  An explicit solution of the spectral problem of the non-local Schr\"odinger
operator obtained as the sum of the square root of the Laplacian and a quartic
potential in one dimension is presented. The eigenvalues are obtained as zeroes
of special functions related to the fourth order Airy function, and closed
formulae for the Fourier transform of the eigenfunctions are derived. These
representations allow to derive further spectral propertiessuch as estimates of
spectral gaps, heat trace and the asymptotic distribution of eigenvalues, as
well as a detailed analysis of the eigenfunctions. A subtle spectral effect is
observed which manifests in an exponentially tight approximation of the
spectrum by the zeroes of the dominating term in the Fourier representation of
the eigenfunctions and its derivative.
