= Quasi-adiabatic continuation
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Quasi-adiabatic continuation transports a separated low-energy subspace along a sufficiently regular path of local <Hamiltonian operators> whose <spectral gap> stays open. In finite <volume>, a suitable <unitary operator> $U(s)$ relates the spectral projectors by $P(s)=U(s)P(0)U(s)^\dagger$.
The generator can be built from a filtered time evolution of $\partial_sH$. <Lieb-Robinson bounds> then control how far a transformed local <observable> spreads, providing a quasi-local evolution rather than a strictly local one. Exact spectral-flow constructions can have tails decaying faster than every power and almost exponentially; an unconditional pure exponential claim is too strong. The method compares gapped phases and transports their low-energy structure. Its parameter evolution need not equal the physical real-time evolution generated by $H(s)$.<Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems by Sven Bachmann and others>[Proposition 2.4 and Theorems 4.5 and 4.8: spectral transport and locality estimates]{ref}
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