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# Appendix G: Numerical evaluation of wavefunction boundary integrals

Boundary methods are a central component of this thesis. Closed integrals of a function over the boundary coordinate are ubiquitous. Generally and a square matrix of integrals

 (G.1)

is required, where the indices label multiple functions. For evaluation of the quantum band profile (Chapters 2 and 3), the local density of states (Chapter 6), the tension and area-norm matrices (Chapter 5), and the Vergini matrix and its derivative (Chapter 6), and are basis functions or eigenstates which oscillate about zero on the length scale , the quantum (de Broglie) free-space wavelength. The deformation function boundary integrals (Chapter 4) do not involve any quantum scale, but are also evaluated using the method below. I will present only the case where boundary integrals over become line integrals over ; the generalization to higher is simple.

My tool for evaluation of an integral on a closed curve is the discretization

 (G.2)

where is the range of , that is, the length of the line integral (billiard perimeter). The points are spread uniformly (equidistant in ) along the closed curve. Because no point is special, no special quadrature [161] weights arise near any endpoints: all weights are equal. More sophisticated and accurate approximations exist for closed line integral evaluation [58], however this is sufficient for my needs and is very simple to code. Its errors will be discussed and tested below.

A single integral (G.2) requires function evaluations of . Naively one might guess that filling a matrix using (G.1) requires evaluations. However, the correct way to compute (G.1) requires only such evaluations: First fill the rectangular matrices and , from which follows

 (G.3)

This matrix multiplication does require operations, but being simple adds and multiplications (and using optimized library code e.g. BLAS), it is very fast and does not affect the scaling. If you like, the matrix multiply `performs' the integration over . In the case where and are the same function, only evaluations are required. Note that if a general weighting function is required in the integrand (G.1), it can easily be incorporated into or , or equivalently be included as a diagonal matrix inserted between and in (G.3).

Subsections

Next: Convergence with number of Up: Dissipation in Deforming Chaotic Previous: Appendix F: Cross correlations
Alex Barnett 2001-10-03