  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 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 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. To clarify whether compact charmonia
survive in the deconfinement phase, we investigate spatial boundary-condition
dependence of the energy of the c\bar c systems above T_c. In fact, for
low-lying c \bar c scattering states, there appears a significant energy
difference \Delta E \equiv E{\rm (APBC)}-E{\rm (PBC)} between periodic and
anti-periodic boundary conditions as \Delta
E\simeq2\sqrt{m_c^2+3\pi^2/L^2}-2m_c (m_c: charm quark mass) 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, we find almost no spatial
boundary-condition dependence 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, which indicates that J/\Psi and
\eta_c would survive as compact c\bar c quasi-bound states below 2T_c.
