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In this paper we used the theory of adiabatic radial oscillations developedby Chandrasekhar to study the conditions for dynamical stability of constantenergy-density stars, or Schwarzschild stars, in the unstudied ultra compactregime beyond the Buchdahl limit, that is, for configurations with radius $R$in the range $R_{\rm S}<R<(9/8)R_{\rm S}$, where $R_{\rm S}$ is theSchwarzschild radius of the star. These recently found analytical solutionsexhibit a negative pressure region in their centre and, in the limit when $R\toR_{\rm S}$, the full interior region of the star becomes filled with negativepressure. Here we present a systematic analysis of the stability of theseconfigurations against radial perturbations. We found that, contrary to theusual expectation found in many classical works, the ultra compactSchwarzschild star is stable against radial oscillations. We computed values ofthe critical adiabatic index $\gamma_{c}$ for several stellar models withvarying radius $R/R_{\rm S}$ and found that it also approaches a finite valueas $R/R_{\rm S} \to 1$

On the radial stability of ultra compact Schwarzschild stars beyond the Buchdahl limit