Numerical Calculation Of The Energy Spectrum Of Quantum Graphs

One-dimensional stationary state problem is one of the most familiar and simplest basic problems in quantum mechanics.While the term "quantum graph" are usually to refer to a direct generalization of this kind of one-dimensional problems.From the mathematical point of view,quantum graph is the investigation of linear differential equations defined on some networks or graphs.From the perspectives of physics and chemistry,quantum graphs can be considered as an ideal models of many realistic systems,e.g.,electronic states in some macromolecules with complex network structures,Anderson localization in solids,propagation of sound or electromagnetic waves in the network,etc.This kind of differential operators usually has an extremely complex energy spectra,and some of them manifest some remarkable chaotic properties.The main purpose of this paper is to investigate the variations of the energy spectrum of quantum graphs with the length of the edge.We will take several typical quantum graphs as examples,calculate their energy spectra by numerical method,and analyze the variation of the energy eigenvalues with the change of the length of a specific edge by controlling variables.Firstly,the paper begins with a brief introduction to some basic concepts and theory concerning quantum graph,such as the Schr?dinger equation defined on quantum graph,and the boundary conditions satisfied by wave functions at a vertex.Then a system of linear equations determine the energy eigenvalues are given by the boundary conditions.We prove that the determinant of coefficient matrix of the system of linear equations must always be real or purely imaginary.It enables us to numerically calculate all the energy eigenvalues of a specific quantum graph within a finite energy interval by the criterion whether the determinant is vanishing.Finally,by taking some typical graphs,such as "8"-like graph,dumbbell-like and two-vertex three-edge quantum graphs as examples,the variation of some specific energy spectra with the change of some edge length is analyzed.It is found that,for a quantum graph with a given number of edges and the form they are connected(i.e.the topological property)remain unchanged,in general,its energy eigenvalue of a specific level decreases with the increase of the length of edge.Meanwhile,for a quantum graph with loops or several edges are closed,when the length of a loop is an integral multiple of some specific wave length,the corresponding energy level of this wavelength is invariant with the change of some edge lengths,the physical mechanism of occurrence of this kind of invariant levels are also discussed.