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Lensed gravitational waves will offer new means to probe the distribution ofmatter in the universe, complementary to electromagnetic signals. Lensedcontinuous gravitational waves provide new challenges and opportunities beyondthose of transient compact binary coalescence. Here we consider continuousgravitational waves emitted by isolated neutron stars and lensed by Sgr A$^*$,the supermassive black hole at the center of our galaxy, a system observable bythe next generation of gravitational wave detectors. We analyze the signaturesof this system in detail, addressing parameter estimation and model selection.Future detectors can distinguish lensed continuous waves and measure theirparameters with precision $\sim 1 - 10\%$ for sources within $2-4$ Einsteinradii of Sgr A$^*$, depending on the source distance, thanks to the relativemotion of the observer-lens-source system. The chances of observingstrongly-lensed neutron stars increase by one order of magnitude relative toprevious estimates, thanks to the possibility of detecting lensed systems atseveral Einstein radii. Multiple images can be resolved with an angularaccuracy $\sim 10$mas, comparable to the best optical telescopes. Imagelocalization probes deviations from axial symmetry and the existence ofcompanions to Sgr A$^*$ in regions complementary to stellar orbits and blackhole imaging. Our methods and many of our results extend to other lenses (e.g.galactic substructure) and sources (e.g. long-lived inspiralling binaries),rendering lensed continuous gravitational waves into versatile probes ofastrophysics and fundamental physics.

Through the lens of Sgr A$^*$: identifying and resolving strongly lensed Continuous Gravitational Waves beyond the Einstein radius