Resonances and phase space geometry

Figure 3: A $ C^1$ 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 $ \xi ^2 + V ( x ) $.
\includegraphics[width=16cm]{spl.eps}

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 $ H_V - \lambda^2$.

The phase space for problems in one dimensional scattering is given by $ \mathbb{R}^2$:

$\displaystyle (x, \xi) \in \mathbb{R}^2,  $   $ x  = $position and $ \xi  = $momentum.$\displaystyle $

The semiclassical quantization takes functions of $ (x, \xi)$ to operators on $ L^2(\mathbb{R})$ according to the rule

$\displaystyle x \mapsto x  $   (multiplication by $ x$),$\displaystyle  \
\xi \mapsto \frac h i   \partial_x.
$

The ``small'' constant $ h$ is mathematician's Planck constant - that is the rescaled parameter of the problem which may have little to do with the actual constant $ \hbar $. 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 $ h = 1$ and take a potential which is large enough to exhibit some qualitative semiclassical phenomena. It is a $ C^1$ potential given in Fig.3. By an energy level we mean the set

$\displaystyle \Sigma_E = \{ (x , \xi) \;:\; \xi^2 + V ( x ) = E \}  .
$

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.
\includegraphics[width=16cm]{rebd.eps}

The bound states, that is solutions of $ (H_V - \lambda^2) u = 0$, with $ u \in L^2(\mathbb{R})$, $ u \neq 0$, 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.
\includegraphics[width=16cm]{shre.eps}

The closed components of energy levels $ \Sigma_E$ with $ E > 0$ cannot support bound states since $ H_V $ 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:

$\displaystyle S_0(E) = \int_\mathbb{R}(V(x) - E)_+^{\frac12} dx,   \log(-\mat...
...olimits \lambda_j) \sim -\frac{S_0(\mathop{\rm Re}\nolimits \lambda_j^2)} h  .$ (25)

In our example $ h = 1$ yet the agreement of the Agmon distance with the widths (imaginary parts) of the two resonance shown in Fig.5 is already quite good:

   
$ -\log(-\mathop{\rm Im}\nolimits \lambda) $ $ S_0(\mathop{\rm Re}\nolimits \lambda^2) $
   
7.9363 7.3281
3.7500 2.2884

Figure: Resonances of a ``well in an island'' potential. The last plots $ (\mathop{\rm Re}\nolimits \lambda_j^2, -\log(-\mathop{\rm Im}\nolimits \lambda_j^2))$ and compares the logarithms of resonance width (imaginary parts) to Agmon distances - see (25).
\includegraphics[width=16cm]{rehs.eps}

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.
\includegraphics[width=16cm]{btre.eps}

Fig.7 focuses on the barrier top resonances generated by the two unstable equilibria. Semiclassicaly, a potential, $ V(x)$, with a nondegenerate global maximum at $ x_0$ has resonances approximately given by

$\displaystyle \lambda_j^2 \simeq V(x_0) - i h (V''(x_0) /2)^{\frac12} (2 k + 1),   k = 0, 1, \cdots,$ (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 $ k = 0$ above.

Figure: Resonances of an approximation of $ V(x) = \cosh(x)^{-2}$ given by splinepot(sech(linspace(-5,5,21)).$ \hat{}\;$2,linspace(-5,5,21)).
\includegraphics[width=16cm]{reck.eps}

In Fig.8 we show a computation for a $ C^1$ approximation of the Eckart barrier potential, $ V(x) = \cosh(x)^{-2}$. In that case resonances are given exactly by

$\displaystyle \lambda = \pm \frac{\sqrt{3}}{2} - \left( k+\frac12 \right) i,  \
k = 0, 1, \cdots,
$

see for instance [12, Appendix]. This agress with (26) if we rescale the potential and take $ h \rightarrow 0$. The approximation gives the resonances close to the real axis with great precision. The other resonances, some generated by the $ C^1$ 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.
\includegraphics[width=16cm]{rere.eps}

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

$\displaystyle V(x) \simeq C_-(x - a)^k,   x \rightarrow a+,   V(x) \simeq C_+(x - b)^p,   x \rightarrow b-.$ (27)

The resonances then have the following asymptotic form

$\displaystyle \pm \frac{\pi}{b - a} n - i A \log n - B + o(1),   n \rightarrow \infty.$ (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

$\displaystyle \char93  \{ \lambda_j \; : \; \vert \lambda_j \vert \leq r \} = \frac{2(b - a)}{\pi} r(1 + o(1))  ,   r \longrightarrow \infty  .$ (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.
\includegraphics[width=16cm]{repa.eps}

David Bindel 2006-10-04