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arXiv:2403.12594 · Reviews in Mathematical Physics

Hamiltonian for a Bose gas with contact interactions

Daniele Ferretti, Alessandro Teta

Published Jun 10, 2025• 3 citations
arXivHTMLPDF

Abstract

In this paper, we study the Hamiltonian for a three-dimensional Bose gas of [Formula: see text] spinless particles interacting via zero-range (also known as contact) interactions. Such interactions are encoded by (singular) boundary conditions imposed on the coincidence hyperplanes, i.e. when the coordinates of two particles coincide. It is well known that imposing the same kind of boundary conditions as in the two-body problem with a point interaction leads to a Hamiltonian unbounded from below (and thus unstable). This is due to the fact that the interaction becomes overly strong and attractive when the coordinates of three or more particles coincide. In order to avoid such instability, we develop a suggestion originally formulated by Minlos and Faddeev in 1962, introducing slightly modified boundary conditions that weaken the strength of the interaction between two particles i and j in two scenarios: (a) a third particle approaches the common position of i and j; (b) another distinct pair of particles approach each other. In all other cases, the usual boundary condition is restored. Using a quadratic form approach, we construct a class of Hamiltonians characterized by such modified boundary conditions, that are self-adjoint and bounded from below. We also compare our approach with the one developed years ago by Albeverio, Høegh-Krohn and Streit using the theory of Dirichlet forms [J. Math. Phys. 18 (1977) 907–917]. In particular, we show that the N-body Hamiltonian defined by Albeverio et al. is a special case of our class of Hamiltonians. Furthermore, we also introduce a Dirichlet form by considering a more general weight function, and we prove that the corresponding N-body Hamiltonians essentially coincide with those constructed via our method.

DOI
10.1142/s0129055x25500205

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We introduce \emph{contact (zero range) interactions } , a special class of self-adjoint extensions of the N-body Schrödinger free hamiltonian $ H_0$ restricted to functions with support away from the \emph{contact manifold} $ Γ\equiv \cup Γ_{i,j} \;\;\; Γ_{i,j}\equiv \{x_i = x_j \; i \not= j \} \;,\; x_i \in R^3 $. These extensions are defined by boundary conditions at $ Γ$. We discuss the spectral properties as function of the masses and the statistics The (Efimov) spectrum is entirely due "conspiracy" of the contact interactions of two pairs. These states are called in Theoretical Physics \emph{trimers} if the two pairs have an element in common, \emph{quadrimers} otherwise. The analysis can be extended to the case in which there is a regular two-body potential, but then the spectral properties cannot be given explicitly. We prove that these interactions are limit (in strong resolvent sense) when $ ε\to 0$ of N-body hamiltonians $ H_ε= H_0+ \sum_{i,j} \frac {1}{ε^3}V_{i,j}( \frac {|x_i-x_j|}ε) ,\; V_{i,j} \in L^1 $. The result is \emph{ independent of the shape of the potential} Strong resolvent convergence implies convergence of spectra.This makes contact interaction a valuable tool in the study of the spectrum of a system of $N$ particles interacting through potentials of very short range.

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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.

