A Lévy random drift destroys compactness for every confining power

Let $V$ be a two-sided semi-selfsimilar Lévy process of index $\alpha\in(0,2)$ on $\mathbb R$ and let $L^{w+V}=\frac12(\Delta-\nabla(w+V)\cdot\nabla)$ act on $L^2(\mathbb R;e^{-(w+V)}\,dx)$. For $w_1=|x|^\delta$ and $w_2=-|x|^\delta$ the unperturbed semigroups are compact exactly when $\delta>1$. Shiozawa and Wang proved that the perturbed semigroups are noncompact, with bottom of the essential spectrum equal to $0$, when $\delta\le 1/\alpha$, and asked what happens when $\delta>1/\alpha$. We show that for $\delta>1/\alpha$ the bottom of the essential spectrum of $-L^{w_1+V}$ is $0$ almost surely as soon as the Lévy measure of $V$ on one of the two half-lines charges $(-\infty,0)$, and that the same holds for $-L^{w_2+V}$ as soon as one of them charges $(0,\infty)$. In particular, when $V$ changes sign and either $\alpha<1$ or the jumps of $V$ go both ways, both semigroups are noncompact for every $\delta>0$. The mechanism is local: a jump of size $m$ at distance $\tau$ from the origin lowers a Rayleigh quotient by a factor $e^{-m/2}$, while the confining power costs only a factor $\tau^{2\delta-2}$, and an $\alpha$-semi-selfsimilar process has jumps of size of order $\tau^{1/\alpha}$ at every scale.