  Two-body open charm strong decays of the $2P$ and $3P$ charmonium states are
studied by the Bethe-Salpeter(BS) method combined with the $^3P_0$ model. The
wave functions and mass spectra of the $2P$ and $3P$ charmonium states are
obtained by solving the BS equation with the relativistic correction. The
strong decay widths and relative ratios of the $2P$ and $3P$ charmonium states
are calculated. Comparing our results with the experimental data, we obtain
some interesting results. Considering the $X^*(3860)$ as the $\chi_{c0}(2P)$,
the total strong decay width is smaller than the experimental data. But the
strong decay width depends on the parameter $\gamma$ in the $^3P_0$ model, and
the mass and width of the $X^*(3860)$ have large errors, we cannot rule out the
possibility that the $X^*(3860)$ is the $\chi_{c0}(2P)$. The $X(4160)$ is a
good candidate for the $\chi_{c0}(3P)$, not only the strong decay width of the
$\chi_{c0}(3P)$ is same as the experimental data, but the relative ratios
$\frac{\Gamma(\chi_{c0}(3P)\to D\bar D)}{\Gamma(\chi_{c0}(3P)\to D^*\bar
D^*)}\approx0.0019<0.09$, and $\frac{\Gamma({\chi_{c0}(3P)\to D\bar
D^*})}{\Gamma({\chi_{c0}(3P)\to D^*\bar D^*})}=0<0.22$ are consistent with the
experimental results of the $X(4160)$. Taking the $X(4274)$ as the
$\chi_{c1}(3P)$, the strong decay width is consistent with the experimental
data, so the $X(4274)$ is a good candidate for the $\chi_{c1}(3P)$. Assigning
the $X(4350)$ as the $\chi_{c2}(3P)$, the corresponding strong decay width is
slightly larger than the experimental data. To identify if the $X(4350)$ is
$\chi_{c2}(3P)$, many more investigations are needed. All of the strong decay
widths and relative ratios of the $2P$ and $3P$ charmonium states can provide
the useful information to discover and confirm these particles in the future.
