It is illustrated by the fitted Regge trajectories for a large majority of mesons that the radial and orbital Regge trajectories for mesons prefer being concave in the (nr, M 2 ) and (l, M 2 ) planes, respectively. The concavity of the meson Regge trajectories is model-independent. The concavity is expected to act as a criterion to choose a newly observed meson or to assign a particle to the unwellestablished state. The concavity of the meson Regge trajectories can assist in model construction and in the choice of the appropriate model to describe mesons. The appropriate models should yield the spectra which can produce the concave Regge trajectories according to the concavity of the meson Regge trajectories. If the large majority of the meson Regge trajectories are concave while a few meson Regge trajectories are convex which neither have been confirmed nor have been completely excluded at present, many existing models should be corrected or even be reconstructed, which will lead to the further understanding of the meson dynamics.

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We present errors of quadrature rules for the nearly singular integrals in the momentum-space bound-state equations and give the critical value of the nearly singular parameter. We give error estimates for the expansion method, the Nyström method, and the spectral method which arise from the near singularities in the momentum-space bound-state equations. We show the relations amongst the near singularities, the odd phenomena in the eigenfunctions, and the unreliability of the numerical solutions.