Mode stability for charged scalar fields away from extremality

Establish mode stability for the charged scalar field equation (g_{M,Q}^{-1})^{\mu\nu} (^{A}D)_{\mu}(^{A}D)_{\nu} \phi = 0 on sub-extremal Reissner–Nordström spacetimes (|Q|<M), without any smallness assumption on the coupling |\mathfrak{q}Q|, by proving that no nontrivial separated mode solutions exist that satisfy the standard ingoing boundary condition at the event horizon and outgoing boundary condition at future null infinity for any real frequency, and in particular excluding exponentially growing modes.

Background

The main results of the paper, which establish precise late-time tails and instability phenomena for charged scalar fields on Reissner–Nordström backgrounds, are unconditional in the extremal and near-extremal regimes where mode stability is known. Away from extremality, the results are proved under an explicit assumption of mode stability on the real axis (a quantitative no-mode statement formalized as a Wronskian bound in Condition 2.7).

The validity of mode stability for the charged scalar field on sub-extremal Reissner–Nordström spacetimes is not established in general. It is known to hold in the extremal and near-extremal cases (proved in the companion work) and for sufficiently small |\mathfrak{q}Q| by continuity from the neutral scalar field. However, for large |\mathfrak{q}Q| in the sub-extremal regime, the presence or absence of real-frequency modes and exponentially growing modes remains unresolved.

Resolving this problem would remove a key conditional assumption and enable unconditional late-time asymptotics for the charged scalar field away from extremality. It is also important for understanding the full nonlinear Einstein–Maxwell–charged scalar field system under small perturbations of sub-extremal Reissner–Nordström initial data in the large-coupling regime.