We discuss the existence of large isolated (non-unit) eigenvalues of the Perron-Frobenius operator for expanding interval maps. Corresponding to these eigenvalues (or `resonances') are distributions which approach the invariant density (or equilibrium distribution) at a rate slower than that prescribed by the minimal expansion rate. We consider the transitional behaviour of the eigenfunctions as the eigenvalues cross this `minimal expansion rate' threshold, and suggest dynamical implications of the existence and form of these eigenfunctions. A systematic means of constructing maps which possess such isolated eigenvalues is presented.