17 −500 −490 −480 −470 0 0.25 0.5 0.75 1  E (scaled)   P(E )  0 0.5 1 1.5 2 0.2 0.4 0.6 0.8 / $_{J}$   C x ( )  −4 −2 0 2 4 6 8 0 0.1 0.2 0.3 0.4  E (scaled)   P(E )  0 0.5 1 1.5 2 0.1 0.2 0.3 / $_{J}$   C x ( )  −100 −50 0 50 100 0 0.005 0.01 0.015 0.02 0.025  E (scaled)   P(E )  0 0.5 1 1.5 2 0.02 0.04 0.06 / $_{J}$   C x ( )  −500 −400 −300 −200 −100 0 100 0 0.01 0.02 0.03  E (scaled)   P(E )  0 0.5 1 1.5 2 0.02 0.04 / $_{J}$   C x ( )  FIG. 5: (Color online) The LDOS (left) and the spectral content of the fluctuations (right) of N = 500 bosons with u = 4, for Zero, Pi, Edge, and TwinFock preparations (top to bottom). The horizontal axes are E − E x and ω/ω$_{J}$ . The lines in the LDOS figures are based on a semiclassical analysis (see text), while the circles are from the exact quantum calculation. Note that due to the mirror symmetry of the Zero preparation the expected frequency should approach 2 ω$_{J}$ , while for the Pi preparation it is bounded from below by 2 ω x (both frequencies are indicted by vertical dashed lines). Note also the outstanding difference between the spectral support of Zero and Pi preparations compared with continuous-like support in the case of Edge and Fock preparations.