^{1,a)}and Françoise Truc

^{1,b)}

### Abstract

We describe the spectral theory of the adjacency operator of a graph which is isomorphic to a regular tree at infinity. Using some combinatorics, we reduce the problem to a scattering problem for a finite rank perturbation of the adjacency operator on a regular tree. We develop this scattering theory using the classical recipes for Schrödinger operators in Euclidian spaces.

We thank the referee for his careful reading of our manuscript and for suggesting many improvements to our initial text.

I. INTRODUCTION

II. THE SETUP: GRAPHS ASYMPTOTIC TO A REGULAR TREE

III. THE SPECTRAL DECOMPOSITION OF THE ADJACENCY MATRIX OF THE TREE AND THE FOURIER-HELGASON TRANSFORM

A. Points at infinity

B. The spectral Riemann surface

C. Calculation of the Green's function

D. The density of states

E. The Fourier-Helgason transform

IV. A SCATTERING PROBLEM FOR A SCHRÖDINGER OPERATOR WITH A COMPACTLY SUPPORTED NON-LOCAL POTENTIAL

A. Formal derivation of the Lippmann-Schwinger equation

B. Existence and uniqueness of the solution for the modified “Lippmann-Schwinger-type” equation

C. The set and the pure point spectrum

D. The deformed Fourier-Helgason transform

1. The relation of the deformed Fourier-Helgason transform with the resolvent

2. End of the proof of Theorem 4.3

V. THE SPECTRALTHEORY FOR A GRAPH ASYMPTOTIC TO A REGULAR TREE

A. Some combinatorics

B. The spectraltheory of Γ

VI. OTHER FEATURES OF THE SCATTERING THEORY IN THE SETTING OF SEC. IV

A. Correlation of scattered plane waves

B. The T-matrix and the S-matrix

C. The S-matrix and the asymptotics of the deformed plane waves

D. Computation of the transmission coefficients in terms of the Dirichlet-to Neumann operator

### Key Topics

- Fourier transforms
- 17.0
- Green's function methods
- 14.0
- Scattering theory
- 8.0
- Electron densities of states
- 6.0
- Fourier transform spectroscopy
- 6.0

## Figures

A graph Γ asymptotic to a regular 2-tree with L = 3; the edge boundary ∂ e Γ0 has 4 edges.

A graph Γ asymptotic to a regular 2-tree with L = 3; the edge boundary ∂ e Γ0 has 4 edges.

A regular tree with q = 2 and some level sets of a Busemann function.

A regular tree with q = 2 and some level sets of a Busemann function.

The surface S, the map from S to , and the double cover of S 0 over I q .

The surface S, the map from S to , and the double cover of S 0 over I q .

A simple example with strictly larger than K.

A simple example with strictly larger than K.

Changing the graph with ν = 0 into : the dashed edges are the new edges, the continuous ones the old edges. The picture is done in the same situation as in Figure 6 .

Changing the graph with ν = 0 into : the dashed edges are the new edges, the continuous ones the old edges. The picture is done in the same situation as in Figure 6 .

The construction in the proof of Theorem 5.1; for the graph Γ one has q = 3, ν = −1, N′ = N″ = 1, m = 1, M = 5.

The construction in the proof of Theorem 5.1; for the graph Γ one has q = 3, ν = −1, N′ = N″ = 1, m = 1, M = 5.

The tree , the ball B n−1, and the end T 1 for n = 3.

The tree , the ball B n−1, and the end T 1 for n = 3.

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