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

Abstract

We develop a spectral cut-off and time-slicing construction for non-autonomous Hamiltonian evolution equations. Let \(H_0\) be a positive self-adjoint reference operator with compact resolvent on a Hilbert space \(\Hilb\), and let $$ P_N=\mathbf{1}_{[0,N]}(H_0). $$ For a time-dependent family of generally unbounded symmetric Hamiltonians (H(t)), we consider the finite-dimensional cut-off Hamiltonians $$ H_N(t)=P_NH(t)P_N. $$ Their time-sliced propagators admit finite-dimensional state-sum representations and, when suitable configuration-space or phase-space kernels are available, oscillatory integral realizations. We establish commutator conditions implying uniform stability of the cut-off dynamics and construct the full unitary propagator as the strong limit of the finite-dimensional propagators. Additional \(H_0\)-regularity yields the quantitative estimate $$ \sup{s,t\in I} |U_N(t,s)P_Nu-U(t,s)u| \leq C(1+N)^{-\sigma}|u|{\mu+\sigma}. $$ For Hamiltonians that are H"older continuous of exponent \(\alpha\) in time, we also prove the joint spectral and time-slicing bound $$ |U{N,M}(t,s)P_Nu-U(t,s)u| \leq C\left((1+N)^{-\sigma}+(1+N)^\mu M^{-\alpha}\right) |u|_{\mu+\sigma}. $$ The assumptions are verified for time-dependent Schr"odinger operators and symmetric first-order pseudodifferential Hamiltonians. In the periodic case, the construction is compatible with finite-order Floquet--Magnus coefficients for unbounded operators.