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The Casimir Effect in the plane-sphere geometry.
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Show that scattering matrices are positive definite #29

Closed michael-hartmann closed 6 years ago

michael-hartmann commented 7 years ago

Try to show that the scattering matrices are positive definite.

First, show that the Mie coefficients al, bl differ in sign. This can be proven using the recurrence relations for PC and probably similar for arbitrary metals.

Then, show that the blocks EE and MM are symmetric and EM = ME and symmetric.

If Id-EE and Id-MM are diagonal dominant, the scattering matrices are positive definite and we can use Choleky decomposition to calculate the determinant.

See Wikipedia article in positive definite matrix.

michael-hartmann commented 7 years ago

The round-trip operator is positive definitie. As the scattering matrix Id-M is probably also positive definite, the eigenvalues of M are bounded by 0<lambda_j<1. Probably Id-M is probably diagonally dominant. This might be shown by summing the first row, l1=m, showing that the sum of all other rows is smaller. This gives an estimate of the spectral radius as M is a positive matrix, see the Perron-Frobenius theorem.

michael-hartmann commented 6 years ago

The scattering matrices 1-M are diagonally-dominant and therefore positive definite.