Adiabatic theorems for linear and nonlinear Hamiltonians

Conditions for the validity of the quantum adiabatic approximation are analyzed. For the case of linear Hamiltonians, a simple and general sufficient condition is derived, which is valid for arbitrary spectra and any kind of time variation. It is shown that in some cases the found condition is necessary and sufficient. The adiabatic theorem is generalized for the case of nonlinear Hamiltonians.

💡 Research Summary

The paper revisits the quantum adiabatic approximation and establishes a unified framework that works for both linear and nonlinear Hamiltonians. For linear systems the authors start from the standard adiabatic theorem, which assumes a non‑degenerate, slowly varying spectrum, and point out that the usual quantitative condition – often expressed as (|\langle m(t)|\dot n(t)\rangle| \ll \Delta_{mn}(t)) – is too restrictive for many realistic scenarios (continuous spectra, abrupt parameter changes, etc.). By analysing the exact Schrödinger dynamics they derive a simple, fully general sufficient condition:

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