We derive an analytical theory of the prestellar core initial mass function based on an extension of the Press-Schechter statistical formalism. With the same formalism, we also obtain the mass spectrum for the non self-gravitating clumps produced in supersonic flows. The mass spectrum of the self-gravitating cores reproduces very well the observed initial mass function and identifies the different mechanisms responsible for its behaviour. The theory predicts that the shape of the IMF results from two competing contributions, namely a power-law at large scales and an exponential cut-off (lognormal form) centered around the characteristic mass for gravitational collapse. The cut-off exists already in the case of pure thermal collapse, provided that the underlying density field has a lognormal distribution. Whereas pure thermal collapse produces a power-law tail steeper than the Salpeter value, dN/dlog Mpropto M^{-x}, with x=1.35, this latter is recovered exactly for the (3D) value of the spectral index of the velocity power spectrum, nsimeq 3.8, found in observations and in numerical simulations of isothermal supersonic turbulence. Indeed, the theory predicts that x=(n+1)/(2n-4) for self-gravitating structures and x=2-n/3 for non self-gravitating structures, where n is the power spectrum index of log(rho). We show that, whereas supersonic turbulence promotes the formation of both massive stars and brown dwarfs, it has an overall negative impact on star formation, decreasing the star formation efficiency. This theory provides a novel theoretical foundation to understand the origin of the IMF and to infer its behaviour in different environments. It also provides a complementary approach and useful guidance to numerical simulations exploring star formation, while making testable predictions.
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