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

Abstract

Let $H_0$ be a discrete periodic Schrödinger operator on ${\ell }^2({\mathbb{Z}}^d)$: $$H_0=-\Delta +V,$$ where $\Delta$ is the discrete Laplacian and $V:{\mathbb{Z}}^d\to \mathbb{C}$ is periodic. We prove that for any $d\ge 3$, the Fermi variety at every energy level is irreducible (modulo periodicity). For $d=2$, we prove that the Fermi variety at every energy level except for the average of the potential is irreducible (modulo periodicity) and the Fermi variety at the average of the potential has at most two irreducible components (modulo periodicity). This is sharp since for $d=2$ and a constant potential $V$, the Fermi variety at $V$-level has exactly two irreducible components (modulo periodicity). We also prove that the Bloch variety is irreducible (modulo periodicity) for any $d\ge 2$. As applications, we prove that when $V$ is a real-valued periodic function, the level set of any extrema of any spectral band functions, spectral band edges in particular, has dimension at most $d-2$ for any $d\ge 3$, and finite cardinality for $d=2$. We also show that $H=-\Delta +V+v$ does not have any embedded eigenvalues provided that $v$ decays super-exponentially.

References in corpus (10)

• Bethe-Sommerfeld Conjecture
• Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues
• Irreducibility of the Bloch Variety for Finite-Range Schrödinger Operators
• Reducible Fermi surface for multi-layer quantum graphs including stacked graphene
• Topics on Fermi varieties of discrete periodic Schrödinger operators
• The Landis conjecture on exponential decay
• Zeroes of the spectral density of the periodic Schroedinger operator with Wigner-von Neumann potential
• Periodic and limit-periodic discrete Schrödinger operators
• Fermi isospectrality for discrete periodic Schrodinger operators
• On spectral bands of discrete periodic operators

Cited by in corpus (7)

• Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues
• Irreducibility of the Bloch Variety for Finite-Range Schrödinger Operators
• Topics on Fermi varieties of discrete periodic Schrödinger operators
• Fermi isospectrality of discrete periodic Schrödinger operators with separable potentials on ${\mathbb{Z}}^2$
• Fermi isospectrality for discrete periodic Schrodinger operators
• Localisation for Delone operators via Bernoulli randomisation
• The curious spectra and dynamics of non-locally finite crystals