  J/Psi and eta_c above the QCD critical temperature T_c are studied in
anisotropic quenched lattice QCD, considering whether the c\bar c systems above
T_c are spatially compact (quasi-)bound states or scattering states. We adopt
the standard Wilson gauge action and O(a)-improved Wilson quark action with
renormalized anisotropy a_s/a_t =4.0 at \beta=6.10 on 16^3\times (14-26)
lattices, which correspond to the spatial lattice volume V\equiv
L^3\simeq(1.55{\rm fm})^3 and temperatures T\simeq(1.11-2.07)T_c. We
investigate the c\bar c system above T_c from the temporal correlators with
spatially-extended operators, where the overlap with the ground state is
enhanced. To clarify whether compact charmonia survive in the deconfinement
phase, we investigate spatial boundary-condition dependence of the energy of
c\bar c systems above T_c. In fact, for low-lying S-wave c \bar c scattering
states, it is expected that there appears a significant energy difference
\Delta E \equiv E{\rm (APBC)}-E{\rm (PBC)}\simeq2\sqrt{m_c^2+3\pi^2/L^2}-2m_c
(m_c: charm quark mass) between periodic and anti-periodic boundary conditions
on the finite-volume lattice. In contrast, for compact charmonia, there is no
significant energy difference between periodic and anti-periodic boundary
conditions. As a lattice QCD result, almost no spatial boundary-condition
dependence is observed for the energy of the c\bar c system in J/\Psi and
\eta_c channels for T\simeq(1.11-2.07)T_c. This fact indicates that J/\Psi and
\eta_c would survive as spatially compact c\bar c (quasi-)bound states below
2T_c. We also investigate a $P$-wave channel at high temperature with maximally
entropy method (MEM) and find no low-lying peak structure corresponding to
\chi_{c1} at 1.62T_c.
