Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2206.09663

Trace formulas for magnetic Schr\"odinger operators on periodic graphs and their applications

Evgeny Korotyaev·Natalia Saburova
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider Schr\"odinger operators with periodic magnetic and electric potentials on periodic discrete graphs. The spectrum of such operators consists of a finite number of bands. We determine trace formulas for the magnetic Schr\"odinger operators. The traces of the fiber operators are expressed as finite Fourier series of the quasimomentum. The coefficients of the Fourier series are given in terms of the magnetic fluxes, electric potentials and cycles in the quotient graph from some specific cycle sets. Using the trace formulas we obtain new lower estimates of the total bandwidth for the magnetic Schr\"odinger operator in terms of geometric parameters of the graph, magnetic fluxes and electric potentials. We show that these estimates are sharp.

Explore related subjects

math.SP

Keep this discovery

BibTeXRIS

Evgeny Korotyaev, Natalia Saburova. 2022-06-20. Trace formulas for magnetic Schr\"odinger operators on periodic graphs and their applications. https://arxiv.org/abs/2206.09663

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

KEEP EXPLORING

Related papers

arXiv2026

Regular hyperbolic tilings have no $\ell^2$ eigenfunctions

We show that the adjacency operator of the $1$-skeleton of any regular tiling of the hyperbolic plane has no nonzero square-integrable eigenfunctions. As a consequence, the same holds for every infinite connected regular graph admitting a proper planar embedding with regular dual.

math.SP↗
arXiv2026

Inverse Heat Source Problems from Boundary Flux and Interior Observations on Sets of Low Hausdorff Dimension

This paper investigates conditional stability for inverse source problems for the heat equation with a known temporal factor and an unknown spatial component in a bounded $C^{1,1}$ domain. We focus on observations supported on sets of low Hausdorff dimension and establish conditional stability in this setting. For boundary observations on compact sets of positive $q$-dimensional Hausdorff content, we establish logarithmic stability from full-time boundary flux observations and double-logarithmic stability from delayed-time boundary flux observations. The admissible dimensional ranges are $q>d-2$ when the observation set is contained in a flat boundary patch and $q>d-1-c_{d+1}$ on a general $C^{1,1}$ boundary, where $c_{d+1}>0$ depends only on the dimension. A key ingredient in deriving these results is a new boundary spectral inequality for the Dirichlet Laplacian, which controls a finite Dirichlet spectral sum through observations of the normal derivative of its elliptic extension on such a boundary set. Our results also cover inverse heat source problems with interior observations on sets of positive $q$-dimensional Hausdorff content for some $q>d-1$, yielding logarithmic stability from full-time observations for general sources in $H_0^1(\Omega)$ and H\"older stability from terminal-time observations for sources in a suitable spectral Gevrey class.

math.SP↗
arXiv2026

Resolvent bounds and eigenvalue estimates of generalized Schr\"odinger operators with complex potentials on compact manifolds

We extend Cuenin's compact-manifold spectral bounds for Schr\"odinger operators with complex potentials to a general pseudodifferential setting. More precisely, we study operators \(P+V\), where \(P\) is a positive self-adjoint elliptic classical pseudodifferential operator of positive order and \(V\) is complex-valued. The main analytic input is a resolvent principle showing that spectral cluster estimates for \(P\) imply \(L^p\)-\(L^{p'}\) resolvent estimates along suitable complex curves. Combined with Sogge's spectral cluster bounds, this yields exterior-region resolvent estimates extending those of Krupchyk and Uhlmann; we also prove direct resolvent bounds in the interior region. On Zoll manifolds, we discuss the sharpness of the resulting spectral bounds.

math.SP↗