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Completeness of the Set of Eigenfunctions via Rayleigh's Quotient

The fact that eigenvalues of the regular Sturm-Liouville problem form a semi-unbounded sequence, i.e., that

is very important. It implies that the set of eigenfunctions of the Sturm-Liouville problem

 (327)

with

is a generalized Fourier basis. In other words, they form a complete basis set for the subspace of of those square-integrable functions which satisfy the given boundary conditions, Eq.(3.27). This subspace is

Recall that a set is said to be complete, if for any vector , the error vector

can be made to have arbitrarily small squared norm by letting , i.e.,

Here

is the th (generalized) Fourier coefficient with the consequence that is perpendicular to the subspace

The subspace induces to be decomposed into the direct sum

Here ( perp'') is the subspace of all vectors perpendicular to

In other words, is the space of all vectors satisfying the set of constraint conditions
 0 0

Our starting point for demonstrating the completeness is the Rayleigh principle. It says that the Rayleigh quotient

satisfies various minimum principles when is restricted to lie on various subspaces , . Indeed, one has

i.e., for any subject to the constraint .

More generally, the st eigenvalue is characterized by

i.e., for any subject to the constraints
 0

The th error vector

satisfies the constraint conditions

Consequently, it satisfies the corresponding Rayleigh inequality

or

We insert the expression for into the left hand side, and obtain

The orthonormality of eigenfunctions and the definition of the generalized Fourier coefficients guarantee that the last two sums cancel. Furthermore, by doing an integration by parts twice, and by observing that the resulting end point terms vanish because of the Dirichlet-Neumann boundary conditions, Eq. 3.27, we obtain

As a consequence the Rayleigh inequality becomes

Without loss of generality one may assume that the lowest eigenvalue . This can always be made to come about by readjusting the and the function in the Sturm-Liouville equation. As a result, the finite sum may be dropped without decreasing the Consequently,

The numerator is independent of . Thus

because is an unbounded sequence. Thus we have

The function is an arbitrary square integrable function satisfying the the given mixed Dirichlet-Neuman end point conditions. Consequently, the Sturm-Liouville eigenfunctions form a (complete) generalized Fourier basis indeed. [references_for_chapter3] [plain]

Next: Green's Function Theory Up: Sturm-Liouville Theory Previous: Discrete Unbounded Sequence of   Contents   Index
Ulrich Gerlach 2010-12-09