  The Casimir energy for a massless scalar field between the closely spaced two
concentric D-dimensional (for D>3) spheres is calculated by using the mode
summation with contour integration in the complex plane of eigenfrequencies and
the generalized Abel-Plana formula for evenly spaced eigenfrequency at large
argument. The sign of the Casimir energy between closely spaced two concentric
D-dimensional spheres for a massless scalar field satisfying the Dirichlet
boundary conditions is strictly negative. The Casimir energy between D-1
dimensional surfaces close to each other is regarded as interesting both by
itself and as the key to describing of stability of the attractive Casimir
force. PACS number(s): 03.70.+k, 11.10.Kk, 11.10.Gh, 03.65.Ge
