Ein einfaches Beispiel ist die Differentialgleichung
−
ψ
″
=
λ
ψ
{\displaystyle -\psi ''=\lambda \psi }
auf dem Intervall
[
0
,
π
]
{\displaystyle [0,\pi ]}
ψ
(
0
)
=
ψ
(
π
)
=
0.
{\displaystyle \psi (0)=\psi (\pi )=0.}
Aufgrund der Randbedingungen wird der periodische Ansatz
ψ
(
x
)
=
a
sin
(
λ
x
)
{\displaystyle \psi (x)=a\sin({\sqrt {\lambda }}x)}
λ
>
0
{\displaystyle \lambda >0}
a
∈
R
{\displaystyle a\in \mathbb {R} }
ψ
(
0
)
=
ψ
(
π
)
=
0
{\displaystyle \psi (0)=\psi (\pi )=0}
a
≠
0
{\displaystyle a\neq 0}
sin
(
λ
π
)
=
0
,
{\displaystyle \sin({\sqrt {\lambda }}\pi )=0,}
λ
π
=
n
π
{\displaystyle {\sqrt {\lambda }}\pi =n\pi }
λ
=
n
2
{\displaystyle \lambda =n^{2}}
n
∈
N
{\displaystyle n\in \mathbb {N} }
λ
n
=
n
2
{\displaystyle \lambda _{n}=n^{2}}
und genügt der Weyl-Asymptotik.
Die Folge der Eigenfunktionen ergibt sich, bis auf die zu bestimmenden Koeffizienten
a
n
{\displaystyle a_{n}}
ψ
n
(
x
)
=
a
n
sin
(
n
x
)
.
{\displaystyle \psi _{n}(x)=a_{n}\sin(n\,x).}
Die Orthonormalbasis der Eigenfunktionen im Hilbertraum
L
2
(
[
a
,
b
]
,
d
x
)
{\displaystyle L^{2}([a,b],\mathrm {d} x)}
w
(
x
)
=
1
{\displaystyle w(x)=1}
trigonometrischen Formel
sin
(
n
x
)
sin
(
m
x
)
=
1
2
(
cos
(
(
n
−
m
)
x
)
−
cos
(
(
n
+
m
)
x
)
)
{\displaystyle \textstyle \sin(nx)\;\sin(mx)={\frac {1}{2}}{\Big (}\cos {\big (}(n-m)x{\big )}-\cos {\big (}(n+m)x{\big )}{\Big )}}
⟨
ψ
n
,
ψ
m
⟩
=
∫
ψ
n
(
x
)
¯
ψ
m
(
x
)
d
x
=
∫
0
π
a
n
sin
(
n
x
)
¯
a
m
sin
(
m
x
)
d
x
=
a
n
a
m
∫
0
π
sin
(
n
x
)
sin
(
m
x
)
d
x
=
a
n
a
m
2
∫
0
π
(
cos
(
(
n
−
m
)
x
)
−
cos
(
(
n
+
m
)
x
)
)
d
x
=
{
a
n
a
m
2
[
1
n
−
m
sin
(
(
n
−
m
)
x
)
−
1
n
+
m
sin
(
(
n
+
m
)
x
)
]
0
π
=
0
wenn
n
≠
m
a
n
2
2
[
x
−
1
2
n
sin
(
2
n
x
)
]
0
π
=
a
n
2
π
2
wenn
n
=
m
=
a
n
2
π
2
δ
n
m
.
{\displaystyle {\begin{aligned}\langle \psi _{n},\psi _{m}\rangle &=\int {\overline {\psi _{n}(x)}}\psi _{m}(x)\mathrm {d} x=\int _{0}^{\pi }{\overline {a_{n}\sin(nx)}}\;a_{m}\sin(mx)\,\mathrm {d} x=a_{n}a_{m}\int _{0}^{\pi }\sin(nx)\sin(mx)\,\mathrm {d} x\\&={\frac {a_{n}a_{m}}{2}}\int _{0}^{\pi }{\bigg (}\cos {\big (}(n-m)x{\big )}-\cos {\big (}(n+m)x{\big )}{\bigg )}\,\mathrm {d} x\\\\&={\begin{cases}{\frac {a_{n}a_{m}}{2}}{\bigg [}{\frac {1}{n-m}}\sin {\big (}(n-m)x{\big )}-{\frac {1}{n+m}}\sin {\big (}(n+m)x{\big )}{\bigg ]}_{0}^{\pi }=0&&{\text{wenn}}\;n\neq m\\\\{\frac {a_{n}^{2}}{2}}{\Bigg [}x-{\frac {1}{2n}}\sin(2nx){\bigg ]}_{0}^{\pi }={\frac {a_{n}^{2}\pi }{2}}&&{\text{wenn}}\;n=m\end{cases}}\\\\&={\frac {a_{n}^{2}\pi }{2}}\delta _{nm}.\end{aligned}}}
Hierbei bedeutet
δ
n
m
{\displaystyle \delta _{nm}}
Kronecker-Delta und die Normierung
⟨
ψ
n
,
ψ
m
⟩
=
δ
n
m
{\displaystyle \langle \psi _{n},\psi _{m}\rangle =\delta _{nm}}
a
n
=
2
π
{\displaystyle a_{n}={\sqrt {\frac {2}{\pi }}}}
ψ
n
(
x
)
=
2
π
sin
(
n
x
)
{\displaystyle \psi _{n}(x)={\sqrt {\frac {2}{\pi }}}\sin(n\,x)}
annehmen.
Die zugehörige Eigenfunktionsentwicklung ist die Fourierreihe mit
Ψ
=
∑
n
=
1
∞
ψ
n
=
∑
n
=
1
∞
2
π
sin
(
n
x
)
.
{\displaystyle \Psi =\sum _{n=1}^{\infty }\psi _{n}=\sum _{n=1}^{\infty }{\sqrt {\frac {2}{\pi }}}\sin(nx).}
![{\displaystyle [0,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2a912eda6ef1afe46a81b518fe9da64a832751)
![{\displaystyle L^{2}([a,b],\mathrm {d} x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea9d3106f77b79317bc26162b5698f352ffd1eff)
![{\displaystyle {\begin{aligned}\langle \psi _{n},\psi _{m}\rangle &=\int {\overline {\psi _{n}(x)}}\psi _{m}(x)\mathrm {d} x=\int _{0}^{\pi }{\overline {a_{n}\sin(nx)}}\;a_{m}\sin(mx)\,\mathrm {d} x=a_{n}a_{m}\int _{0}^{\pi }\sin(nx)\sin(mx)\,\mathrm {d} x\\&={\frac {a_{n}a_{m}}{2}}\int _{0}^{\pi }{\bigg (}\cos {\big (}(n-m)x{\big )}-\cos {\big (}(n+m)x{\big )}{\bigg )}\,\mathrm {d} x\\\\&={\begin{cases}{\frac {a_{n}a_{m}}{2}}{\bigg [}{\frac {1}{n-m}}\sin {\big (}(n-m)x{\big )}-{\frac {1}{n+m}}\sin {\big (}(n+m)x{\big )}{\bigg ]}_{0}^{\pi }=0&&{\text{wenn}}\;n\neq m\\\\{\frac {a_{n}^{2}}{2}}{\Bigg [}x-{\frac {1}{2n}}\sin(2nx){\bigg ]}_{0}^{\pi }={\frac {a_{n}^{2}\pi }{2}}&&{\text{wenn}}\;n=m\end{cases}}\\\\&={\frac {a_{n}^{2}\pi }{2}}\delta _{nm}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2dde480c8eca3979069abaf17c70c79a7d4da4d)
