We study the $3.4-4.4$ fundamental rovibrational band of H$_3^+$, a key tracer of the ionization of the molecular interstellar medium (ISM), in a sample of 12 local ($d$$<$400\,Mpc) (ultra)luminous infrared galaxies ((U)LIRGs) observed with JWST NIRSpec. The P, Q, and R branches of the band are detected in 13 out of 20 analyzed regions within these (U)LIRGs, which increases the number of extragalactic H$_3^+$ detections by a factor of 6. For the first time in the ISM, the H$_3^+$ band is observed in emission; we detect this emission in three regions. In the remaining ten regions, the band is seen in absorption. The absorptions are produced toward the $3.4-4.4$ hot dust continuum rather than toward the stellar continuum, indicating that they likely originate in clouds associated with the dust continuum source. The H$_3^+$ band is undetected in Seyfert-like (U)LIRGs where the mildly obscured X-ray radiation from the active galactic nuclei might limit the abundance of this molecule. For the detections, the H$_3^+$ abundances, $N($H$_3^+$) H (0.5--5.5)times 10$^ imply relatively high ionization rates, $ H_2 $, of between 3times 10$^ $ and $>$4times 10$^ $, which is lower than the molecular outflow velocities measured using other tracers such as OH\,119 or rotational CO lines. This suggests that H$_3^+$ traces gas close to the outflow-launching sites before it has been fully accelerated. We used nonlocal thermodynamic equilibrium models to investigate the physical conditions of these clouds. In seven out of ten objects, the H$_3^+$ excitation is consistent with inelastic collisions with H$_2$ in warm translucent molecular clouds kin $sim 250--500\,K and H_2 )$sim 10$^ $). In three objects, dominant infrared pumping excitation is required to explain the absorptions from the (3,0) and (2,1) levels of H$_3^+$ detected for the first time in the ISM.
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