Special Function Theory

We shall now reconsider the eigenvalue problem

$$Lu=\lambda u ~~,$$

but we take $\lambda$ to be a degenerate eigenvalue. This means that we take $\lambda$ to have more than one eigenvector. These eigenvectors span a subspace, the eigenspace of $L$ corresponding to $\lambda$ . 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 $T$ , with the following three properties:

(i)

The domain of $T$ coincides with that of $L$ ,

(ii)

the transformation $T$ commutes with $L$ , i.e.

$$TL=LT ~~,$$

and

(iii)

the eigenvalues of $T$ are non-degenerate.

A transformation with these properties determines a unique eigenbasis for each eigenspace of the original eigenvalue problem. Indeed, let $u$ be an eigenvector of $T$ :

$$Tu=\tau u ~~.$$

Then

$$T(Lu)=L(Tu)=\tau~Lu~~,$$

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

$$Lu=\lambda u ~~.$$

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

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

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

• the operator $L$ with the Laplacian on $E^2$ , the Euclidean two-dimensional plane,
• the operator $T$ with the generator of translations in $E^2$ ,
• the operator $S$ with the generator of rotations in $E^2$ ,
• the eigenvectors of $T$ with the plane-wave solutions to the Helmholtz equation,
• the eigenvectors of $S$ 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.