Bounds on spectral norms and barcodes

We investigate the relations between algebraic structures, spectral invariants, and persistence modules, in context of monotone Lagrangian Floer homology with Hamiltonian term. Firstly, we use newly introduced method filtered continuation elements to prove that norm controls barcode perturbation submanifold, up shift, bottleneck distance. Moreover, show it satisfies Chekanov type low-energy intersection phenomena, non-degeneracy theorems. Secondly, introduce a new averaging for bounding from above, apply produce precise uniform bounds on certain closed symplectic manifolds. Finally, by using theory our are fact sharp some cases. Along way calculation quantum submanifolds, answer question Usher.