegative operators V ( p) and…”