We describe the spectraltheory 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.
Received 06 September 2012Accepted 22 April 2013Published online 03 June 2013
We thank the referee for his careful reading of our manuscript and for suggesting many improvements to our initial text.
Article outline: 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
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