2026/05/31 by Jean-Pierre Magnot
Mathematics · Physics and Astronomy · #math.SP #math-ph #math.FA #math.MP #msc:47D06 #msc:35Q41 #msc:81S40 #msc:81Q05 #msc:81Q10 #msc:35S30 #msc:47G30 #msc:58J40
arxiv created 2026/08/01 · arxiv updated 2026/08/04
We develop a spectral cut-off and time-slicing construction for non-autonomous Hamiltonian evolution equations. Let \(H0\) be a positive self-adjoint reference operator with compact resolvent on a Hilbert space \(\Hilb\), and let PN=1[0,N](H0). For a time-dependent family of generally unbounded symmetric Hamiltonians (H(t)), we consider the finite-dimensional cut-off Hamiltonians HN(t)=PNH(t)PN. 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 \(H0\)-regularity yields the quantitative estimate sups,t∈ I |UN(t,s)PNu-U(t,s)u| ≤ 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 |UN,M(t,s)PNu-U(t,s)u| ≤ C((1+N)-σ+(1+N)μM-α) |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.