  We consider the Dirichlet eigenvalues of the Laplacian among a Poissonian
cloud of hard spherical obstacles of fixed radius in large boxes of
$\mathbb{R}^d$, $d \ge 2$. In a large box of side-length $2l$ centered at the
origin, the lowest eigenvalue is known to be typically of order $(\log
l)^{-2/d}$. We show here that with probability arbitrarily close to $1$ as $l$
goes to infinity, the spectral gap stays bigger than $\sigma (\log l)^{-(1 +
2/d)}$, where the small positive number $\sigma$ depends on how close to $1$
one wishes the probability. Incidentally, the scale $(\log l)^{-(1+ 2/d)}$ is
expected to capture the correct size of the gap. Our result involves the proof
of new deconcentration estimates. Combining this lower bound on the spectral
gap with the results of Kerner-Pechmann-Spitzer, we infer a type-I generalized
Bose-Einstein condensation in probability for a Kac-Luttinger system of
non-interacting bosons among Poissonian spherical impurities, with the sole
macroscopic occupation of the one-particle ground state when the density
exceeds the critical value.
