We prove that the Frobenius-Perron operator U of the cusp map F:[-1,1] →[-1,1], F(x)= 1 -2 √|x| (which is an approximation of the Poincaré section of the Lorenz attractor) has no analytic eigenfunctions corresponding to eigenvalues different from 0 and 1. We also prove that for any q ∈ (0,1) the spectrum of U in the Hardy space in the disk {z ∈ C:|z - q| < 1 + q} is the union of the segment [0,1] and some finite or countably infinite set of isolated eigenvalues of finite multiplicity.