B_{c} and heavy meson spectroscopy in the local approximation of the Schrodinger equation with relativistic kinematics

We present bound state masses of the self-conjugate and non-self-conjugate mesons in the context of the Schr\"{o}dinger equation taking into account the relativistic kinematics and the quark spins. We apply the usual interaction by adding the spin dependent correction. The pseudoscalar and vector decay constants of the $B_{c}$ meson and the unperturbed radial wave function at the origin are also calculated. We have obtained a local equation with a complete relativistic corrections to a class of three attractive static interaction potentials of the general form $% V(r)=-Ar^{-\beta}+\kappa r^{\beta}+V_{0},$ with $\beta =1,1/2,~3/4$ which can also be decomposed into scalar and vector parts in the form $% V_{V}(r)=-Ar^{-\beta}+(1-\epsilon)\kappa r^{\beta}$ and $% V_{S}(r)=\epsilon \kappa r^{\beta}+V_{0};$ where $0\leq \epsilon \leq 1.$ The energy eigenvalues are carried out up to the third order approximation using the shifted large-N-expansion technique.