We obtain an exact analytic expression for the dynamical structure factor of one-dimensional quantum gas of hard rods. Our result is valid for an arbitrary many-body state of the system, with finite temperature states and the ground state being important special cases that we analyze in detail. We demonstrate that the expression obeys fundamental relations such like the $f$-sum rule and the detailed balance. We also reveal the hidden fermionic structure behind the correlator. In the static limit we show that it can be written in terms of universal functions, which, at zero temperature, coincide with the level spacing distribution function of the Gaussian unitary ensemble. Our Letter provides a full and exact characterization of a dynamic correlation function in a strongly correlated interacting quantum many-body system.

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• Quantum statistical mechanics
• 1-dimensional systems
• Strongly correlated systems
• Ultracold gases
• Exact solutions for many-body systems
• Integrable systems
• Random matrix theory