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A remark on the absence of eigenvalues in continuous spectra for discrete Schrödinger operators on periodic lattices

Kazunori Ando, Hiroshi Isozaki, Hisashi Morioka

TL;DR

This work extends Rellich–Vekua-type uniqueness from continuous Helmholtz operators to discrete Schrödinger operators on periodic lattices with exponentially decaying potentials. By constructing a growth bound via an increasing height function and employing Paley–Wiener-type arguments together with a momentum-space analyticity framework, the authors prove a Rellich-type theorem that rules out eigenvalues embedded in the essential spectrum for a broad lattice class (square, triangular, hexagonal, ladders). They also develop a momentum-space analytic picture and address unique continuation properties, showing UCP holds under certain finite-graph conditions but can fail on lattices like Kagome due to compactly supported eigenvectors. The results have implications for inverse scattering and spectral theory on lattices, clarifying when embedded eigenvalues can be excluded and highlighting the role of lattice geometry in UCP and spectral conclusions.

Abstract

We prove a Rellich-Vekua type theorem for Schrödinger operators with exponentially decreasing potentials on a class of lattices including square, triangular, hexagonal lattices and their ladders. We also discuss the unique continuation theorem and the non-existence of eigenvalues embedded in the continuous spectrum.

A remark on the absence of eigenvalues in continuous spectra for discrete Schrödinger operators on periodic lattices

TL;DR

This work extends Rellich–Vekua-type uniqueness from continuous Helmholtz operators to discrete Schrödinger operators on periodic lattices with exponentially decaying potentials. By constructing a growth bound via an increasing height function and employing Paley–Wiener-type arguments together with a momentum-space analyticity framework, the authors prove a Rellich-type theorem that rules out eigenvalues embedded in the essential spectrum for a broad lattice class (square, triangular, hexagonal, ladders). They also develop a momentum-space analytic picture and address unique continuation properties, showing UCP holds under certain finite-graph conditions but can fail on lattices like Kagome due to compactly supported eigenvectors. The results have implications for inverse scattering and spectral theory on lattices, clarifying when embedded eigenvalues can be excluded and highlighting the role of lattice geometry in UCP and spectral conclusions.

Abstract

We prove a Rellich-Vekua type theorem for Schrödinger operators with exponentially decreasing potentials on a class of lattices including square, triangular, hexagonal lattices and their ladders. We also discuss the unique continuation theorem and the non-existence of eigenvalues embedded in the continuous spectrum.

Paper Structure

This paper contains 22 sections, 11 theorems, 105 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Growth property of eigenfunctions
  3. Lattice Schrödinger operators
  4. Periodic lattice
  5. Examples
  6. Square lattice
  7. Triangular lattice
  8. Hexagonal lattice.
  9. Kagome lattice.
  10. Ladder.
  11. Momentum representation and Fermi surface
  12. Rellich type uniqueness theorem
  13. Analyticity of eigenfunction in momentum space
  14. Unique continuation property
  15. UCP
  16. ...and 7 more sections

Key Result

Lemma 2.2

Given a subset $\Omega \subset \mathcal{V}$, assume that there exists an increasing height function $h : \Omega \to {\bf Z}$. Let $\widehat{ V} (v)$ be a $\bf C$-valued bounded function on $\Omega$. Then, there exists a constant $C _0 > 0$ which depends only on $C$ in (muvgvwdegvbound) and $\sup_{v holds for any $\widehat{ u}$ satisfying $(\widehat{ \Delta}_{\Gamma} + \widehat{ V}(v))\widehat{u}

Figures (7)

  • Figure 1: Two-dimensional square lattice. $p_1 = 0$, ${\bf v}_1 = (1,0)$, ${\bf v}_2 = (0,1)$. $\mathbb{D}_h (x)$ is the domain of dependence for $(\widehat{H}-\lambda )\widehat{u}=0$ on the square lattice.
  • Figure 2: Trianglar lattice. $p_1 = 0$, ${\bf v}_1 = (1,0)$, ${\bf v}_2 = (1/2,\sqrt{3}/2)$. $\mathbb{D}_h (x)$ is the domain of dependence for $(\widehat{H}-\lambda )\widehat{u}=0$ on the triangle lattice.
  • Figure 3: Hexagonal lattice on ${\bf R}^2$. $p_1 = (1,0)$, $p_2 = (2,0)$. ${\bf v}_1 = (3/2,-\sqrt{3}/2)$, ${\bf v}_2 = (3/2,\sqrt{3}/2)$.
  • Figure 4: The domain of dependence for $(\widehat{H}-\lambda )\widehat{u}=0$ on the hexagonal lattice. The domain $\mathbb{D}_h (x)$ with (1) is the case where $x$ is the left-end point of a horizontal edge. The domain $\mathbb{D}_h (x)$ with (2) is the case where $x$ is the right-end point of a horizontal edge.
  • Figure 5: Kagome lattice on ${\bf R}^2$. $p_1 = (0,0)$, $p_2 = (1/2,0)$, $p_3 = (1/4,\sqrt{3}/4)$. ${\bf v}_1 = (1/2,\sqrt{3} /2)$, ${\bf v}_2 = (-1/2,\sqrt{3}/2)$.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Definition 5.1
  • ...and 7 more