Spectrum bottom and largest vacuity

The paper considers a scalar second-order elliptic operator with a non-negative random potential term V(x) = ∑ _{X ∈?} W(x?X) ≥ 0 corresponding to a Poisson cloud ? = {X} of “soft obstacles.” The operator acts on functions vanishing outside a large cubic open “box”rQ ⁰ = (?½r, ½r) ^d ?? ^d , d≥ 2. The paper develops a method of estimating from below the spectrum bottom of the operator through the volume of the largest connected set that can be made of smaller “blocks” containing relatively few obstacles. In the case of constant coefficients, the principal eigenvalue λ_{?, V} (r?) of (?? + V) in r? ⁰ is shown to satisfy, with high probability, the estimate where λ_?,* is the infimum of principal value of operator with zero potential term V≡ 0 under the Dirichlet condition on the boundary of a regular set of volume not exceeding one.