Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1804.07816

Unique continuation and lifting of spectral band edges of Schrödinger operators on unbounded domains (With an Appendix by Albrecht Seelmann)

Ivica Nakić·Matthias Täufer·Martin Tautenhahn·Ivan Veselic·Albrecht Seelmann
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.SPmath-phmath.APmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ivica Nakić, Matthias Täufer, Martin Tautenhahn, Ivan Veselic, Albrecht Seelmann. 2020-08-17. Unique continuation and lifting of spectral band edges of Schrödinger operators on unbounded domains (With an Appendix by Albrecht Seelmann). https://arxiv.org/abs/1804.07816

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum

We obtain a representation of the critical almost Mathieu family as a Jacobi matrix that has a singularity. This allows us to prove that the Hausdorff dimension of its spectrum is not larger than 1/2 for all irrational frequencies, solving a long-standing problem. Other corollaries include two very short proofs of zero measure of the spectrum (e.g. Problem 5 in B. Simon's list of the 21'st century problems). We also obtain continuity of the measure of the spectrum for general singular Jacobi matrices, and prove a similar Hausdorff dimension result for the quantum graph graphene.

math.SP↗
arXiv2026

Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes

This article is concerned with mixed Steklov--Robin eigenvalues on bounded domains in $\mathbb{R}^{n}, n \geq 2$, with Lipschitz boundary. Specifically, we consider domains with symmetry of order $4$ containing a spherical hole. We obtain isoperimetric inequalities for the $k$-th Steklov-Robin eigenvalues for each $k \in \{2, 3, \dots, n+1\}$. We provide examples to emphasize the fact that the symmetry assumptions, on the family of domains considered, are crucial.

math.SP↗
arXiv2026

The Y-partition is the optimal three-partition for the disc and the harmonic oscillator

We prove that the Y-partition into three equal sectors is the minimal spectral three-partition both for the Dirichlet Laplacian on the unit disc and for the planar harmonic oscillator $-Δ+|x|^2$, with minimal energies $j_{3/2,1}^{2}$ and $5$; for the disc, this confirms a conjecture of Helffer and Hoffmann-Ostenhof. The minimizing regular strong partition is unique up to rotation, and every open minimizing partition has cells with the Dirichlet form domains of the sectors. The proof is a positive radial transplantation to the sphere that preserves segregation and matches the angular-energy measures of the separated model states; the three-lune theorem of Helffer, Hoffmann-Ostenhof, and Terracini then gives the lower bound. The transplantation lowers the shifted quadratic form by a nonnegative defect with strictly positive radial weight; in the equality case, spherical equipartition makes the defects vanish, which separates variables, and a Poincaré inequality on the circle identifies the sectors.

math.SP↗