  In this work, we are concerned with the stability and convergence analysis of
the second order BDF (BDF2) scheme with variable steps for the molecular beam
epitaxial model without slope selection. We first show that the variable-step
BDF2 scheme is convex and uniquely solvable under a weak time-step constraint.
Then we show that it preserves an energy dissipation law if the adjacent
time-step ratios $r_k:=\tau_k/\tau_{k-1}<3.561.$ Moreover, with a novel
discrete orthogonal convolution kernels argument and some new discrete
convolutional inequalities, the $L^2$ norm stability and rigorous error
estimates are established, under the same step-ratios constraint that ensuring
the energy stability., i.e., $0<r_k<3.561.$ This is known to be the best result
in literature. We finally adopt an adaptive time-stepping strategy to
accelerate the computations of the steady state solution and confirm our
theoretical findings by numerical examples.
