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| These plots show computed resonances of \[H_L=-d^2/dx^2 + V_1(x) + e^{-cL}\delta(x-L)\] for \(1\le L \le 50\), where
\[ V_1(x) = \frac{1}{2}\delta(x-3) + 3\delta(x-2) + \delta(x). \] On each top panel, the red circles are the resonances of \(H_1=-d^2/dx^2 + V_1(x)\), the green circles are the reflectionless points of \(H_1\) and the blue points are the resonances of \(H_L\). A schematic picture of the potential is shown on the lower panel.
The first plot shows what happens when \(c=0.0\). Here infinitely many resonances converge to the real axis. In addition there is a resonance converging to each reflectionless point (green circle). For the second plot \(c=1.6\) so we know that infinitely many resonances converge to the line \(\hbox{Im} (k) = -0.8\). Above this line, resonances of \(H_L\) converge to the resonances (red circles) of \(H_1\), while below the line resonances of of \(H_L\) converge to the reflectionless points (green circles) of \(H_1\). The resonances were computed using the rootfinding procedure ''Analytic'' in Maple and assembled into a video using MATLAB. |