相关论文: Reducible Fermi Surfaces for Non-symmetric Bilayer…

We construct two types of multi-layer quantum graphs (Schr\"odinger operators on metric graphs) for which the dispersion function of wave vector and energy is proved to be a polynomial in the dispersion function of the single layer. This…

We prove that the Fermi surface of a connected doubly periodic self-adjoint discrete graph operator is irreducible at all but finitely many energies provided that the graph (1) can be drawn in the plane without crossing edges (2) has…

Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$ with arbitrary positive integers $q_l$, $l=1,2,\cdots,d$. Let $\Delta_{\rm discrete}+V$ be the discrete Schr\"odinger operator on $\mathbb{Z}^d$, where…

Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$, where $q_l\in \mathbb{Z}_+$, $l=1,2,\cdots,d$. Let $\Delta+V$ be the discrete Schr\"odinger operator, where $\Delta$ is the discrete Laplacian on…

The aim of this article is to present a complete system of Floquet spectral invariants for the discrete Schr\"odinger operators with periodic potentials on periodic graphs. These invariants are polynomials in the potential and determined by…

In this paper, we consider the discrete periodic Schr\"odinger operators $\Delta+V$ on $\Z^d$, where $V$ is $\Gamma$-periodic with $\Gamma=q_1 \mathbb{Z}\oplus q_2\mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$ and positive integers $q_j$,…

It is known that, if a locally perturbed periodic self-adjoint operator on a combinatorial or quantum graph admits an eigenvalue embedded in the continuous spectrum, then the associated eigenfunction is compactly supported--that is, if the…

We derive Floquet theory from quantum geometry. We identify quasienergy folding as a consequence of a broken gauge group of the adiabatic gauge potential $U(1){\mapsto}\mathbb{Z}$. Fixing instead the gauge freedom using the…

Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z} $ with $q_1\in \mathbb{Z}_+$ and $q_2\in\mathbb{Z}_+$. Let $\Delta+X$ be the discrete periodic Schr\"odinger operator on $\mathbb{Z}^2$, where $\Delta$ is the discrete Laplacian and…

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 group reveals the central spectral objects, Bloch and…

We investigate the operator $-\Delta -\alpha \delta (x-\Gamma)$ in $L^2(\mathbb{R}^3)$, where $\Gamma$ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion…

We study the problem of how the Floquet property manifests for periodic Schr\"{o}dinger operators which are known to have multiple of asymptotic spectral solutions. The main conclusions are made for elliptic potentials, we demonstrate that…

We consider Schr\"odinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We obtain two-sided estimates of the total bandwidth for the Schr\"odinger operators in terms of…

Recent experimental advances in Floquet engineering and controlling dissipation in open systems have brought about unprecedented flexibility in tailoring novel phenomena without any static and Hermitian analogues. It can be epitomized by…

We present a method to estimate the number of irreducible components of the Fermi varieties of periodic Schr\"odinger operators on graphs in terms of suitable asymptotics. Our main theorem is an abstract bound for the number of irreducible…

For three-dimensional non-interacting multi-band metals, we show that important information about the shape and the quantum geometry of Fermi surfaces is encoded in the subleading logarithmic term of bipartite charge fluctuations. This…

We consider Schr\"odinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schr\"odinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a…

We study the spectrum of some periodic differential operators, in particular the periodic Schr\"{o}dinger operator acting on infinite $n$-cubic graphs. Using Floquet-Bloch theory, we derive and analyze on the dispersion relations of the…

A massive quantum particle on a two-dimensional curved surface experiences a surface-geometry induced attractive potential that is characterized by the radii of curvature at a given point. With bilayer graphene sheets and carbon…

The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which…