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# Special Function Theory

We shall now reconsider the eigenvalue problem

but we take to be a degenerate eigenvalue. This means that we take to have more than one eigenvector. These eigenvectors span a subspace, the eigenspace of corresponding to . This subspace has a basis of eigenvectors, but its choice is not unique.

In spite of this we ask: Is there a way of constructing a basis which is dictated by objective criteria (for our purposes, by geometry and/or physics) and not by subjective preferences?

The answer to this question is yes'' whenever one can identify a linear transformation, call it , with the following three properties:

(i)
The domain of coincides with that of ,
(ii)
the transformation commutes with , i.e.

and
(iii)
the eigenvalues of are non-degenerate.
A transformation with these properties determines a unique eigenbasis for each eigenspace of the original eigenvalue problem. Indeed, let be an eigenvector of :

Then

i.e., is again an eigenvector of corresponding to the same eigenvalue . The non-degeneracy of implies that is a multiple of ; in other words,

Thus is also an eigenvector of . Conversely, if belongs to the -eigenspace of , then also belongs to this subspace. The set of all the eigenvectors of which lie in this -subspace form a basis for this subspace. This basis is orthonormal if and are hermitian. The elements of this -determined basis are uniquely labelled by the real eigenvalues and, of course, by the subspace label . A set of commuting linear transformations, such as and , whose eigenvalues uniquely label their common eigenvectors, is called a complete set of commuting operators.

The operator is not unique. Suppose there is another hermitian operator, say , which together with forms another complete set of commuting operators. This means that one now has two orthonormal bases for the -eigenspace of , one consisting of the eigenvectors of , the second consisting of the eigenvectors of . Furthermore, these two bases are related by a unitary transformation, i.e. by a rotation in the complex eigenspace of .

One of the most far reaching applications of this geometrical framework consists of identifying

• the operator with the Laplacian on , the Euclidean two-dimensional plane,
• the operator with the generator of translations in ,
• the operator with the generator of rotations in ,
• the eigenvectors of with the plane-wave solutions to the Helmholtz equation,
• the eigenvectors of with the cylinder (Bessel) solutions to the Helmholtz equation, and
• the unitary transformation with the Fourier series representation of a plane wave in terms of the Bessel solutions.

Subsections

Next: The Helmholtz Equation Up: LINEAR MATHEMATICS IN INFINITE Previous: Fourier Sine Theorem   Contents   Index
Ulrich Gerlach 2010-12-09