Spectral minimal partitions of unbounded domains

We study the problem of constructing $k$-spectral minimal partitions of domains in $d$ dimensions, where the energy functional to be minimized is a $p$-norm ($1 \le p \le \infty$) of the infimum of the spectrum of a suitable Schrödinger operator $-Δ+V$, with Dirichlet conditions on the boundary of the partition elements (cells). The main novelty of this paper is that the domains may be unbounded, including of infinite volume. First, we prove a sharp upper bound for the infimal energy among all $k$-partitions by a threshold value which involves the infimum $Σ$ of the essential spectrum of the Schrödinger operator on the whole domain as well as the infimal energy among all $k-1$-partitions. Strictly below such threshold, we develop a concentration-compactness-type argument showing optimal partitions exist, and each cell admits ground states (i.e., the infimum of the spectrum on each cell is a simple isolated eigenvalue). Second, for $p