BlackTHUNDER strikes twice: rest-frame Balmer-line absorption and high Eddington accretion rate in a Little Red Dot at $z=7.04$

JWST has revealed a population of 'Little Red Dots' (LRDs): compact, red objects at redshifts z=2-9 with 'v'-shaped spectral energy distributions, broad permitted lines, and, often, hydrogen Balmer absorption. We use NIRSpec/IFS data from the BlackTHUNDER survey to study the H$\alpha$ line in the LRD Abell2744-QSO1 at z=7.04, which is a confirmed AGN due to time-variable equivalent width (EW) in its broad emission lines. The H$\alpha$ spectral profile is non-Gaussian, requiring at least two Gaussian components. We also detect a narrow-line Gaussian component, and strong H$\alpha$ absorption (EW relative to the continuum $\sim 22_{-7}^{+12} \mathring{\mathrm{A}}$), confirming a connection between the strong Balmer break and line absorption. The absorber is at rest with respect to broad H$\alpha$, suggesting that the gas cannot be interpreted as an inflow or outflow, forming instead a long-lived structure. Its velocity dispersion is $\sigma_{\rm abs}=110_{-10}^{+20}$ km s$^{-1}$, consistent with the value inferred from the analysis of the Balmer break. Based on H$\alpha$, we infer a black hole mass of log($M_{\rm BH}/{\rm M_\odot}$)=7.2, smaller but close to the previous estimates based on H$\beta$. The Eddington ratio is 0.09. Combining the high signal-to-noise ratio of the narrow H$\alpha$ line with the spectral resolution R=3,700 of the G395H grating, we infer a narrow-line intrinsic dispersion $\sigma_{\rm n}=22_{-6}^{+5}$ km s$^{-1}$, which places a stringent constraint on the black-hole-to-dynamical-mass ratio of this system to be $M_{\rm BH}/M_{\rm dyn}$=0.15-1.2, confirming the overmassive nature of the black hole and potentially leaving little room for a host galaxy.