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rower as the screening effect increases (the two metal- Mott insulator phase boundaries are shown for model (i) at ω$_{0}$ = 5; for the other cases we show the stability re- gion of the metallic phase). If the coupling strength is increased into the unphysical overscreening regime, one finds a first order transition to a bipolaronic insulator. 

The right hand panels of Fig. 2 present the phase dia- grams as a function of screened interaction and coupling strength (parametrized as ratio of unscreened to screened interaction). For U/U$_{scr}$ ∼ 2-3, typical of values found in RPA-type calculations, the critical screened interaction is 1 2 to 2 3 of the U$_{c}$$_{2}$ defined in the unscreened model. The dependence on screening frequency is weak. 

We next consider the effect of screening on the elec- tron spectral functions. In the model without screening the spectral function is characterized by peaks at ω = 0 (if the model is in the metallic phase) and ω ≃ ± U/ 2. Figure 3 shows that the situation is quite different in the screened case. The top panel presents the changes that occur as the screening strength is varied at fixed bare interaction U = 10 and screening frequency ω$_{0}$ = 3. A multipeak structure is evident, with a broad high energy feature at an energy set by a combination of the bare U and λ , a lower energy sideband and, in the metallic case, a (split) peak at ω = 0. While the peak position of the lower energy sidebands varies roughly in parallel with the screened U its energy is a parameter-dependent fraction of U$_{scr}$/ 2 (approximately 60%). 

The lower panel of Fig. 3 presents the variation of the 

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spectral function with screening frequency for fixed bare and screened interactions U = 8 > U$_{c}$$_{2}$ , U$_{scr}$ = 3 < U$_{c}$$_{2}$ . For reference the spectrum in the absence of screening is also shown. At the lowest value of the screening fre- quency, ω$_{0}$ = 1, the model is in its insulating phase and the spectrum is a broadened version of that of the un- screened insulator. For all other screening frequencies, the model is in the metallic phase. We see that the po- sitions of both the lower energy sideband and the high energy peak increase as the screening frequency is in- creased. At the highest screening frequency the lower energy portion of the spectrum begins to approach that expected in the antiadiabatic limit, with a peak near U$_{scr}$/ 2. We therefore interpret the lower energy feature as the “screened U ” sideband, but note that in general its energy does not yield a good measure of U$_{scr}$ . Even at the highest screening frequency ( ω$_{0}$ = 4 D = 16), substantial spectral weight exists at high energies ( ω ≈ ω$_{0}$ ). Figure 4 shows spectra computed for the Ohmic screen- 

ing model. The spectra are qualitatively similar to those 

