Related papers: Absolutely Continuous Spectrum for Quantum Trees
We study basic spectral features of graph Laplacians associated with a class of rooted trees which contains all regular trees. Trees in this class can be generated by substitution processes. Their spectra are shown to be purely absolutely…
We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by…
We discuss the dynamical implications of the recent proof that for a quantum particle in a random potential on a regular tree graph absolutely continuous spectrum occurs non-perturbatively through rare fluctuation-enabled resonances. The…
A tree-strip of finite cone type is the product of a tree of finite cone type with a finite set. We consider random Schr\"odinger operators on these tree strips, similar to the Anderson model. We prove that for small disorder the spectrum…
We study a class of rooted trees with a substitution type structure. These trees are not necessarily regular, but exhibit a lot of symmetries. We consider nearest neighbor operators which reflect the symmetries of the trees. The spectrum of…
We analyze spectral properties of a quantum graph in the form of a ring chain with a $\delta$ coupling in the vertices exposed to a homogeneous magnetic field perpendicular to the graph plane. We find the band spectrum in the case when the…
We consider the Laplacian on a rooted metric tree graph with branching number $ K \geq 2 $ and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely…
The aim of this paper is twofold. First, we study eigenvalues and eigenvectors of the adjacency matrix of a bond percolation graph when the base graph is finite and well approximated locally by an infinite regular graph. We relate…
We study the point spectrum of a periodic quantum tree equipped with a Schr\"odinger type differential operator with delta-type vertex conditions, using subsets of the compact graph that generates the tree. We prove analogs of existing…
We will consider cross products of finite graphs with a class of trees that have arbitrarily but finitely long line segments, such as the Fibonacci tree. Such cross products are called tree-strips. We prove that for small disorder random…
We characterize the spectrum of the transition matrix for simple random walk on graphs consisting of a finite graph with a finite number of infinite Cayley trees attached. We show that there is a continuous spectrum identical to that for a…
We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then…
In this paper we study the adjacency spectrum of families of finite rooted trees with regular branching properties. In particular, we show that in the case of constant branching, the eigenvalues are realized as the roots of a family of…
On an infinite, radial metric tree graph we consider the corresponding Laplacian equipped with self-adjoint vertex conditions from a large class including $\delta$- and weighted $\delta'$-couplings. Assuming the numbers of different edge…
In this paper we start a systematic study of quantum field theory on random trees. Using precise probability estimates on their Galton-Watson branches and a multiscale analysis, we establish the general power counting of averaged Feynman…
We investigate spectral properties of quantum graphs in the form of a periodic chain of rings with a connecting link between each adjacent pair, assuming that wave functions at the vertices are matched through conditions manifestly…
We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a…
We prove the almost sure existence of absolutely continuous spectrum at low disorder for the Anderson model on the simplest example of a product of a regular tree with a finite graph. This graph contains loops of unbounded size.
We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. Moreover, we impose that each vertex has at least one forward neighbor. Then, we show that the spectrum of the Laplace operator…
We prove persistence of absolutely continuous spectrum for the Anderson model on a general class of tree-like graphs.