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We study a local-to-global inequality for spectral invariants of Hamiltonianswhose supports have a ``large enough" tubular neighborhood on semipositivesymplectic manifolds. In particular, we present the first examples of such aninequality when the Hamiltonians are not necessarily supported in domains withcontact type boundaries, or when the ambient manifold is irrational. Thisextends a series of previous works studying locality phenomena of spectralinvariants. A main new tool is a lower bound, in the spirit of Sikorav, for theenergy of Floer trajectories that cross the tubular neighborhood against thedirection of the negative-gradient vector field.

A local-to-global inequality for spectral invariants and an energy dichotomy for Floer trajectories