Contact interactions, self-adjoint extensions, and low-energy scattering
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909146953023488 |
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| author | DeSena, Daniel R. Tiburzi, Brian C. |
| author_facet | DeSena, Daniel R. Tiburzi, Brian C. |
| contents | Low-energy scattering is well described by the effective-range expansion. In quantum mechanics, a tower of contact interactions can generate terms in this expansion after renormalization. Scattering parameters are also encoded in the self-adjoint extension of the Hamiltonian. We briefly review this well-known result for two particles with s-wave interactions using impenetrable self-adjoint extensions, including the case of harmonically trapped two-particle states. By contrast, the one-dimensional scattering problem is surprisingly intricate. We show that the families of self-adjoint extensions correspond to a coupled system of symmetric and antisymmetric outgoing waves, which is diagonalized by an SU(2) transformation that accounts for mixing and a relative phase. This is corroborated by an effective theory computation that includes all four energy-independent contact interactions. The equivalence of various one-dimensional contact interactions is discussed and scrutinized from the perspective of renormalization. As an application, the spectrum of a general point interaction with a harmonic trap is solved in one dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15290 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Contact interactions, self-adjoint extensions, and low-energy scattering DeSena, Daniel R. Tiburzi, Brian C. Quantum Physics Quantum Gases Nuclear Theory Low-energy scattering is well described by the effective-range expansion. In quantum mechanics, a tower of contact interactions can generate terms in this expansion after renormalization. Scattering parameters are also encoded in the self-adjoint extension of the Hamiltonian. We briefly review this well-known result for two particles with s-wave interactions using impenetrable self-adjoint extensions, including the case of harmonically trapped two-particle states. By contrast, the one-dimensional scattering problem is surprisingly intricate. We show that the families of self-adjoint extensions correspond to a coupled system of symmetric and antisymmetric outgoing waves, which is diagonalized by an SU(2) transformation that accounts for mixing and a relative phase. This is corroborated by an effective theory computation that includes all four energy-independent contact interactions. The equivalence of various one-dimensional contact interactions is discussed and scrutinized from the perspective of renormalization. As an application, the spectrum of a general point interaction with a harmonic trap is solved in one dimension. |
| title | Contact interactions, self-adjoint extensions, and low-energy scattering |
| topic | Quantum Physics Quantum Gases Nuclear Theory |
| url | https://arxiv.org/abs/2403.15290 |