Microscopic Derivation of Time-dependent Point Interactions
Abstract
We study the dynamics of the three-dimensional polaron - a quantum particle coupled to bosonic fields - in the quasi-classical regime. In this case the fields are very intense and the corresponding degrees of freedom can be treated semiclassically. We prove that in such a regime the effective dynamics for the quantum particles is approximated by the one generated by a time-dependent point interaction, i.e., a singular time-dependent perturbation of the Laplacian supported in a point. As a by-product, we also show that the unitary dynamics of a time-dependent point interaction can be approximated in strong operator topology by the one generated by time-dependent Schrödinger operators with suitably rescaled regular potentials.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2019
- DOI:
- 10.48550/arXiv.1904.11012
- arXiv:
- arXiv:1904.11012
- Bibcode:
- 2019arXiv190411012C
- Keywords:
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- Mathematical Physics
- E-Print:
- minor revision, to appear in SIAM J. Math. Anal., pdfLaTex, 28 pages