Spectral Cut-off Oscillatory Integrals for Non-Autonomous Hamiltonian Evolution Equations

We develop a spectral cut-off construction of real-time oscillatory integrals associated with non-autonomous Hamiltonian evolution equations. Let $H_0$ be a positive self-adjoint reference operator on a Hilbert space $\text{Hilb}$, and let $P_N={\mathbf{1}}_{[0,N]}(H_0)$ be its spectral projections. For a time-dependent family of generally unbounded Hamiltonians $H(t)$, we consider the finite-dimensional cut-off Hamiltonians $$H_N(t)=P_{N}H(t)P_N\mathrm{.}$$ The corresponding propagators are represented by time-sliced finite dimensional oscillatory integrals. Under suitable $H_0$-relative regularity and stability assumptions, we prove convergence of these cut-off oscillatory amplitudes to the strong solution of the original Hamiltonian evolution equation $$\text{ii}{\mathrm{\partial }}_{t}u(t)=H(t)u(t)\mathrm{.}$$ In the periodic case, the same construction yields finite-dimensional effective Hamiltonians and provides a natural bridge with the Floquet–Magnus expansion for unbounded operators. We also discuss how spectral cut-offs may later be used to define renormalized traces of real-time amplitudes.