Supermassive black hole (SMBH) coalescences are ubiquitous in the history of the Universe and often exhibit strong accretion activities and powerful jets. These SMBH mergers are also promising candidates for future gravitational wave detectors such as Laser Space Inteferometric Antenna (LISA). In this work, we consider neutrino counterpart emission originating from the jet-induced shocks. The physical picture is that relativistic jets launched after the merger will push forward inside the premerger disk wind material, and then they subsequently get collimated, leading to the formation of internal shocks, collimation shocks, forward shocks and reverse shocks. Cosmic rays can be accelerated in these sites and neutrinos are expected via the photomeson production process. We formulate the jet structures and relevant interactions therein, and then evaluate neutrino emission from each shock site. We find that month-to-year high-energy neutrino emission from the postmerger jet after the gravitational wave event is detectable by IceCube-Gen2 within approximately five to ten years of operation in optimistic cases where the cosmic-ray loading is sufficiently high and a mildly super-Eddington accretion is achieved. We also estimate the contribution of SMBH mergers to the diffuse neutrino intensity, and find that a significant fraction of the observed very high-energy ($E_ ugtrsim1$ PeV) IceCube neutrinos could originate from them in the optimistic cases. In the future, such neutrino counterparts together with gravitational wave observations can be used in a multimessenger approach to elucidate in greater detail the evolution and the physical mechanism of SMBH mergers.
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