Figure 3:
A double barrier potential with the plot generated using
matlab command
spline([-2,-1,0,1,2],40*[0,0,1,-2,1,0,0]) and the phase space
portrait for the classical Hamiltonian
.
In this section we will give a qualitative description of
geometric origins of various types of resonances. This is
based on semiclassical analysis of the correspondence
between the phase space dynamics and the behaviour of
solutions of
.
The phase space for problems in one dimensional scattering
is given by
:
position and momentum.
The semiclassical quantization takes functions of to operators on
according to the rule
(multiplication by ),
The ``small'' constant is mathematician's Planck constant -
that is the rescaled parameter of the problem which may have little to
do with the actual constant . We refer to [9] and [10]
for introduction to semiclassical analysis, and to [30] for the
theory of resonances in the semiclassical limit. A classical
physics treatment is [20] while a modern physical
perspective can for be found, for example, in [15]. For recent
work on semiclassical scattering on the line one should consult [16]
and [26].
Here we will simply put and take a potential which is large
enough to exhibit some qualitative semiclassical phenomena. It is a
potential given in Fig.3. By an energy level we
mean the set
Figure 4:
The bound states and resonances for the potential given in
Fig.3. The colour coding gives an approximate classical/quantum
correspondence between the bound states and energy level satisfying
Bohr-Sommerfeld quantization conditions.
The bound states, that is solutions of
,
with
, , are well approximated by
WKB methods using Bohr-Sommerfeld rules - see [20]. That
correspondence is shown in colour in Fig.4.
Figure 5:
Resonances close to the real axis. The real part is related to the
bounded components of the energy levels and the imaginary part is related
to tunneling between the bounded and unbounded energy levels.
The closed components of energy levels with cannot
support bound states since cannot have positive eigenvalues.
However, they support shape resonances. The real parts are
semiclassically close to the energy levels given by Bohr-Sommerfeld rules,
while the imaginary parts are approximated using the tunneling distance
also known as the Agmon distance:
(25)
In our example yet the agreement of the Agmon distance with
the widths (imaginary parts) of the two resonance shown in
Fig.5 is already quite good:
7.9363
7.3281
3.7500
2.2884
Figure:
Resonances of a ``well in an island'' potential.
The last plots
and compares the logarithms of resonance width (imaginary parts)
to Agmon distances - see (25).
It will be interesting to compare the more precise results of
[13, Sect.10] with numerical results. For the moment, we show
another example obtained using the following code (see Fig.6):
x = [-2.5,-2,-1,-0,75,-0.5,0,0.5,0.75,1,2,2.5]
V = [0,0.2,[1,0.5625,0.25,0,0.25,0,0,5625,1]+0.1,0.2,0]
splinepot(36*V,2*x)
Figure 7:
Barrier top resonances corresponding to two unstable equilibria.
The actual real parts are corrected due to quantum effects.
Fig.7 focuses on the barrier top resonances generated
by the two unstable equilibria. Semiclassicaly, a potential, , with a
nondegenerate global maximum at has resonances approximately given by
(26)
provided some other conditions are satisfied - see [6]. In
Fig.7 we see that a nearly degenerate resonances are
generated by the two ``barrier tops'' - they correspond to above.
Figure:
Resonances of an approximation of
given by
splinepot(sech(linspace(-5,5,21)).2,linspace(-5,5,21)).
In Fig.8 we show a computation for a approximation
of the Eckart barrier potential,
. In that case
resonances are given exactly by
see for instance [12, Appendix].
This agress with (26) if we rescale the potential and
take
. The approximation gives the resonances close
to the real axis with great precision. The other resonances, some generated
by the singularities at the boundaries (see below) mask
the very unstable resonances deeper in the complex [4].
Figure 9:
Regge resonances generated by reflections by the singularities
at the end points of the support of the potential and lying on
logarithmic curves.
In Fig.9 we show resonances with larger real parts. They dominate
large energy asymptotics of resonances and are generated by the reflection
by the singularities at the end points of the support of the potential.
This heuristic analysis is known to be correct if the potential behaves as
follows
(27)
The resonances then have the following asymptotic form
(28)
We will call them Regge resonances (as opposed to
related but different Regge poles) since such asymptotic formulæ
origin in the work of Tullio Regge [27]. For simple potentials
satisfying (27) (see Fig.1),
asymptotics (28) are visible almost immediately but for complicated
potentials, large energies are necessary -
see Fig.2 and Fig.10.
As was also observed by Regge, (28) gives a counting law for
resonances
(29)
For arbitrary potentials (which may not satisfy (27)),
(29) was proved in [34], and different new proofs
were provided in [11] and [29].
Figure 10:
Resonances of a period array of wells obtained using
squarepot(10*[1,-1,1,-1,1,-1,1],[-7,-5,-3,-1,1,3,5,7]*0.25).
The rich structure of the potential (compared to, say, Fig.1)
makes the asymptotic formulæ (28) valid for large energies
only.