9.7 Anisotropic Kepler System
323
For f > 2, the eigenvalues are real and we can write λ ± = e ±u , where u =
ln[
1
2 (−f +
f 2 − 4)]. Then we find
F (1) = (1 − e
+u )(1 − e
−u ) = −4 sinh
2
u
2
.
(9.137)
For the case f > 2, the orbits in the neighborhood of the γ th periodic orbit are
unstable.
For f < 2, the eigenvalues are complex. We can write λ ± = e ±iu , where
cos(u) = −
f
2 . Then
F (1) = (1 − e
+iu )(1 − e
−iu ) = 4 sin
2
u
2
.
(9.138)
For the case f < 2, the orbits in the neighborhood of the γ th periodic orbit are
stable.
We can see that F (1) is independent of the initial coordinates and the integration
over
dτ γ gives the period, τ γ , of the γ th periodic orbit. Thus, for d = 2 we obtain
g osc (e) ≈ −
i
¯
h
γ (stable)
τ γ
2 sin(
u γ
2 )
exp
i
¯
h
S γ (e) −
iπn γ
2
−
i
¯
h
γ (unstable)
τ γ
2 sinh(
u γ
2 )
exp
i
¯
h
S γ (e) −
iπn γ
2
,
(9.139)
where the first term gives the contribution from stable periodic orbits and the second
term gives contributions from unstable periodic orbits. Equation (9.139) is called
the Gutzwiller trace formula. (See Littlejohn 1991 for a discussion of other trace
formulas.)
The method of stationary phase may not be the best way to compute the
contribution from stable periodic orbits if they are not isolated but form families
in local regions of the phase space. Then one may need to use methods similar to
those used in previous sections. The Gutzwiller trace formula has not been applied
successfully to nonintegrable systems that have a mixture of stable and unstable
orbits in their phase space. But it has been applied successfully to systems that
contain only unstable orbits. In the next section, we will describe how Gutzwiller
applied his result to the anisotropic Kepler system.
9.7 Anisotropic Kepler System
The anisotropic Kepler system consists of a bound electron of charge −q in the
Coulomb field of a donor impurity of charge +q in a semiconductor such as silicon
323
For f > 2, the eigenvalues are real and we can write λ ± = e ±u , where u =
ln[
1
2 (−f +
f 2 − 4)]. Then we find
F (1) = (1 − e
+u )(1 − e
−u ) = −4 sinh
2
u
2
.
(9.137)
For the case f > 2, the orbits in the neighborhood of the γ th periodic orbit are
unstable.
For f < 2, the eigenvalues are complex. We can write λ ± = e ±iu , where
cos(u) = −
f
2 . Then
F (1) = (1 − e
+iu )(1 − e
−iu ) = 4 sin
2
u
2
.
(9.138)
For the case f < 2, the orbits in the neighborhood of the γ th periodic orbit are
stable.
We can see that F (1) is independent of the initial coordinates and the integration
over
dτ γ gives the period, τ γ , of the γ th periodic orbit. Thus, for d = 2 we obtain
g osc (e) ≈ −
i
¯
h
γ (stable)
τ γ
2 sin(
u γ
2 )
exp
i
¯
h
S γ (e) −
iπn γ
2
−
i
¯
h
γ (unstable)
τ γ
2 sinh(
u γ
2 )
exp
i
¯
h
S γ (e) −
iπn γ
2
,
(9.139)
where the first term gives the contribution from stable periodic orbits and the second
term gives contributions from unstable periodic orbits. Equation (9.139) is called
the Gutzwiller trace formula. (See Littlejohn 1991 for a discussion of other trace
formulas.)
The method of stationary phase may not be the best way to compute the
contribution from stable periodic orbits if they are not isolated but form families
in local regions of the phase space. Then one may need to use methods similar to
those used in previous sections. The Gutzwiller trace formula has not been applied
successfully to nonintegrable systems that have a mixture of stable and unstable
orbits in their phase space. But it has been applied successfully to systems that
contain only unstable orbits. In the next section, we will describe how Gutzwiller
applied his result to the anisotropic Kepler system.
9.7 Anisotropic Kepler System
The anisotropic Kepler system consists of a bound electron of charge −q in the
Coulomb field of a donor impurity of charge +q in a semiconductor such as silicon