2.6.7 Three Classes of EVPs
For the EVP (2.41) we define three sub-classes, dependent on the operator L:
1. The general EVP: L contains at least one derivative, n > 0.
2. The one-term EVP: L is single-termed and n ! 0.
3. The special EVP: L is single-termed and free of derivatives, n = 0.
2.6.8 The Rayleigh Quotient
For any test function u(x) of a semidefinite and self-adjoint EVP, we define the
associated Rayleigh quotient R[u] by:
R u
½ Š ¼
R b
a uKu dx
R b
a uLu dx
; u 2 u TF :
ð2:50Þ
The Rayleigh quotient is a functional, that is, a ‘function of a function’: It’s
argument is a function (here u(x)) – not an independent variable as for a usual
function.
2.7 Properties of Eigenvalues and Eigenfunctions
The above definitions may now be put into use in a series of theorems concerning
mathematical properties of eigenvalues and eigenfunctions.
2.7.1 Real-Valueness of Eigenvalues
It is often essential to ascertain whether the eigenvalues of a given EVP are
real-valued; Here is a sufficient condition:
Theorem 2.3. The eigenvalues of a semidefinite and self-adjoint EVP with
real coefficients are all real.
Proof. Assume the EVP Ku = kLu possesses a complex-valued eigenpair (k, u).
The complex conjugated pair ( k,
u) is then also an eigenpair. The functions u and
u
are both test functions, since every eigenfunction is also a test function. If the EVP
is self-adjoint we know that
2.6 Concepts of Differential EVPs
67