The inverse Laplace transform as the ultimate tool for transverse mass   spectra
  New high statistics data from the second generation of ultrarelativistic heavy-ion experiments open up new possibilities in terms of data analysis. To fully utilize the potential we propose to analyze the $m_\perp$-spectra of hadrons using the inverse Laplace transform. The problems with its inherent ill-definedness can be overcome and several applications in other fields like biology, chemistry or optics have already shown its feasability. Moreover, the method also promises to deliver upper bounds on the total information content of the spectra, which is of big importance for all other means of analysis. Here we compute several Laplace inversions from different thermal scenarios, both analytically and numerically, to test the efficiency of the method. Especially the case of a two component structure, related to a possible first order phase transition to a quark gluon plasma, is closer investigated and it is shown that at least a signal to noise ratio of $10^4$ is necessary to resolve two individual components. 
Ekkard Schnedermann