In the framework of spectral sum rules in the planar limit of quantum chromodynamics, we propose two new methods for calculating the spectra of light mesons based on using linear radial Regge trajectories and the simplest quark-antiquark operators interpolating meson states. Both methods predict a resonance near $500$ MeV in the scalar-isoscalar channel, which hypothetically corresponds to the lightest scalar hadron, the $\sigma$-meson. This can mean that the quark-antiquark component is strongly dominating in its structure even if the $\sigma$-meson is a tetraquark state. In one of the methods, we obtain a reasonable agreement with experimental data using only two input parameters: the phenomenological value of the gluon condensate and the weak decay constant of the pion. In this case, the predicted quark condensate value agrees well with contemporary lattice computation results.