Siegert pseudostates: Completeness and time evolution

Within the theory of Siegert pseudostates, it is possible to accurately calculate bound states and resonances. The energy continuum is replaced by a discrete set of states. Many questions of interest in scattering theory can be addressed within the framework of this formalism, thereby avoiding the need to treat the energy continuum. For practical calculations it is important to know whether a certain subset of Siegert pseudostates comprises a basis. This is a nontrivial issue, because of the unusual orthogonality and overcompleteness properties of Siegert pseudostates. Using analytical and numerical arguments, it is shown that the subset of bound states and outgoing Siegert pseudostates forms a basis. Time evolution in the context of Siegert pseudostates is also investigated. From the Mittag-Leffler expansion of the outgoing-wave Green’s function, the time-dependent expansion of a wave packet in terms of Siegert pseudostates is derived. In this expression, all Siegert pseudostates—bound, antibound, outgoing, and incoming—are employed. Each of these evolves in time in a nonexponential fashion. Numerical tests underline the accuracy of the method.

Figure 1

The complex $k$ spectrum for a particle in the step potential of Eq. (1) with $V_0=5$ and $a=10$. The spectrum is entirely discrete. Ten bound states $[\mathrm{Re}\left(k\right)=0,\mathrm{Im}\left(k\right)>0]$ are present, as well as nine antibound $[\mathrm{Re}\left(k\right)=0,\mathrm{Im}\left(k\right)<0]$. The positive $\mathrm{Re}\left(k\right)$ branch of the spectrum shows the complex wave numbers of the outgoing Siegert pseudostates, the negative $\mathrm{Re}\left(k\right)$ branch the incoming.

Figure 2

(a) The integration contour in the complex $k$ plane employed to evaluate Eq. (63). The integration proceeds along the imaginary axis from $\mathrm{i}\infty$ to the origin and from there along the real axis to $+\infty$. This contour corresponds to integration from $-\infty$ to $+\infty$ on the physical sheet of the energy Riemann surface. (b) If $t<0$, then the contour may be closed in the first quadrant of the $k$ plane. (c) The integration path in (a) may be evaluated for $t>0$ using one closed contour and one that runs from $-\infty$ to $+\infty$.

Figure 3

Time evolution of the wave packet described at $t=0$ by Eq. (70); $k_0=15$. In the $r$ interval between 0 and 10, the potential is constant [$V_0=5$ and $a=10$ in Eq. (1)].

Figure 4

The same parameters as in Fig. 3, but with $k_0=5$. Notice that there are significant physical reflections as soon as the wave packet hits the discontinuity at $r=a$.

Figure 5

The integration contour used to evaluate Eq. (B2) depends on the nature of the Siegert pseudostate considered. Arrows indicate the direction of the integration path in the complex $s$ plane. The circle symbolizes the location of the pole of the integrand on the right-hand side of Eq. (B2). (a) Bound state. (b) Outgoing Siegert pseudostate. (c) Antibound state. (d) Incoming Siegert pseudostate.