Mathematical Methods for Physicists: A concise introduction - Site Map

STURM±LIOUVILLE SYSTEMS
the system. In general there is one eigenfunction to each eigenvalue. This is the
non-degenerate case. In the degenerate case, more than one eigenfunction may
correspond to the same eigenvalue. The eigenfunctions form an orthogonal set
with respect to the density function p…x† which is generally 0.Thus by suitable
normalization the set of functions can be made an orthonormal set with respect to
p…x† in a x b. We now proceed to prove these two general claims.
Property 1
If r…x† and q…x† are real, the eigenvalues of a Sturm±Liouville
system are real.
We start with the Sturm±Liouville equation (7.104) and the boundary conditions
(7.104a):
d dy
r…x† ‡‰q…x†‡p…x†Šy ˆ 0; a x b;
dx dx
k 1 y…a†‡k 2 y 0 …a† ˆ0; l 1 y…b†‡l 2 y 0 …b† ˆ0;
and assume that r…x†; q…x†; p…x†; k 1 ; k 2 ; l 1 ,andl 2 are all real, but and y may be
complex. Now take the complex conjugates
d d y
r…x† ‡‰q…x†‡ p…x†Šy
dx dx
ˆ 0;
…7:105†
k 1 y…a†‡k 2 y 0 …a† ˆ0; l 1 y…b†‡l 2 y 0 …b† ˆ0;
…7:105a†
where y and are the complex conjugates of y and , respectively.
Multiplying (7.104) by y, (7.105) by y, and subtracting, we obtain after
simplifying
d
dx r…x†…yy 0 yy 0
† ˆ… †p…x†yy:
Integrating from a to b, and using the boundary conditions (7.104a) and (7.105a),
we then obtain
… †
Z b
a
p…x† y 0 2 dx ˆ r…x†…yy 0 yy 0 †j b a ˆ 0:
Since p…x† 0ina x b, the integral on the left is positive and therefore
ˆ , that is, is real.
Property 2
The eigenfunctions corresponding to two di€erent eigenvalues
are orthogonal with respect to p…x† in a x b.
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