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 $$P_N=1_{[0,N]}(H_0).$$ For a time-dependent family of generally unbounded symmetric Hamiltonians (H(t)), we consider the finite-dimensional cut-off Hamiltonians $$H_N(t)=P_{N}H(t)P_N.$$ 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 $$\sup s,t\in I|U_N(t,s)P_{N}u-U(t,s)u|\le C(1+N{)}^{-\ddot{I}ƒ}|u|\hat{I}¼+\ddot{I}ƒ.$$ For Hamiltonians that are H"older continuous of exponent \(α\) in time, we also prove the joint spectral and time-slicing bound $$|UN,M(t,s)P_{N}u-U(t,s)u|\le C\left((1+N{)}^{-\ddot{I}ƒ}+(1+N{)}^{\hat{I}}¼M^{-\hat{I}\pm }\right)|u{|}_{\hat{I}¼+\ddot{I}ƒ}.$$ 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.