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# Representation by a Helmholtz basis

I choose a basis set composed entirely of solutions to (5.5) in . Therefore they are -dependent (equivalently, energy-dependent), and this means we are joining the Class B methods. Since the basis functions do not obey the BCs we can choose the BCs for which the analytic forms of the functions are simplest, namely, free space. The eigenfunctions can be chosen real-valued everywhere, so we can choose entirely real basis functions. Typical basis functions at wavenumber include: Real Plane Waves (RPWs), Evanescent Plane Waves (EPWs), and angular momentum states (e.g. the singular ) based at various origins (which if they are singular, must be outside , or placed on as in the BIM basis set). Each of these forms is oscillatory with wavenumber . Following Heller's PWDM, the default' basis will be RPWs uniformly spaced in angle, containing both sin and cos terms, and suitably symmetrized for the billiard shape. The basis size will be discussed below. Helmholtz basis sets will be further discussed in Sections 5.3.1 and 6.1.3, and examples described in Appendix J.

Substitution of the -dependent basis representation (5.11)

into (5.10) and (5.7) gives (5.12)

where the -dependent positive definite matrices are given by (5.13)

The matrix can be evaluated using methods in Appendix G, and the evaluation of is discussed in Section 5.4.

The condition (5.12) is equivalent  to finding the maximum-eigenvalue solution to the generalized eigenproblem (5.14)

The maximum eigenvalue is , and is then simply the tension corresponding to the best' solution. The basis coefficients of this best solution are returned in the eigenvector .

Essentially, the method proposed in this chapter is then searching for local minima in the inverse largest eigenvalue of (5.14), as a function of . An example sweep in is plotted in Fig. 5.2. However there are implementation issues without which this specification is all but useless. The rest of the chapter is devoted to such issues.  Subsections   Next: Numerical rank of the Up: Chapter 5: Improved sweep Previous: Definition of the billiard
Alex Barnett 2001-10-03