Ground-state domination conjecture for the Friedrichs spectrum

Let $P$ be a symmetric critical operator in $\boldsymbol{\Omega}$, and let $\phi_0$ be a ground state of $P$. For $\lambda>0$, let $\phi_\lambda$ solve

$$(P-\lambda)u=0$$

in $\Omega$ and suppose that there is a constant $C>0$ such that

$$|\phi_\lambda(x)|\leq C\phi_0(x) \qquad \forall x\in\Omega.$$

Ground-state domination conjecture. Then $\lambda$ belongs to the spectrum of the Friedrichs extension of $P$ on $L^2(\Omega,\mathrm{d}x)$. This conjecture links pointwise domination by a ground state to spectral inclusion for the Friedrichs extension; the supplied text gives no resolution status.