EMS series of congress reports•Apr 19, 2017•8 citations
In this paper, we study the Hamiltonian for a three-dimensional Bose gas of [Formula: see text] spinless particles interacting via zero-range (also known as contact) interactions. Such interactions are encoded by (singular) boundary conditions imposed on the coincidence hyperplanes, i.e. when the coordinates of two particles coincide. It is well known that imposing the same kind of boundary conditions as in the two-body problem with a point interaction leads to a Hamiltonian unbounded from below (and thus unstable). This is due to the fact that the interaction becomes overly strong and attractive when the coordinates of three or more particles coincide. In order to avoid such instability, we develop a suggestion originally formulated by Minlos and Faddeev in 1962, introducing slightly modified boundary conditions that weaken the strength of the interaction between two particles i and j in two scenarios: (a) a third particle approaches the common position of i and j; (b) another distinct pair of particles approach each other. In all other cases, the usual boundary condition is restored. Using a quadratic form approach, we construct a class of Hamiltonians characterized by such modified boundary conditions, that are self-adjoint and bounded from below. We also compare our approach with the one developed years ago by Albeverio, Høegh-Krohn and Streit using the theory of Dirichlet forms [J. Math. Phys. 18 (1977) 907–917]. In particular, we show that the N-body Hamiltonian defined by Albeverio et al. is a special case of our class of Hamiltonians. Furthermore, we also introduce a Dirichlet form by considering a more general weight function, and we prove that the corresponding N-body Hamiltonians essentially coincide with those constructed via our method.
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.
We consider a Hamiltonian describing three quantum particles in dimension one interacting through two-body short-range potentials. We prove that, as a suitable scale parameter in the potential terms goes to zero, such a Hamiltonian converges to one with zero-range (also called delta or point) interactions. The convergence is understood in the norm resolvent sense. The two-body rescaled potentials are of the form vσε(xσ)=ε−1vσ(ε−1xσ), where σ = 23, 12, 31 is an index that runs over all the possible pairings of the three particles, xσ is the relative coordinate between two particles, and ε is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials vσ with ασδσ, where δσ is the Dirac delta-distribution centered on the coincidence hyperplane xσ = 0 and ασ=∫Rvσdxσ. To prove the convergence of the resolvents, we make use of Faddeev’s equations.
We introduce \emph{contact (zero range) interactions } , a special class of self-adjoint extensions of the N-body Schrödinger free hamiltonian $ H_0$ restricted to functions with support away from the \emph{contact manifold} $ Γ\equiv \cup Γ_{i,j} \;\;\; Γ_{i,j}\equiv \{x_i = x_j \; i \not= j \} \;,\; x_i \in R^3 $. These extensions are defined by boundary conditions at $ Γ$. We discuss the spectral properties as function of the masses and the statistics The (Efimov) spectrum is entirely due "conspiracy" of the contact interactions of two pairs. These states are called in Theoretical Physics \emph{trimers} if the two pairs have an element in common, \emph{quadrimers} otherwise. The analysis can be extended to the case in which there is a regular two-body potential, but then the spectral properties cannot be given explicitly. We prove that these interactions are limit (in strong resolvent sense) when $ ε\to 0$ of N-body hamiltonians $ H_ε= H_0+ \sum_{i,j} \frac {1}{ε^3}V_{i,j}( \frac {|x_i-x_j|}ε) ,\; V_{i,j} \in L^1 $. The result is \emph{ independent of the shape of the potential} Strong resolvent convergence implies convergence of spectra.This makes contact interaction a valuable tool in the study of the spectrum of a system of NNN particles interacting through potentials of very short range.
Introduction The one-dimensional Bose gas with point-like interaction of the particles (the quantum variant of the nonlinear Schrödinger equation) is one of the main and most important models which can be Solved by means of the Bethe Ansatz. This model has been thoroughly investigated (and. We shall start with the construction of eigenfunctions of the Hamiltonian in a finite volume. Quantities interesting from the physical point of view (in the thermodynamic limit at zero temperature) are considered; the thermodynamics at finite temperatures is investigated in detail as well. A number of essential ideas which will be applied to other models are introduced. The construction of eigenfunctions of the Hamiltonian is explained in section 1. Their explicit form and, in particular, the two-particle reducibility, are common features of the models solvable by means of the Bethe Ansatz method. Periodic boundary conditions are imposed on the wave function in section 2; the Bethe equations for the particle' momenta are introduced and analyzed. Taken in the logarithmic form, these equations realize the extremum condition of a certain functional, the corresponding action being called the Yang-Yang action. The transition to the thermodynamic limit is considered in section 3. In that same section the ground state of the gas is constructed. The density of the particle distribution in momentum space and the energy of the ground state are calculated. The method of the transition to the thermodynamic limit described in this section is rather general and may be applied to any model solvable by means of the Bethe Ansatz. In section 4, excitations above the ground state are constructed and their main characteristics (energy, momentum and scattering matrix) are determined by means of the dressing equations. The ground state of the model is the Dirac sea (also called the Fermi sphere).
The present paper is devoted to the study of the few–body quantum mechanical problem with singular finite rank cluster interactions. The corresponding Hamiltonians play an important role in mathematical physics, since few body Hamiltonians with more regular interactions present considerable difficulties which make impossible a detailed analytic study [ 13 ]. Also a numerical study of such Hamiltonians using Faddeev equations present very hard problems, since the interaction between the particles does not vanish at large distances. On the other hand few-body problems with singular cluster interaction are useful and intensively studied in statistical physics, since the corresponding Hamiltonians can be analyzed in detail even if the number of particles is very large. Models describing one dimensional particles are of particular interest, since the eigenfunctions of the many body Hamiltonians can often be calculated using Bethe Ansatz [ 14 ]. In fact the well-known Yang-Baxter equation was first written in connection with the study of system of several one-dimensional particles with pairwise delta interactions. Similar methods were used in atomic physics to study collisions of several particles [ 12 ] (in three dimensions). Few-body systems in applications are usually investigated by combining the classical description of the dynamics of heavy atoms with the quantum description of the dynamics of electrons and other light particles. The first attempt to construct a three body Hamiltonian describing three quantum particles in R 3 interacting through pairwise delta functional potentials is due to G.V. Skorniakov and K.A.Ter-Martirosian [ 29 ]. R.A.Minlos and L.D.Faddeev proved that the corresponding Hamiltonian is not bounded from below and therefore cannot be used in 278 the originally intended physical applications [ 24 , 25 ]. The operator was defined using the method of self–adjoint perturbations used by F.A.Berezin and L.D.Faddeev for investigating Schödinger operators with delta potential in R 3 [ 10 ]. Almost three decades later semibounded three-body operators in dimension three with generalized two-body interactions were constructed by extending the standard Hilbert space of square integrable functions in R 9 [ 17 , 18 , 19 , 27 , 30 ]. The most interesting model considered used the theory of generalized extensions suggested by B.Pavlov [ 7 , 26 ]. Different aspects of these models were analyzed recently [ 8 , 23 , 20 ]. A general approach to these operators is described in [ 7 , 22 ]. Let us mention that a realization of a many-body lower bounded Hamiltonian with point interactions for particles in R 3 has been obtained in the original Hilbert space by using the theory of Dirichlet forms [ 2 ]. The few-body systems of two dimensional particles with two-body interactions was studied in [ 11 ], where it was proven that the corresponding Hamiltonian is semibounded.
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.