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arXiv · 2605.26899

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

Jean-Pierre Magnot
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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 $$ \sup{s,t\in I} |U_N(t,s)P_Nu-U(t,s)u| \leq C(1+N)^{-σ}|u|{μ+σ}. $$ For Hamiltonians that are H"older continuous of exponent \(α\) in time, we also prove the joint spectral and time-slicing bound $$ |U{N,M}(t,s)P_Nu-U(t,s)u| \leq C\left((1+N)^{-σ}+(1+N)^μM^{-α}\right) |u|_{μ+σ}. $$ 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.

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Jean-Pierre Magnot. 2026-08-01. Spectral Cut-off Oscillatory Integrals for Non-Autonomous Hamiltonian Evolution Equations. https://arxiv.org/abs/2605.26899

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