相关论文: Discrete Schrodinger equation on graphs: An effect…

We derive a formula for the level spacing probability distribution in quantum graphs. We apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases. Moreover, we derive an…

In the broad range of studies related to quantum graphs, quantum graph spectra appear as a topic of special interest. They are important in the context of diffusion type problems posed on metric graphs. Theoretical findings suggest that…

This paper presents a simple model that mimics quantum mechanics (QM) results in terms of probability fields of free particles subject to self-interference, without using Schroedinger equation or complex wavefunctions. Unlike the standard…

In this paper, we propose a circuit design for implementing quantum walks on complex networks. Quantum walks are powerful tools for various graph-based applications such as spatial search, community detection, and node classification.…

Here I present a new discrete model of quantum mechanics for relativistic 1-electron systems, in which particle movement is described by a directed space-time graph with attached 4-spinors, but without any continuous wave functions. These…

We show that all scaling quantum graphs are explicitly integrable, i.e. any one of their spectral eigenvalues $E_n$ is computable analytically, explicitly, and individually for any given $n$. This is surprising, since quantum graphs are…

The question of what conditions should be set at the nodes of a discrete graph for the wave equation with discrete time is investigated. The variational method for the derivation of these conditions is used. A parallel with the continuous…

Sampling algorithms, hypergraph degree sequences, and polytopes play a crucial role in statistical analysis of network data. This article offers a brief overview of open problems in this area of discrete mathematics from the point of view…

The proposed theory of causally structured discrete fields studies integer values on directed edges of a self-similar graph with a propagation rule, which we define as a set of valid combinations of integer values and edge directions around…

We present an alternative application of discrete Morse theory for two-particle graph configuration spaces. In contrast to previous constructions, which are based on discrete Morse vector fields, our approach is through Morse functions,…

We describe some basic tools in the spectral theory of Schr\"odinger operator on metric graphs (also known as "quantum graph") by studying in detail some basic examples. The exposition is kept as elementary and accessible as possible. In…

Deep Equilibrium Models (DEQs) replace a stack of explicit layers with a single operator whose fixed point defines the output, giving the expressive power of an arbitrarily deep network at the memory cost of a single layer. Quantum Deep…

In some previous works, the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters of the graph was established. However, a specific local coordinate chart on the…

We treat the stationary (cubic) nonlinear Schr\"odinger equation (NSLE) on simplest graphs. Formulation of the problem and exact analytical solutions of NLSE are presented for star graphs consisting of three bonds. It is shown that the…

We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on…

In the present paper, we precisely conduct a q-calculus method for the numerical solutions of PDEs. A nonlinear Schrodinger equation is considered. Instead of the classical discretization methods we consider subdomains according to…

We study the small amplitude linearization of the Korteweg de Vries equation on the line with a local defect scattering waves represented by a metric graph domain adjoined at one point. For a representative collection of examples, we derive…

In this paper we present a general framework for solving the stationary nonlinear Schr\"odinger equation (NLSE) on a network of one-dimensional wires modelled by a metric graph with suitable matching conditions at the vertices. A formal…

Quantum graphs were introduced to model free electrons in organic molecules using a self-adjoint Hamiltonian on a network of intervals. A second graph quantization describes wave propagation on a graph by specifying scattering matrices at…

Based on the numerical solution of the quantum lattice Boltzmann method in curved space, we predict the onset of a quantized alternating current on curved graphene sheets. Such numerical prediction is verified analytically via a set of…