The weight function for the Stieltjes integral definition of the spectral density is substantially that it is the value at the origin of a wave function of unit amplitude at infinity. In order to better grasp this concept, Figure 10 repeats the resonance graph of Figure 5 (and those following it), having been truncated at the origin with a perspective showing the starting amplitude in clear outline.
Figure 11 exhibits another kind of graph which would be of interest, namely the result of showing only the trace across the origin featured in Figure 10, but for varying masses. The result could run from zero mass, without any classically forbidden region at all, to a large enough mass that it would correspond to actual electrons in atoms. That would be the limit which interested Sewell  and Titchmarsh .
The peaks in these functions, especially in the limit of large masses, suggest the interpretation of the Dirac harmonic oscillator in terms of complex energies and associated eigenfunctions.