Antibound states

The second type of solutions are those for which k is purely imaginary and negative. These states are called antibound states and have the asymptotic behavior of 4>( x > L) a e+ww. 

In what follows, in order to emphasize the relationship of resonant states with the continuous spectra, the discussion will be restricted to potentials where bound and antibound states are absent. This also simplify the mathematical content. It is important to stress, however, that the approach is complete general. 

Furthermore, It should be noted that the presence of a bound state (but not the antibound state) has also been seen in Raman scattering experiments (Giintherodt et al. 1983, 1985). No phonon softening was observed, however. This discrepancy to neutron scattering results are attributed to different typical probing times for those techniques. 

In general, expansions involving resonant states may be divided in two broad classes a first class that involves bound, resonant, and continuum states, where usually resonant states are considered in the interval (0,oo) [13, 17, 21, 24], and a second class that refers fo expansions defined in the interval (0, a), where a stands for fhe radius of the finite range potential, that involves the full set of bound, antibound (virtual), and resonant states. Here, the expansions are purely discrete and follow using Gauchy s infe-gral theorem [15, 16, 18, 25]. Some of these expansions are discussed in Ref. [26]. 

G. Garcia-Calderon, On the overcompletness of the set of bound, antibound and resonant states, Lett. Nuovo Cimento 33 (1982) 253. 

---