Periodic Jacobi matrices on trees

Abstract

We begin the systematic study of the spectral theory of periodic Jacobi matrices on trees including a formal definition. The most significant result that appears here for the first time is that these operators have no singular continuous spectrum. We review important previous results of Sunada and Aomoto and present several illuminating examples. We present many open problems and conjectures that we hope will stimulate further work.

Remarks on periodic Jacobi matrices on trees

“…Example 3.1. The free Laplacian on a degree d homogeneous tree has a Green's function (diagonal matrix element of the resolvent) that is well-known and computed, for example, in [3,Example 7.1]. [3, (7.3)] says that…”

“…The rg-model [3, Example 7.2] mentioned at the end of Section 2 was originally emphasized by Aomoto [1] because he showed, by proving one of its Green's functions had a pole, it has a point eigenvalue at 0 which is outside the continuous spectrum of H. Earlier, Godsil-Mohar [10] had also noted the model had a spectral measure with a pure point at zero and computed the weight it contributes to the IDS. [3] wrote down an explicit zero energy eigenvector, namely view the corresponding tree T r,g (with r > g) as a tree centered at a single red vertex at level 0, with g vertices at level 1, then g(r − 1) vertices at level 2, each level 1 vertex linked to r − 1 level 2 vertices, etc. Thus, level 2k − 1; k = 1, 2, .…”

“…This paper is a contribution to a growing recent literature on periodic Jacobi matrices on trees. Specifically, we were motivated by the work of Avni, Breuer and Simon [3] (see also [2]) whose notation and ideas we will follow; in particular, we refer the reader to that paper for the definitions from graph theory that we will use. We note also the relevance of some recent preprints by a group at Berkeley [8,4].…”

“…There are three big general theorems in the subject: (1) a result of Sunada [21] on labelling of gaps in the spectrum that implies the spectrum of H is at most p bands where p is the number of vertices in the underlying finite graph G, (2) a result of Aomoto [1] stating that if G has a fixed degree then H has no point spectrum, and (3) a result of Avni-Breuer-Simon [3] that there is no singular spectrum because matrix elements of the resolvent are algebraic functions. Besides a very few additional theorems, the subject at this point is mainly some interesting examples and lots of conjectures and questions.…”

“…Section 2 discusses quantitative estimates on the distance from σ to the top of the spectrum of H. Section 3 will show that in a number of examples, σ can be viewed as an antibound state and we make a general conjecture that it is a pole of the analytically continued Green's function. Section 4 studies an example of a finite graph with non-fixed degree for which Aomoto [1] showed that H has a point eigenvalue and ABS [3] found an explicit eigenfunction. We prove that this eigenfunction and its translates span the eigenspace.…”