The persistence of a relative Rabinowitz-Floer complex

We give a quantitative refinement of the invariance of the Legendrian contact homology algebra in general contact manifolds. We show that in this general case, the Lagrangian cobordism trace of a Legendrian isotopy defines a DGA stable tame isomorphism which is similar to a bifurcation invariance-proof for a contactization contact manifold. We use this result to construct a relative version of the Rabinowitz-Floer complex defined for Legendrians that also satisfies a quantitative invariance, and study its persistent homology barcodes. We apply these barcodes to prove several results, including: displacement energy bounds for Legendrian submanifolds in terms of the oscillatory norms of the contact Hamiltonians; a proof of Rosen and Zhang's non-degeneracy conjecture for the Shelukhin--Chekanov--Hofer metric on Legendrian submanifolds; and, the non-displaceability of the standard Legendrian real-projective space inside the contact real-projective space.