Diploma thesis

4.3 Densities
Figs. 4.2(a)–4.2(l) show qualitatively an analogous behaviour for the energies of the harmonic
potential as for the square well except the fact that here the real part of the energy yields ε R =
2m−1 instead of m 2 . The imaginary part of the energy is always non-positive which confirms that
our imaginary potential model really describes a damping effect and thus dissipation. Furthermore
we can separate all states in two groups of n-states and k-states again, where we resumed the
notation of the last chapter which means lim c→∞ ε k I = 0 and lim c→∞ ε n I = −∞. Also the well
known fusion of always two adjoining k-states is a property of the harmonic potential and an
effect of its natural symmetry, as well as the interchange of states being k- or n-state at some
waists ω crit . Observing the imaginary parts of the k-states reveals again a minimum so we can
see that after it ε k I indeed increases to zero. Unfortunately we can not derive the analogon of this
for the n-states that is a maximum of ε n I + c. This is related to the behaviour of the left hand
side of (4.37) and (4.80) which yields an increasing effort of the numerical calculation for large
values of c and ω for the n-states. So the numerical evaluation of the n-states for large waists and
dissipations can unfortunately not be included in this diploma thesis.
Nevertheless we know from the last chapter that the limit ω → ∞ yields finite values for ε n R and
there is some additional evidence that also c → ∞ is followed by finite results. We can take for
instance the obvious analogy of all results to the square well potential where a finite saturation
value exists which we even can calculate. Moreover we found out that the imaginary potential
does not directly affect the real part of the energy but yields the same effect as a potential barrier
confining the n-states in a square well potential with the width 2w for c → ∞. Thus the ε n R
become the real energies of states of exactly this potential well and should be therefore finite. The
general situation actually did not change since the harmonic potential becomes a square well one
for c → ∞ and w = const. < ∞. So the assumption, that also in the harmonic case a kind of
independent potential well containing states with finite energy eigenvalues develops, seems to be
reasonable. We will have a deeper look at exactly this issue while discussing the corresponding
densities of the energies.
4.3 Densities
Taking (4.35) and (4.78) we can calculate via (4.21) the densities for given c and ω:
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