Hydromagnetic Resonators - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 105

d 2

d z 2 + k 2 Q ( z )=0 ,

(5.9)

with the boundary condition
Q ( z )

c
ωX ∓

d Q ( z )
d z

∓ i =0 . (5.10)

z =0; l z

Here the upper sign refers to z = 0, the lower to z = l z .
For the perfect conductive walls ( X ∓ →∞ in (5.10)) (5.9) with condi-

tions (5.10) is the self-adjoint Sturm-Liouville problem. It is known (see, e.g.,
[10], [15]) that in this case the spectrum consists of a discrete set of real pos-
itive eigenvalues k n . The corresponding normalized eigenfunctions Q n ( z )are
mutually orthonormal functions:

Q n Q m = δ nm ,

where δ nm is the Kronecker symbol,
δ nm = 1 ,

= m,

0 , n = m

and

l z

Q n ( z ) Q ∗ m ( z )d z,

Q n Q m =

0

where the asterisk denotes complex conjugation. An arbitrary function F ( z )
satisfying boundary conditions (5.10) can be presented as convergent series

F ( z )= ∞ F n Q n ( z ) ,

n =1

where F n are Fourier coecients.
At finite Σ P , the boundary problem (5.9)-(5.10) is non-self-conjugate. It
has a discrete spectrum of complex eigenvalues k n and eigenfunctions Q n ( z ),
n =0 , 1 , 2 ,... . The eigenfunctions in this case are not mutually orthogonal.
However, there exist a set of bi-orthonormal functions

=
δ nm . Functions G m ( z ) are found from the conjugate boundary problem. It is
obtained in this case from (5.9)-(5.10) by replacing i in (5.10) by

{ G m ( z ) } : Q n G m

− i :

d 2

d z 2 + µ G ( z )=0 ,

G ( z )

c
ωX ∓

d G ( z )
d z

± i =0 .

z =0; l z

Eigenvalues k n ,µ n and eigenfunctions G n ( z ) ,Q n ( z ) are complex conjugate:
µ n = k n ∗ , G n ( z )= Q n ( z ) ,