Contact interactions, self-adjoint extensions, and low-energy scattering

• URL: http://arxiv.org/abs/2403.15290v1
• Date: Fri, 22 Mar 2024 15:39:56 GMT
• Title: Contact interactions, self-adjoint extensions, and low-energy scattering
• Authors: Daniel R. DeSena, Brian C. Tiburzi,
• Abstract summary: We show that 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. As an application, the spectrum of a general point interaction with a harmonic trap is solved in one dimension.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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