indices $ i i $) $$\rho (w) \,=\,   \frac{w } {2  {\tilde g}_5^2}  \, \frac{ e^{- k^2 z_h^ 2 }    } { z_h}  \,
\vert F( \omega , z_h) \vert^ 2  \,,
\label{spectralf}$$ where we defined the dimensionless coupling $ {\tilde g}_5^2 =  g_5^2 /R $, as in the zero temperature case. This is the object that will describe the thermal behavior of the heavy vector mesons. In the next section we present the results of the numerical calculations of $\rho$.

 Spectral functions for charmonium and bottomonium S-wave states
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We solved numerically equation (\[eqmotionbtb\]) for the bulk to boundary propagator $v (\omega, z) $, written in terms of the function $ F$ as in eq. (\[vdecomp\]), with the boundary conditions described in the previous section. The parameters used are the zero temperature ones, from ref. [@Braga:2015jca], namely a flavor independent UV cutoff $  1/z_0 = 12.5 $ GeV and flavor dependent soft wall parameters with values $ k_c = 1.2 $ GeV for charmonium and $  k_b = 3.4 $ GeV for bottomonium S-wave states.

The spectral function (\[spectralf\]) was calculated for different temperatures. An important non trivial fact emerged from the analysis of the large frequency asymptotic behavior. 