By applying the Bohr-Sommerfeld quantization approach to the quadratic form of the spinless Salpeter-type equation (QSSE), we show that the obtained Regge trajectories for the mesons consisting of unequally massive quarks take the form M 2 = β (c l l + π n r + c 0 )2/3 + c 1 , which have the same form as the Regge trajectories for charmonia and bottomonia. Then we apply the obtained Regge trajectories to fit the spectra of the strange mesons, the heavy-light mesons (the D, D s , B and B s mesons) and the bottom-charmed mesons. The fitted Regge trajectories are in agreement with the experimental data and the theoretical predictions, which demonstrates that the newly proposed Regge trajectories can be applied universally to the light mesons, the heavy-light mesons and the heavy mesons. By fitting the spectra of the mesons composed of different quarks, the concavity of these Regge trajectories are illustrated, which is of cardinal significance for the potential models.

In this paper, we present one new form of the Regge trajectories for heavy quarkonia which is obtained from the quadratic form of the spinless Salpeter-type equation (QSSE) by employing the Bohr-Sommerfeld quantization approach. The obtained Regge trajectories take the parameterized form M 2 = β(c l l + π n r + c 0 ) 2/3 + c 1 , which are different from the present Regge trajectories. Then we apply the obtained Regge trajectories to fit the spectra of charmonia and bottomonia. The fitted Regge trajectories are in good agreement with the experimental data and the theoretical predictions.

We take (µ ± e ∓ ) systems and consider the states with quantum number J P = 0 − as examples, to explore the different contents of the instantaneous Bethe-Salpeter (BS) equation and its analog, Breit equation, by solving them exactly. The results show that the two equations are not equivalent, although they are analogous. Furthermore, we point out that the Breit equation contains extra unphysical solutions, so it should be abandoned if one wishes to have an accurate description of the bound states for the instantaneous interacting binding systems. [5,6] un-properly. To explore the different contents i.e. the un-equivalence of the two equations, in the paper we restrict ourselves to the bound states of the systems (µ ± e ∓ ) systems with quantum number J P = 0 − (S-wave) to solve the two equations exactly and to examine the obtained solutions accordingly.BS equation for a fermion-antifermion system has the general formulation [4]:(1) where χ P (q) is the BS wave function, P is the total momentum, q is relative momentum, and V (P, q, k) is the kernel of the equation, p 1 , p 2 are the momenta of the constituents 1 and 2 respectively. The total momentum P and the relative momentum q are related to the momenta p 1 , p 2 as follows:If the kernel V (P, k, q) of the BS equation has the behavior: V (P, q, k)| P =0 = V ( q, k) ( P = 0 in center mass frame of the concerned bound state), the BS equation is called as an 'instantaneous BS equation'.For instance, in Coulomb gauge, the terms corresponding to the possible transverse-photon exchange between the two components in BS kernel for (µ ± e ∓ ) systems are considered as higher order, so the 'lowest order' BS equation kernel for the systems has the form:Namely, there are some physical systems whose bound states are described by an instantaneous BS equation indeed. * Not post-mail address.In Refs. [1,7], we showed how to solve an instantaneous BS equation exactly, and also showed the authors of Ref. [2,3] how to mislead the problem to its analog: Breit equation. For present convenience, let us repeat the main procedure in Refs. [1,7].Firstly, as done by the authors of Refs. [2,3], the 'instantaneous BS wave function' ϕ P ( q) asis introduced, then the BS equation Eq.(1) can be rewritten asHere S(1) f (p 1 ) and Sf (−p 2 ) are the propagators of the fermion and anti-fermion respectively andThe propagators (in C.M.S. i.e. P = 0) can be decomposed as:, where J = 1 for the fermion (i = 1) and J = −1 for the anti-fermion (i = 2). It is easy to check± can be considered as 'energy' projection operators, and 'complete' for the projection. For below discussions let us introduce the notations ϕ ±± P ( q) as:Because of the completeness of the projection for Λ ± , we have:

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We employ the Nyström method with the extended Simpson's rule to solve numerically the Schrödinger equation in momentum space with the screened Cornell potential. Two peculiar phenomena, furcation and deviation of the first point, occur in the obtained eigenfunctions. By applying the Landé subtraction method to remove or relieve the singularities in the integrands, the furcation phenomenon disappears, the deviation of the first point is weakened, and the obtained eigenvalues have high accuracy.