Diploma thesis

3 Complex square well potential
ω → 0 at all as we can see by comparing with Figs. 3.2(b) – 3.2(k). So from this point of view
both real and imaginary part do not show any tendency for ω → 0 to coincide with the calculated
limit lim ω→0 ε k 0 = k 2 = m 2 at all. This looks inconsistent but at a later point of this discussion we
will give an interpretation that provides an appropriate explanation of this.
First we spend some time on looking at the particular strength of dissipation the maxima and
minima of the imaginary part occur at. Let us call it c crit which seems to depend on the particular
waist ω and state m we are considering. From Figs. 3.3(a) – 3.4(b) we can read off that c crit
obviously increases for ω ≈ 0 with decreasing ω and for ω ≈ π with increasing ω, respectively.
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Moreover, considering the symmetric states in Fig. 3.2(a) shows that the real part reaches its
biggest slope at c crit that means it has an inflection point right there. A closer look on the
antisymmetric states in Fig. 3.2(a) and on the n-states in Fig. 3.2(l) as well provides the same
insight. So it seems that c crit denotes the dissipation, where the energy has its largest deviation
from the limits we just discussed, that is for ε R being constant and for ε k I,0 as well as for ε n I,∞
being equal to zero.
Many things we have introduced and discussed so far can be generalized to arbitrary waists
0 ≤ ω ≤ π , so we continue with the regime where all areas yield similar extensions.
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3.2.2 Three areas regime
Now let us take a look at the remaining pictures, that is the regime 0.1 ≤ ω ≤ 1.1, where both
types of states are coexisting among the lowest six energy levels. First we can generalize the
concept of c crit since ε k I,0 and ε n I,∞ + c reveal also in this case minima and maxima.
Figure 3.5: ε k I,0 ≤ 0 (minima) and ε n I,∞ + c ≥ 0 (maxima) plotted for ω = 0.7.
Considering Figs. 3.2(b) – 3.2(k) yields that for intermediate waists c crit also represents an inflection
point of the real part of the energy, so its largest slope. Thus we can conclude that the critical
dissipation exhibits the largest deviation from the limits ω 0 and ω π in the same way as we
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found out in the end of the previous subsection.
Since there occur only k-states for ω 0 and only n-states for ω π in the course of continuous
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rearrangement a kind of continuous transfer of lower k-states with higher n-states for increasing
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