Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues

@article{Liu2022IrreducibilityOT,
  title={Irreducibility of the Fermi variety for discrete periodic Schr{\"o}dinger operators and embedded eigenvalues},
  author={Wencai Liu},
  journal={Geometric and Functional Analysis},
  year={2022},
  pages={1-30}
}
  • Wencai Liu
  • Published 8 June 2020
  • Mathematics
  • Geometric and Functional Analysis
Let $$H_0$$ H 0 be a discrete periodic Schrödinger operator on $$\ell ^2(\mathbb {Z}^d)$$ ℓ 2 ( Z d ) : $$\begin{aligned} H_0=-\Delta +V, \end{aligned}$$ H 0 = - Δ + V , where $$\Delta $$ Δ is the discrete Laplacian and $$V:\mathbb {Z}^d\rightarrow \mathbb {C}$$ V : Z d → C is periodic. We prove that for any $$d\ge 3$$ d ≥ 3 , the Fermi variety at every energy level is irreducible (modulo periodicity). For $$d=2$$ d = 2 , we prove that the Fermi variety at every energy level except for the… 
Fermi isospectrality for discrete periodic Schrodinger operators
. Let Γ = q 1 Z ⊕ q 2 Z ⊕· · ·⊕ q d Z , where q l ∈ Z + , l = 1 , 2 , · · · , d . Let ∆+ V be the discrete Schr¨odinger operator, where ∆ is the discrete Laplacian on Z d and the potential V : Z d →
On spectral bands of discrete periodic operators
. We consider discrete periodic operator on Z d with respect to lattices Γ ⊂ Z d of full rank. We describe the class of lattices Γ for which the operator may have a spectral gap for arbitrarily small
Revisiting the Christ–Kiselev’s multi-linear operator technique and its applications to Schrödinger operators
Based on Christ–Kiselev’s multi-linear operator techniques, we prove several spectral results of perturbed periodic Schrödinger operators, including WKB type solutions, sharp transitions of
Topics on Fermi varieties of discrete periodic Schrödinger operators
  • Wencai Liu
  • Mathematics
    Journal of Mathematical Physics
  • 2022
This is a survey of recent progress on the irreducibility of Fermi varieties, rigidity results and embedded eigenvalue problems of discrete periodic Schrödinger operators.

References

SHOWING 1-10 OF 60 REFERENCES
Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators
In this paper, we consider the Schrodinger equation, $$Hu = - {u^"} + \left({V\left(x \right) + {V_0}\left(x \right)} \right)u = Eu,$$ where V0(x) is 1-periodic and V(x) is a decaying
Criteria for Embedded Eigenvalues for Discrete Schrödinger Operators
  • Wencai Liu
  • Mathematics
    International Mathematics Research Notices
  • 2019
In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n-1})+V(n)u(n). \end{equation*}$$We view $H$ as a perturbation of the free operator
The Landis conjecture on exponential decay
Consider a solution $u$ to $\Delta u +Vu=0$ on $\mathbb{R}^2$, where $V$ is real-valued, measurable and $|V|\leq 1$. If $|u(x)| \leq \exp(-C |x| \log^{1/2}|x|)$, $|x|>2$, where $C$ is a sufficiently
Reducible Fermi surfaces for non-symmetric bilayer quantum-graph operators
  • S. Shipman
  • Mathematics
    Journal of Spectral Theory
  • 2019
This work constructs a class of non-symmetric periodic Schrodinger operators on metric graphs (quantum graphs) whose Fermi, or Floquet, surface is reducible. The Floquet surface at an energy level is
On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators
AbstractThe article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co- compact free action of the
Fermi isospectrality for discrete periodic Schrodinger operators
. Let Γ = q 1 Z ⊕ q 2 Z ⊕· · ·⊕ q d Z , where q l ∈ Z + , l = 1 , 2 , · · · , d . Let ∆+ V be the discrete Schr¨odinger operator, where ∆ is the discrete Laplacian on Z d and the potential V : Z d →
Generic properties of dispersion relations for discrete periodic operators
An old problem in mathematical physics deals with the structure of the dispersion relation of the Schrodinger operator $-\Delta+V(x)$ in $R^n$ with periodic potential near the edges of the spectrum.
Discrete Bethe–Sommerfeld Conjecture
In this paper, we prove a discrete version of the Bethe–Sommerfeld conjecture. Namely, we show that the spectra of multi-dimensional discrete periodic Schrödinger operators on $${\mathbb{Z}^d}$$Zd
An overview of periodic elliptic operators
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
Bethe–Sommerfeld Conjecture
Abstract.We consider Schrödinger operator −Δ + V in $${\mathbb{R}}^d (d \geq 2)$$ with smooth periodic potential V and prove that there are only finitely many gaps in its spectrum.
...
...