  The hot plasma filling galaxy clusters emits copious X-ray radiation. The
classic unheated and unperturbed cooling flow model predicts dramatic cooling
rates and an isobaric X-ray spectrum with constant differential luminosity
distribution. The observed cores of clusters (and groups) show instead a strong
deficit of soft X-ray emission: $dL_{\rm x}/dT \propto (T/T_{\rm
hot})^{\alpha=2\pm1}$. Using 3D hydrodynamic simulations, we show that such
deficit arises from the tight self-regulation between thermal instability
condensation and AGN outflow injection: condensing clouds boost the AGN
outflows, which quench cooling as they thermalize through the core. The
resultant average distribution slope is $\alpha \simeq 2$, oscillating within
the observed $1<\alpha<3$. In the absence of thermal instability, the X-ray
spectrum remains isothermal ($\alpha > 8$), while unopposed cooling drives a
too shallow slope, $\alpha<1$. AGN outflows deposit their energy inside-out,
releasing more heat in the inner cooler phase; radially distributed heating
alone induces a declining spectrum, $1<\alpha<2$. Turbulence further steepens
the spectrum and increases the scatter: the turbulent Mach number in the hot
phase is subsonic, while it becomes transonic in the cooler phase, making
perturbations to depart from the isobaric mode. Such increase in $d\ln P/d\ln
T$ leads to $\alpha\approx3$. Self-regulated AGN outflow feedback can address
the soft X-ray problem through the interplay of heating and turbulence.
