Siegert pseudostates: Completeness and time evolution

Abstract

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 timedependent 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.

General properties of the evolution of unstable states at long times

“…(Here ℜ (z) and ℑ (z) denote the real and imaginary parts of z respectively). Note that relations (20) and (19) establish a direct connection between the amplitude a(t) for the state |u and the exact effective Hamiltonian h u (t) governing the time evolution in the one-dimensional subspace H ∋ |u . Thus the use of the relation (20) is one of the most effective tools for the accurate analysis of the early-as well as the long-time properties of the instantaneous energy and decay rate for a given qausistationary state |u(t) .…”

“…The next step is to use the relation (20). So one should now divide (24) by (10) and then collect together all components of the same order with respect to ( t ).…”

Slow Electrons Generated by Intense High-Frequency Laser Pulses

“…Recently, one of us has developed a method which enables one to exactly incorporate the outgoing-wave boundary conditions [11,13,14]. The method is based on the expansion of the solution to the TDSE in terms of the Siegert states; it generalizes earlier applications of Siegert states for time propagation of wave packets [15][16][17] to nonstationary systems. The first applications of this method to the laser-atom interaction problem [9][10][11] have demonstrated its ability to produce very accurate highly resolved photoelectron spectra.…”

Bound states of dipolar molecules studied with the Berggren expansion method

“…The complex-energy GamowSiegert states [53,59] states have been used in various contexts in nuclear, atomic, and molecular physics [60][61][62][63][64][65][66][67][68][69][70][71][72][73]. Some recent applications of Gamow-Siegert states, also in the context of a CC formalism relevant to the problem of dipole anions, can be found in, e.g., Refs.…”