Related papers: Quantum Ergodicity on Graphs : from Spectral to Sp…
We study joint eigenfunctions of the spherical Hecke algebra acting on $L^2(\Gamma_n \backslash G / K)$ where $G = \text{PGL}(3, F)$ with $F$ a non-archimedean local field of arbitrary characteristic, $K = \text{PGL}(3, O)$ with $O$ the…
The nodal domains of eigenvectors of the discrete Schrodinger operator on simple, finite and connected graphs are considered. Courant's well known nodal domain theorem applies in the present case, and sets an upper bound to the number of…
We consider the ensemble of adjacency matrices of Erd\H{o}s-R\'{e}nyi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\equiv p(N)$. We rescale the matrix so that its bulk…
We extend the approach from [arXiv:2110.15301] to prove windowed spectral projection estimates and a generalized Weyl law for the (Weyl) quantized baker's map on the torus. The spectral window is allowed to shrink in the semiclassical…
The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with $T$ gates whose underlying graph has treewidth $d$ can be simulated deterministically in…
In this paper we investigate when linearly independent eigenfunctions of the Schr\''odinger operator may have the same modulus. General properties are established and the one-dimensional case is treated in full generality. The study is…
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 consider discrete Schr\"odinger operators with periodic potentials on periodic graphs perturbed by guided non-positive potentials, which are periodic in some directions and finitely supported in other ones. The spectrum of the…
We investigate the controllability of an infinite-dimensional quantum system: a quantum particle confined on a Thick Quantum Graph, a generalisation of Quantum Graphs whose edges are allowed to be manifolds of arbitrary dimension with…
A continuous-time quantum walk is modelled using a graph. In this short paper, we provide lower bounds on the size of a graph that would allow for some quantum phenomena to occur. Among other things, we show that, in the adjacency matrix…
We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by…
Based on the main result presented in a recent paper, we derive Ambarzumian-type theorems for Schr\"odinger operators defined on quantum graphs. We recover existing results such as the classical theorem by Ambarzumian and establish some…
We study energy functionals associated with non-local quasi-linear Schr\"odinger operators, and develop a ground state representation. Our main focus is on infinite graphs but we also consider non-local quasi-linear Schr\"odinger operators…
For sequences of quantum ergodic eigenfunctions, we define the quantum flux norm associated to a codimension $1$ submanifold $\Sigma$ of a non-degenerate energy surface. We prove restrictions of eigenfunctions to $\Sigma$, realized using…
An infinite graph is quasi-transitive if its vertex set has finitely many orbits under the action of its automorphism group. In this paper we obtain a structure theorem for locally finite quasi-transitive graphs avoiding a minor, which is…
The Green's function of the discrete Sch\"odinger operator on a finite graph is considered. This setting reproduces Laplacian and signless Laplacian by adjusting appropriate potentials. We show two ways of the expression for the Green's…
We summarize recent works on the stability under disorder of the absolutely continuous spectra of random operators on tree graphs. The cases covered include: Schr\"odinger operators with random potential, quantum graph operators for trees…
We introduce the notion of Benjamini-Schramm convergence for quantum graphs. This notion of convergence, intended to play the role of the already existing notion for discrete graphs, means that the restriction of the quantum graph to a…
In this paper, we consider discrete time quantum walks on graphs with coin focusing on the decentralized model, where the coin operation is allowed to change with the vertex of the graph. When the coin operations can be modified at every…
The continuum limit of loop quantum gravity is still an open problem. Indeed, no proper dynamics in known to start with and we still lack the mathematical tools to study its would-be continuum limit. In the present PhD dissertation, we will…