Eigenvalue Problems

Abstract

Eigenvalue problems are omnipresent in physics. Important examples are the time independent Schrödinger equation in a finite orthogonal basis or the harmonic motion of a molecule around its equilibrium structure. Most important are ordinary eigenvalue problems, which involve the solution of a homogeneous system of linear equations with a Hermitian (or symmetric, if real) matrix. Matrices of small dimension can be diagonalized directly by determining the roots of the characteristic polynomial and solving a homogeneous system of linear equations. The Jacobi method uses successive rotations to diagonalize a matrix with a unitary transformation. A very popular method for not too large symmetric matrices reduces the matrix to tridiagonal form which can be diagonalized efficiently with the QL algorithm. Some special tridiagonal matrices can be diagonalized analytically. Krylov-space algorithms are the choice for matrices of very large dimension. We discuss especially the methods by Arnoldi and Lanczos.

Problems

Problem 10.1 Computer Experiment: Disorder in a Tight-Binding Model

We consider a two-dimensional lattice of interacting particles. Pairs of nearest neighbors have an interaction V and the diagonal energies are chosen from a Gaussian distribution

$$\begin{aligned} P(E)=\frac{1}{\varDelta \sqrt{2\pi }}e^{-E^{2}/2\varDelta ^{2}}. \end{aligned}$$

(10.135)

The wave function of the system is given by a linear combination

$$\begin{aligned} \psi =\sum _{ij}C_{ij}\psi _{ij} \end{aligned}$$

(10.136)

where on each particle (i, j) one basis function $\psi _{ij}$ is located. The nonzero elements of the interaction matrix are given by

$$\begin{aligned} H(ij|ij)=E_{ij} \end{aligned}$$

(10.137)

$$\begin{aligned} H(ij|i\pm 1,j)=H(ij|i, j\pm 1)=V. \end{aligned}$$

(10.138)

The Matrix H is numerically diagonalized and the amplitudes $C_{ij}$ of the lowest state are shown as circles located at the grid points. As a measure of the degree of localization the quantity

$$\begin{aligned} \sum _{ij}|C_{ij}|^{4} \end{aligned}$$

(10.139)

is evaluated. Explore the influence of coupling V and disorder $\varDelta$.