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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.26899 |
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| _version_ | 1866914604627525632 |
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| author | Magnot, Jean-Pierre |
| author_facet | Magnot, Jean-Pierre |
| contents | 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 \(\Hilb\), 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 \[
H_N(t)=P_NH(t)P_N . \] 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 \[
\ii \partial_t u(t)=H(t)u(t). \] 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26899 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral Cut-off Oscillatory Integrals for Non-Autonomous Hamiltonian Evolution Equations Magnot, Jean-Pierre Spectral Theory Mathematical Physics Functional Analysis 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 \(\Hilb\), 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 \[ H_N(t)=P_NH(t)P_N . \] 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 \[ \ii \partial_t u(t)=H(t)u(t). \] 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. |
| title | Spectral Cut-off Oscillatory Integrals for Non-Autonomous Hamiltonian Evolution Equations |
| topic | Spectral Theory Mathematical Physics Functional Analysis |
| url | https://arxiv.org/abs/2605.26899 |