A Lévy random drift destroys compactness for every confining power

Let V be a two-sided semi-selfsimilar Lévy process of index α∈(0,2) on R and let Lw+V=12(Δ-∇(w+V)·∇) act on L²( R;e-(w+V)\,dx). For w₁=|x|^δ and w₂=-|x|^δ the unperturbed semigroups are compact exactly when δ>1. Shiozawa and Wang proved that the perturbed semigroups are noncompact, with bottom of the essential spectrum equal to $0$, when δ≤ 1/α, and asked what happens when δ>1/α. We show that for δ>1/α the bottom of the essential spectrum of -Lw₁+V is $0$ almost surely as soon as the Lévy measure of V on one of the two half-lines charges (-∞,0), and that the same holds for -Lw₂+V as soon as one of them charges (0,∞). In particular, when V changes sign and either α<1 or the jumps of V go both ways, both semigroups are noncompact for every δ>0. The mechanism is local: a jump of size m at distance τ from the origin lowers a Rayleigh quotient by a factor e-m/2, while the confining power costs only a factor τ2δ-2, and an α-semi-selfsimilar process has jumps of size of order τ1/α at every scale.