We study flat bands of periodic graphs in a Euclidean space. These are infinitely degenerate
eigenvalues of the corresponding adjacency matrix, with eigenvectors of compact support …

Abstract Let Γ= q 1 Z⊕ q 2 Z⊕⋯⊕ qd Z with arbitrary positive integers ql, l= 1, 2,⋯, d. Let Δ
discrete+ V be the discrete Schrödinger operator on Z d, where Δ discrete is the discrete …

Abstract Let Γ= q 1 Z⊕ q 2 Z⊕…⊕ qd Z Γ=q_1Z⊕q_2Z⊕...⊕q_dZ, where ql∈ Z+ q_l∈Z_+,
l= 1, 2,…, dl=1,2,...,d, are pairwise coprime. Let Δ+ V Δ+V be the discrete Schrödinger …

This article shows that for a large class of discrete periodic Schrödinger operators, most
wavefunctions resemble Bloch states. More precisely, we prove quantum ergodicity for a …

Let $\Delta+ V $ be the discrete Schr\" odinger operator, where $\Delta $ is the discrete
Laplacian on $\mathbb {Z}^ d $ and the potential $ V:\mathbb {Z}^ d\to\mathbb {C} $ is …

For periodic graph operators, we establish criteria to determine the overlaps of spectral band
functions based on Bloch varieties. One criterion states that for a large family of periodic …

We study the spectra of operators on periodic graphs using methods from combinatorial
algebraic geometry. Our main result is a bound on the number of complex critical points of …

We use methods from algebra and discrete geometry to study the irreducibility of the
dispersion polynomial of a discrete periodic operator associated to a periodic graph after …

A periodic linear graph operator acts on states (functions) defined on the vertices of a graph
equipped with a free translation action. Fourier transform with respect to the translation …

We use B\'{e} zout's theorem and Bernstein-Khovanskii-Kushnirenko theorem to analyze the
level sets of the extrema of the spectral band functions of discrete periodic Schr\" odinger …