Regularized Zero-Range Hamiltonian for a Bose Gas with an Impurity

• URL: http://arxiv.org/abs/2202.12765v1
• Date: Fri, 25 Feb 2022 15:34:06 GMT
• Title: Regularized Zero-Range Hamiltonian for a Bose Gas with an Impurity
• Authors: Daniele Ferretti and Alessandro Teta
• Abstract summary: We study the Hamiltonian for a system of N identical bosons interacting with an impurity. We introduce a three-body force acting at short distances. The effect of this force is to reduce to zero the strength of the zero-range interaction between two particles.
• Score: 77.34726150561087
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the Hamiltonian for a system of N identical bosons interacting with an impurity, i.e., a different particle, via zero-range forces in dimension three. It is well known that, following the standard approach, one obtains the Ter-Martirosyan Skornyakov Hamiltonian which is unbounded from below. In order to avoid such instability problem, we introduce a three-body force acting at short distances. The effect of this force is to reduce to zero the strength of the zero-range interaction between two particles, i.e., the impurity and a boson, when another boson approaches the common position of the first two particles. We show that the Hamiltonian defined with such regularized interaction is self-adjoint and bounded from below if the strength of the three-body force is sufficiently large. The method of the proof is based on a careful analysis of the corresponding quadratic form.

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