  Within the framework of AdS/CFT duality, excited states of the conformal
field living at the global AdS boundary of a four-dimensional spacetime
Einstein gravity are investigated analytically in the probe limit where the
field equations are linearized. At asymptotically large values, the threshold
chemical potential for the appearance of excited condensate states are
discrete, equal spacing, with the gap approaches zero logarithmically in the
limit $T\rightarrow 0$. Remarkably, numerical results show that, this behavior
applies even for states as low as for the first or the second excited state of
the condensate. This is especially significant on the liquid side of the black
hole van der Waals - like phase transition (small or zero topological charge)
where there seems to be no gap between the ground state and the first excited
state at zero temperature. We postulate that, at the exact limit $T = 0$ where
the gap is zero, the spectrum of threshold chemical potentials becomes
continuous, all excited states of the condensate are activated above a finite
chemical potential, suggesting a new quantum phase transition as a function of
the chemical potential. Previous studies have largely missed this continuous
spectrum of excited states in the $T\rightarrow 0$ limit. This fact should be
taken into account carefully in AdS/CFT duality studies.
