Measuring the Quality of the Solution

Before visualizing the eigenfunctions, we should determine values of the parameters , , , , , and working precisions appropriate for the energy values under consideration (say ).

The function solutionQuality carries out numerical integrations along the domain boundaries to quantify the quality of the solutions.

The function multiParameterSolutionQuality constructs and solves the matrices and then measures the quality of the solutions for a set of parameters. The actual solution of the linear system of overdetermined equations is achieved using the function PseudoInverse. The function multiParameterSolutionQuality returns lists of the form:

The value integratedBoundaryValuesDifferences is the result of applying solutionQuality to the solution and reflectionAndTransmissionCoefficientMinius1 is . The last quantity is expected to vanish for the exact solution.

We now carry out some experiments to determine appropriate parameters. We start by determining the number of Fourier modes versus the number of basis states. The following inputs show that 5 to 10 more Fourier modes give good results.

We continue by determining the number of basis states needed in the arms and in the center disk. Obviously, the more basis states we take into account, the better the resulting solution. But after considering four states in the arms and about 30 in the center disk, the solution gets better only slowly. For mainly graphical purposes, we obtain a satisfactory quantitative solution.

We end by finding the precision needed for the calculations. About 40 digits seems appropriate for and . (High-precision results with a smaller precision can give wrong results because matrix arithmetic is carried out in fixed precision with precision estimation a posteriori.)

Here is a high-quality solution where the reflection-transmission coefficient identity is fulfilled to 0.3%.

The last inputs show that for energies , the following are appropriate: , , , and a working precision of about . We definitely need more Fourier modes than basis states: about 10 more seems a good choice. With these parameters we obtain an average function value of along the Dirichlet part of the center disk. The quality of the reflection and transmission coefficients decreases slightly with an increasing number of Fourier modes (because the error gets more homogeneously distributed among the expansion coefficients) and typically is in the order of a few percent or less.