Persson-type spectral minimal partition principle for domains

Let $\Omega \subset \mathbb{R}^d$, $d \geq 2$, be a domain, possibly equal to $\mathbb{R}^d$, and let $V:\Omega \to \mathbb{R}$ be a sufficiently smooth potential bounded from below. Let $\Sigma(\Omega)$ denote the infimum of the essential spectrum of the Schrödinger operator $-\Delta+V$, with Dirichlet conditions on $\partial\Omega$ when $\Omega \neq \mathbb{R}^d$. For a $k$-partition $(\Omega_i)_{i=1}^k$ of $\Omega$ into open, connected, pairwise disjoint sets, let $\lambda(\Omega_i)$ be the infimum of the spectrum of $-\Delta+V$ on $\Omega_i$ with Dirichlet boundary conditions, and define

$$\mathcal L_k^D(\Omega)=\inf\max_{i=1,\ldots,k}\lambda(\Omega_i),$$

where the infimum is over all such $k$-partitions. Persson-type spectral minimal partition principle. One has $\mathcal L_k^D(\Omega)\leq\Sigma(\Omega)$ for every $k\geq1$; if $\mathcal L_k^D(\Omega)<\Sigma(\Omega)$ for some $k\geq1$, then a $k$-partition realizes $\mathcal L_k^D(\Omega)$; in particular, if some $k$-partition has spectral energy at most $\Sigma(\Omega)$, then a $k$-partition realizes $\mathcal L_k^D(\Omega)$. These assertions are proposed for domains because Persson-type characterizations are known in the Euclidean setting, while the corresponding domain problem had not been studied and is left untreated here.