We aim to clarify the mathematical status of the leaky modes in an unbounded magnetohydrodynamic (MHD) plasma. The initial value problem of a one-dimensional unbounded MHD plasma is solved by means of a Laplace transform in time. It is shown that the MHD operator remains self-adjoint in the sense that a self-adjoint extension of the minimal MHD operator exists and that the eigenfrequencies should therefore be real. However, the classical picture of the MHD spectrum is to be extended with a slow and a fast leaky continuous spectrum when unbounded spatial domains are considered. The inversion integral for the Laplace transform is evaluated in such a way as to recast the continuum contribution in terms of a complete spectral representation. This involves the construction of the spectral measure associated with the continuous spectra. In the case of a slab structure, the spectral measure is not a monotonic function of the frequency, but peaks appear in the spectral measure around specific frequencies. These frequencies correspond to the discrete damped leaky modes described in the context of solar physics. The spectral measure can be interpreted physically in terms of the energy carried in and out by the incoming and outgoing waves. (C) 2007 American Institute of Physics.