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## Degenerate Eigenvalues

Every eigenvalue of the eigenvalue equation

is highly degenerate. In fact, each eigenvalue is infinitely degenerate. This means that for one and the same eigenvalue , there is an infinite set of eigenfunctions, namely,

or

These solutions form a basis for the subspace of solutions to the Helmholtz equation

Any solution to this equation is a unique superposition of the basis elements. We shall refer to this subspace as the eigenspace of the (degenerate) eigenvalue .

A matrix, and more generally an operator, is diagonal relative to its eigenvector basis. The Helmholtz operator can, therefore, be viewed as an infinite diagonal matrix

with degenerate eigenvalues along the diagonal.

The question now is, how does one tell the difference between the eigenfunctions having the same eigenvalue ? Physically one says that these eigenfunctions are plane waves propagating into different directions. However, one also would like to express the difference algebraically.

Next: Complete Set of Commuting Up: The Helmholtz Equation Previous: Cartesian versus Polar Coordinates   Contents   Index
Ulrich Gerlach 2010-12-09