In the case of inhomogeneous BCs, given σ 1 ¼ u(0) and σ 2 ¼ _
u 0
ð Þ we can decide
additional terms S(t) using (10.190) such that
S t
ð Þ ¼ σ 2 þ σ 1
r
2m
1
ω
e
À
r
2m t sin ωt þ σ 1 e
À
r
2m t cos ωt:
ð10:217Þ
This term arises from (10.190). Thus, from (10.212) and (10.217), u(t) + S(t) gives a
unique solution for the SOLDE with inhomogeneous BCs. Notice that S(t) does not
depend on the external field.
Phase:
7LPHV
ಥ
(a)
3RVLWLRQP
Phase: 0
7LPHV
ಥ
(b)
3RVLWLRQP
Fig. 10.5 Example of a damped oscillation as a function of t. The data are taken from (10.214). (a)
Overall profile. (b) Profile enlarged near the origin
424
10 Introductory Green’s Functions
u 0
ð Þ we can decide
additional terms S(t) using (10.190) such that
S t
ð Þ ¼ σ 2 þ σ 1
r
2m
1
ω
e
À
r
2m t sin ωt þ σ 1 e
À
r
2m t cos ωt:
ð10:217Þ
This term arises from (10.190). Thus, from (10.212) and (10.217), u(t) + S(t) gives a
unique solution for the SOLDE with inhomogeneous BCs. Notice that S(t) does not
depend on the external field.
Phase:
7LPHV
ಥ
(a)
3RVLWLRQP
Phase: 0
7LPHV
ಥ
(b)
3RVLWLRQP
Fig. 10.5 Example of a damped oscillation as a function of t. The data are taken from (10.214). (a)
Overall profile. (b) Profile enlarged near the origin
424
10 Introductory Green’s Functions