
1204 BOUNDARY VALUE PROBLEMS. GREEN’S FUNCTION
2
◦
. Distinct eigenfunctions y
n
(x) and y
m
(x) are orthogonal with weight s(x) on the inter-
val x
1
≤ x ≤ x
2
:
Z
x
2
x
1
s(x)y
n
(x)y
m
(x) dx = 0 for n 6= m.
3
◦
. If the conditions
q(x) ≥ 0, α
1
β
1
≤ 0, α
2
β
2
≥ 0 (17.1.2.5)
are satisfied, there are no negative eigenvalues. If q ≡ 0 and β
1
= β
2
= 0, then λ
1
= 0
is the least eigenvalue, with the corresponding eigenfunction ϕ
1
= const . Otherwise, all
eigenvalues are positive, provided that conditions (17.1.2.5) are satisfied.
Other general and special properties of the Sturm–Liouville problem (17.1.2.4) are
given in Section 3.8.9; various asymptotic and approximate formulas for the ...