can provide us with good results even in cases when Fock states fail. Or, second, we may need all excited states at once, such as in the calculation of the optical conductivity σ$_{reg}$ ( ω ). In this case, the fundamental problem of any ED calculation is that a finite Hilbert space does not allow for the representation of infinitely many excited states. No basis construction can overcome this obstruction. It is however possible to include sufficiently many states in the Hilbert space, such that the Hilbert space truncation shows up only through a kind of numerical noise, while the relevant physical structures are resolved. For exam- ple, the equally spaced peaks in the lower right panel in Fig. 9 are physically meaningful, while the tiny spikes in the middle right panel are numerical noise. The SOS construction guarantees the correct envelope and shape of σ$_{reg}$ ( ω ). For example, in the middle left panel in Fig. 9, a calculation with Fock states instead of SOS results in a curve that is shifted to smaller energies. Again, the Fock states cannot account for the substantial oscillator elongation in the groundstate and excited states in the adiabatic regime. To summarize, integrated or averaged properties of excited states are obtained correctly with SOS, but not with Fock states. For correct spectral prop- erties, e.g. peak positions at high resolution, additional effort is needed in any case. The SOS construction can be modified in various ways. For example, if oscillator states with large positive and negative elongations should be represented, one can ap- ply a global shift ˆ S ( ξ ), constructing the SOS from coher- ent states | mσ + ξ 〉 $_{c}$. The additional parameter ξ provides the necessary freedom to deal with such a situation. We introduced the SOS mainly for oscillator ground- states. 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