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Spectral Properties of Schrödinger Operators on Perturbed Lattices

Kazunori Ando, Hiroshi Isozaki, Hisashi Morioka

Abstract

We study the spectral properties of Schrödinger operators on perturbed lattices. We shall prove the non-existence or the discreteness of embedded eigenvalues, the limiting absorption principle for the resolvent, construct a spectral representation, and define the S-matrix. Our theory covers the square, triangular, diamond, Kagome lattices, as well as the ladder, the graphite and the subdivision of square lattice.

Spectral Properties of Schrödinger Operators on Perturbed Lattices

Abstract

We study the spectral properties of Schrödinger operators on perturbed lattices. We shall prove the non-existence or the discreteness of embedded eigenvalues, the limiting absorption principle for the resolvent, construct a spectral representation, and define the S-matrix. Our theory covers the square, triangular, diamond, Kagome lattices, as well as the ladder, the graphite and the subdivision of square lattice.

Paper Structure

This paper contains 37 sections, 59 theorems, 426 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Basic properties of graph
  3. Vertices and edges
  4. Laplacian on the periodic graph
  5. Preliminary facts
  6. Examples of periodic lattices
  7. Square lattice
  8. Triangular lattice
  9. Hexagonal lattice
  10. Kagome lattice
  11. Diamond lattice
  12. Higher-dimensional diamond lattice
  13. Subdivision of $d$-dimensional Square Lattice ${\bf Z}^d$, $d\ge2$
  14. Ladder of $d$-dimensional Square Lattice in ${\bf R}^{d+1}$
  15. Graphite in ${\bf R}^3$
  16. ...and 22 more sections

Key Result

Lemma 2.1

(1) $\bigcup_{a \in \bf C}S_{a,sng}^{\bf C}(a_d) = \left(\pi{\bf Z}\right)^d\cap {\bf T}^d_{\bf C}$. (2) $SV(a_d) = \{-d, - d + 2, \cdots, d-2, d\}$. (3) $a_d({\bf T}^d) = [-d,d]$. (4) For $-d < a < d$, each connected component of $S_{a,reg}^{\bf C}(a_d)$ intersects with ${\bf T}^d$, and the inters

Figures (18)

  • Figure 1: Square lattice
  • Figure 2: Triangular lattice
  • Figure 3: Hexagonal lattice
  • Figure 4: Kagome lattice
  • Figure 5: Subdivision of $2$-dimensional square lattice
  • ...and 13 more figures

Theorems & Definitions (62)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • ...and 52 more