The gravitational lensing effect of gravitational waves (GWs) has been extensively discussed as a probe of small-mass dark matter halos, which can provide missing information about dark matter. We propose a multiband observation of lensed GWs from a compact binary to observe both geometrical optics (GO) and wave optics (WO) effects from the same source. This method is expected to be advantageous in breaking parameter degeneracies between a GW source and a dark matter halo acting as a lens. We assume DECi-hertz Interferometer Gravitational wave Observatory (DECIGO) or B-DECIGO as a space-based detector observing the early inspiral phase, and the Einstein Telescope (ET) as a ground-based detector observing the merger phase. We perform a Fisher analysis of multi-band detection for a source with masses $m_1=30M_{\odot },m_2=20M_{\odot }$ at redshift $z=1.5$, and a lens with mass $3\times {10}^3{M}_{\odot }$ at redshift $z=1.0$. With this setup, the GO effect appears in the ET frequency band, and the WO effect in that of DECIGO. For the halo density profile, we adopt singular isothermal sphere, cored isothermal sphere (CIS), and Navarro-Frenk-White models. We find that multiband observation resolves parameter degeneracies and significantly reduces errors in estimated parameters. For the CIS model, in particular, we show that, by combining ET and DECIGO observations, the lens mass error improves by about 72% and 37% compared to ET and DECIGO alone, respectively. Similarly, the impact parameter error is reduced by about 65% and 71%, and the core size error by 34% and 68%, respectively. From these results, we conclude that the multiband observation of GWs from compact binaries improves the estimation of the lens object properties by breaking the parameter degeneracy.

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• Dark matter
• Gravitational lenses
• Gravitational waves