10.7 Eigenvalue Problems
We often encounter eigenvalue problems in mathematical physics. Of these, those
related to Hermitian differential operators have particularly interesting and important
features. The eigenvalue problems we have considered in Part I are typical illustrations. Here we investigate general properties of the eigenvalue problems.
Returning to the case of homogeneous BCs, we consider a following homogeneous SOLDE:
a x
ð Þ
d
2 u
dx 2 þ b x
ð Þ
du
dx
þ c x
ð Þu ¼ 0:
ð10:5Þ
Defining a following differential operator L x such that
L x ¼ a x
ð Þ
d
2
dx 2 þ b x
ð Þ
d
dx
þ c x
ð Þ,
ð10:55Þ
we have a homogeneous equation
L x u x
ð Þ ¼ 0:
ð10:218Þ
Putting a constant Àλ instead of c(x), we have
a x
ð Þ
d
2 u
dx 2 þ b x
ð Þ
du
dx
À λu ¼ 0:
ð10:219Þ
If we define a differential operator L x such that
L x ¼ a x
ð Þ
d
2
dx 2 þ b x
ð Þ
d
dx
À λ,
ð10:220Þ
we have a homogeneous equation
L x u ¼ 0
ð10:221Þ
to express (10.219). Instead, if we define a differential operator L x such that
L x ¼ a x
ð Þ
d
2
dx 2 þ b x
ð Þ
d
dx
,
we have the same homogeneous equation
10.7 Eigenvalue Problems
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