# Spectral Differentiation Matrices for the Numerical Solution of Schroedinger's equation

This page contains additional material related to the paper "Spectral Differentiation Matrices for the Numerical Solution of Schroedinger's Equation", submitted to the Conference Proceedings of the Workshop on the Physics of Non-Hermitian Operators, that was held in Stellenbosch, South Africa, November 2005.

This research was sponsored by the National Research Foundation in South Africa, under grant FA2005032300018.

The paper is concerned with the numerical solution of

```
-y''(x) + p(x) y = E y,

```
on the real line (assuming decaying boundary conditions).

Below are some MATLAB M-files, some of which are listed in the paper, for computing spectra of various potentials p(x) . Download the main code and execute by typing its name in MATLAB. The following are auxilliary codes from DMSUITE that should be downloaded once. (For the complete DMSUITE collection, see the local site or the MathWorks site).

### Illustration 1

This is the computation of the spectrum of the quadratic oscillator p(x) = x^2 , as given in Table 1 of the paper. M-file: table1.m

### Illustration 2

This is the computation of the real spectrum of the PT-symmetric potential p(x) = x^4+iax , as given in Table 2 of the paper. It reproduces Figure 2 of the paper. M-file: table2.m

The code of Table 2 can be combined with MATLAB's fminsearch to minimize the distance between E0 and E1. This computes the critical value of a=3.169036141. Main M-file: finda.m Auxilliary: acrit.m

### Illustration 3

This is the computation of the real spectrum of the QES potential p(z) = -z^4-2bz^2-6iz , which reproduces Figure 3 of the paper. The code is omitted in the paper, but for the sake of numbering we'll assume it was given in a Table 3. M-file: table3.m

Andre Weideman