We study the Hamiltonian for a system of N identical bosons interacting with an impu- rity, 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 prob- lem, 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.
Zero-Range Hamiltonian for a Bose Gas with an Impurity / Ferretti, D.; Teta, A.. - In: COMPLEX ANALYSIS AND OPERATOR THEORY. - ISSN 1661-8254. - 17:(2023). [10.1007/s11785-023-01358-4]
Zero-Range Hamiltonian for a Bose Gas with an Impurity
Teta A.
2023
Abstract
We study the Hamiltonian for a system of N identical bosons interacting with an impu- rity, 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 prob- lem, 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.File | Dimensione | Formato | |
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