  We apply the theory of nonlinear resonance capture to the problem of a black
hole binary (BHB) orbiting a supermassive black hole (SMBH) while embedded in
the accretion disk of an active galactic nucleus (AGN). If successful,
resonance capture can trigger dramatic growth in the BHB eccentricity, with
important consequences for the BHB merger timescale, as well as for the
gravitational wave (GW) signature of such an eccentric merger. This resonance
capture may occur when the orbital period around the SMBH (the "outer binary")
and the apsidal precession of the BHB (the "inner binary") are in a 1:1
commensurability. This effect is analogous to the phenomenon of lunar evection
resonance in the early Sun-Earth-Moon system, with the distinction that in the
present case, the BHB apsidal precession is due to general relativity, rather
than rotationally-induced distortion. In contrast to the case of lunar
evection, however, the inner binary undergoes orbital decay driven by GW
emission, rather than orbital expansion driven by tidal dissipation. This
distinction fundamentally alters the three-body dynamics, forbidding resonance
capture, and limiting eccentricity growth. However, if the BHB migrates through
of a gaseous AGN disk, the change in the outer binary can counterbalance the
suppressing effect of BHB decay, permitting evection resonance capture and the
production of eccentric BHB mergers. We compute the likelihood of resonance
capture assuming an agnostic distribution of parameters for the three bodies
involved and for the properties of the AGN disk. We find that intermediate-mass
ratio BHBs (involving an intermediate-mass black hole and a stellar-mass black
hole) are the most likely to be captured into evection resonance and thus
undergo an eccentric merger. We also compute the GW signature of these mergers,
showing that they can enter the LISA band while eccentric.
