PeriodicSpectra2d

Matlab routine to compute the spectrum of a Schrödinger operators with periodic potential defined on R². The implementation is based on a method recently developed in https://arxiv.org/abs/2104.09575. A Floquet-Bloch transform is used to replace R² by a bounded domain, then eigenvalues are identified as the zeros of a certain operator-valued function of the form I-K(z). This operator is discretized by choosing a Fourier basis. As proved in the article, the method is guaranteed to converge. The code takes the potential function as an input and returns a set of complex numbers, which approximates the spectrum of the Schrödinger operator.

The main building blocks of the routine are

• Main.m - Main script that returns approximation of the spectrum;
• potential(x,y) - returns value of potential function for input x,y in [0,1]²;
• compute_potential_matrix(a,N) - Computes operator matrix of the potential in Fourier representation. Input: vector of frequencies k, Fourier coefficients a and size of potential matrix N.
• newton_method(z_start, maxiter, tol) - performs Newton iteration until |det(I-K(z))|<tol or maxiter steps have been taken.

Any comments or queries are welcome at https://frank-roesler.github.io/contact/