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Regularized Quadratic Forms for a Three Boson System with Zero-Range Interactions

Giulia Basti, Rodolfo Figari, Alessandro Teta

Published Jan 1, 2018• 4 citations
OpenAlex

Abstract

We present two possible strategies to obtain a lower bounded Hamiltonian for three bosons interacting through zero-range interactions. First, we investigate a family of zero-range Hamiltonians defined in a Hilbert space of tensorial wave functions. Then, we examine the regularizing effect of an ultraviolet cutoff on the boundary conditions satisfied by functions in the form domain of zero-range Hamiltonians of a three boson system.

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We present two possible strategies to obtain a lower bounded Hamiltonian for three bosons interacting through zero-range interactions. First, we investigate a family of zero-range Hamiltonians defined in a Hilbert space of tensorial wave functions. Then, we examine the regularizing effect of an ultraviolet cutoff on the boundary conditions satisfied by functions in the form domain of zero-range Hamiltonians of a three boson system.
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.
In this note we discuss the quantum mechanical three-body problem with pairwise zero-range interactions in dimension three. We review the state of the art concerning the construction of the corresponding Hamiltonian as a self-adjoint operator in the bosonic and in the fermionic case. Exploiting a quadratic form method, we also prove self-adjointness and boundedness from below in the case of three identical bosons when the Hilbert space is suitably restricted, i.e., excluding the "s-wave“ subspace.
In this note we discuss the quantum mechanical three-body problem with pairwise zero-range interactions in dimension three. We review the state of the art concerning the construction of the corresponding Hamiltonian as a self-adjoint operator in the bosonic and in the fermionic case. Exploiting a quadratic form method, we also prove self-adjointness and boundedness from below in the case of three identical bosons when the Hilbert space is suitably restricted, i.e., excluding the "s-wave" subspace.
Abstract We study the Hamiltonian for a system of three identical bosons in dimension three interacting via zero-range forces. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan–Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide. The analysis is based on the construction of a suitable quadratic form which is shown to be closed and bounded from below. Then, domain and action of the corresponding Hamiltonian are completely characterized and a regularity result for the elements of the domain is given. Furthermore, we show that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled non-local interactions, also called separable potentials, with a suitably renormalized coupling constant.
We present the mathematical construction of the physically relevant quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered (the unitarity regime). The subject has several precursors in the mathematical literature. We proceed through an operator-theoretic construction of the self-adjoint extensions of the minimal operator obtained by restricting the free Hamiltonian to wave-functions that vanish in the vicinity of the coincidence hyperplanes: all extensions thus model an interaction precisely supported at the spatial configurations where particles come on top of each other. Among them, we select the physically relevant ones, by implementing in the operator construction the presence of the specific short-scale structure suggested by formal physical arguments that are ubiquitous in the physical literature on zero-range methods. This is done by applying at different stages the self-adjoint extension schemes à la Kreĭn–Višik–Birman and à la von Neumann. We produce a class of canonical models for which we also analyze the structure of the negative bound states. Bosonicity and zero range combined together make such canonical models display the typical Thomas and Efimov spectra, i.e. sequence of energy eigenvalues accumulating to both minus infinity and zero. We also discuss a type of regularization that prevents such spectral instability while retaining an effective short-scale pattern. Besides the operator qualification, we also present the associated energy quadratic forms. We structured our analysis so as to clarify certain steps of the operator-theoretic construction that are notoriously subtle for the correct identification of a domain of self-adjointness.
To regularize the three-body problem, Minlos and Faddeev suggested a modification of zero-range model, which diminishes interaction at the triple-collision point. The analysis reveals that this regularization results in four alternatives depending on the regularization parameter σ. Explicitly, Efimov or Thomas effects remain for σ<σc\sigma < {{\sigma }_{c}}σ<σc​ , the additional boundary conditions of two types should be imposed at the triple-collision point for σc⩽σ<σe{{\sigma }_{c}} \leqslant \sigma < {{\sigma }_{e}}σc​⩽σ<σe​ and σe<σ<σr{{\sigma }_{e}} < \sigma < {{\sigma }_{r}}σe​<σ<σr​ , and the problem is regularized for σ⩾σr\sigma \geqslant {{\sigma }_{r}}σ⩾σr​ . Critical values σc<σe<σr{{\sigma }_{c}} < {{\sigma }_{e}} < {{\sigma }_{r}}σc​<σe​<σr​ separating different alternatives are determined both for a two-component three-body system and for three identical bosons.