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We construct a relativistic potential quark model of $D$, $D_s$, $B$, and$B_s$ mesons in which the light quark motion is described by the Dirac equationwith a scalar-vector interaction and the heavy quark is considered a localsource of the gluon field. The effective interquark interaction is described bya combination of the perturbative one-gluon exchange potential$V_{\mathrm{Coul}}(r)=-\xi/r$ and the long-range Lorentz-scalar andLorentz-vector linear potentials $S_{\mathrm{l.r.}}(r)=(1-\lambda)(\sigmar+V_0)$ and $V_{\mathrm{l.r.}}(r)=\lambda(\sigma r+V_0)$, where$0\leqslant\lambda<1/2$. Within the quasiclassical approximation, we obtainsimple asymptotic formulas for the energy and mass spectra and for the meanradii of $D$, $D_s$, $B$, and $B_s$ mesons, which ensure a high accuracy ofcalculations even for states with the radial quantum number $n_r\sim 1$. Weshow that the fine structure of P-wave states in heavy-light mesons isprimarily sensitive to the choice of two parameters: the strong-couplingconstant $\alpha_s$ and the coefficient $\lambda$ of mixing of the long-rangescalar and vector potentials $S_{\mathrm{l.r.}}(r)$ and $V_{\mathrm{l.r.}}(r)$.The quasiclassical formulas for asymptotic coefficients of wave function atzero and infinity are obtained.

The quasiclassical theory of the Dirac equation with a scalar-vector interaction and its applications in the theory of heavy-light mesons