5 Bessel function compensates the logarithmic suppression by ω ( E ). Consequently, as seen below, the fluctuations associated with on-separatrix motion differ dramatically, depending on where the SCS wave-packet is launched. Analytic results are obtained in App. D, also for the Zero and TwinFock preparations. In the latter case the result is: whereas the Wigner function of an eigenstate is semiclas- sically approximated by a micro-canonical distribution: In general none of the above listed initial states is an eigenstate of the BHH, but rather a superposition of BHH eigenstates. Consequently their Wigner function deforms over time (see e.g. Fig. 3) and the expectation values of observables become time dependent (see e.g.Fig. 4), as discussed in later sections. As shown in Fig. 5, the above semiclassical expressions agree well with the exact numerical results. P( E ) ∝ ω ( E ) NK [ 1 − ( 2 E NK ) 2 ] − 1 / 2 . (41) ρ ( ν ) W ( n, ϕ ) ≈ ω ( E$_{ν}$ ) δ ( H ( ϕ, n ) − E$_{ν}$ ) . (37) V. THE INITIAL PREPARATION AND ITS EIGENSTATE EXPANSION - LOCAL DENSITY OF STATES Having set the stage by defining the phase-space rep- resentation of eigenstates in Sect. III and of the initial conditions in Sect. IV, the evolution of any initial prepa- ration is uniquely determined by its eigenstate expansion. Thus, in order to analyze the dynamics, we now evaluate the local density of states (LDOS) with respect to the preparation ψ of the system: The LDOS determines the spectral content of the dy- namics. Consider first the characteristic oscillation fre- quency of dynamical variables. Away from the separa- trix, it is given by the classical estimate ω$_{osc}$ ≈ ω ( E ). For example, for a Zero preparation, we have ω$_{osc}$ = ω$_{J}$ . While the classical frequency vanishes on the separtrix, we still obtain a finite result for near-separatrix prepa- rations (Pi,Edge) because the wave-packet has a finite width, and because ω$_{x}$ provides a lower bound on ω$_{osc}$ . In any case the result becomes h dependent. To be specific, consider the Pi and the Edge preparations separately. In both cases the width of the wave-packet is Δ n = √ N/ 2. In case of the Pi preparation it occupies an energy range Δ E = U Δ n 2 ∝ N , while in the Edge preparation case Δ E = v$_{n}$ Δ n N 1 / $^{2}$, where v$_{n}$ ≈ ω$_{J}$ is identified as the velocity of the phase space points in this region. Using Eq. (29) we find If ψ is mirror symmetric the above expression should be multiplied either by 0 or by 2 in the case of odd/even eigenstates. In Fig. 5, we plot the LDOS associated with Pi, Edge, Zero and TwinFock preparations. The applica- bility of semiclassical methods to calculate the LDOS has been numerically demonstrated for the case of a three site (trimer) model in [35]. By contrast, the simpler two site (dimer) model under consideration, offers an opportunity for deriving exact expressions by substituting Eqs. (35)- (37) into Eq. (38) and evaluating the integrals under the appropriate approximations. For clarity, we summarize below the main results of this analytic evaluation, with the details given in App. D. where the additional factor of “2” applies to the Edge preparation: This factor is due to the different depen- dence of Δ E on N in the two respective cases. On top we might have an additional factor of 2 due to mirror symmetry (see discussion of the dynamics at the end of the next section). Note that as u/N exceeds unity, the distinction between Pi, Zero, and Edge preparations blurs because the wave-packets becomes wider than the width of the separatrix region, until at the Fock regime, all three consist of the same island levels. Thus in this limit, we get for ω$_{osc}$ essentially the same result as in the (negligable K ) Fock regime: the width of the wave-packet is Δ n = √ N/ 2, and the dispersion relation ω ( E ) = 2 Un gives the standard separated-condensates phase-diffusion frequency [17, 18], Consider first the Pi and Edge preparations. The Wigner distributions of both lie on the separatrix and are hence concentrated around the energy E$_{x}$ . Yet, their line shapes are strikingly different: For an Edge prepa- ration we have, ω$_{osc}$ ≈ { 1 2 } × [ log ( N u )] − 1 ω$_{J}$, (42) P( E$_{ν}$ ) = |〈 E$_{ν}$ | ψ 〉| 2 = trace( ρ ( ν $^{)}$ρ( ψ $^{)}$) = ∫ ρ ( ν ) W (Ω) ρ ( ψ ) W (Ω) d Ω h . (38) P( E ) ∝ ω ( E ) ω$_{J}$ exp [ − 1 N ( E − E$_{x}$ ω$_{J}$ ) 2 ] , (39) ω$_{osc}$ ≈ ( u N ) 1 / $^{2}$ω$_{J}$ = $_{U}$√$_{N.}$ (43) P( E ) ∝ ω ( E ) ω$_{J}$ Bessel [ E − E$_{x}$ NU ] (40) It should be clear that, both, in Eq. (42) and in Eq. (43) u/N should be accompanied with a numerical prefactor that should be adjusted because the notion of “width” is somewhat ill defined and in general depends on the pre- cise details of the numerical procedure, which we discuss in the next section. featuring a dip at E ∼ E$_{x}$ . The calculation in the Pi case leads to (the exact expression can be found in App. D). This latter line shape features a peak at E ∼ E$_{x}$ , because the