Next: The scaling method in Up: Solving for the scaling Previous: Solving for the scaling

### Generalized eigenproblem

The problem is solved by a forming a generalized eigenequation,

 (6.25)

which finds the eigenstates of the -derivative of tension, treating the tension quadratic form as a norm which is held constant. The presence of a norm based on the wavefunction at means that null-space vectors can be excluded by the method of Section 5.3.2 with truncation set at about . It is possible to replace this tension norm by other norms, for instance the exact or Dirichlet from the previous chapter, however they will not be quasi-diagonal in the scaling eigenfunction basis. This destroys much of the benefit of the quasi-diagonality of in this basis. The power of (6.25), as realised by VS, is that both matrices are quasi-diagonal in this basis. Therefore, simultaneous diagonalization (that is, solving the generalized eigenproblem) of and in the computational basis returns a very good approximation to the transformation into the desired eigenfunction basis.

From (6.20) to lowest order the diagonal elements of are and those of are . The ratio gives the generalized eigenvalue . The prediction for the eigenwavenumber is , giving to lowest order (see Section 6.2 for higher orders),

 (6.26)

The method will compute all the scaling eigenfunctions within a wavenumber range of up to about 1, for a system size . This corresponds to whose rescaled boundaries lie within about wavelength of the original boundary. The predicted as a function of are shown in Fig. 6.4 (bottom).

Next: The scaling method in Up: Solving for the scaling Previous: Solving for the scaling
Alex Barnett 2001-10-03