Efimov effect for a three-particle system with two identical fermions

We consider a three-particle quantum system in dimension three composed of two identical fermions of mass one and a different particle of mass $m$. The particles interact via two-body short range potentials. We assume that the Hamiltonians of all the two-particle subsystems do not have bound states with negative energy and, moreover, that the Hamiltonians of the two subsystems made of a fermion and the different particle have a zero-energy resonance. Under these conditions and for $m<m^* = (13.607)^{-1}$, we give a rigorous proof of the occurrence of the Efimov effect, i.e., the existence of infinitely many negative eigenvalues for the three-particle Hamiltonian $H$. More precisely, we prove that for $m>m^*$ the number of negative eigenvalues of $H$ is finite and for $m<m^*$ the number $N(z)$ of negative eigenvalues of $H$ below $z<0$ has the asymptotic behavior $N(z) \sim \mathcal C(m) |\log|z||$ for $z \rightarrow 0^-$. Moreover, we give an upper and a lower bound for the positive constant $\mathcal C(m)$.Comment: 26 page

On the quantum mechanical three-body problem with zero-range interactions

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

A Second Order Upper Bound for the Ground State Energy of a Hard-Sphere Gas in the Gross–Pitaevskii Regime

We prove an upper bound for the ground state energy of a Bose gas consisting of N hard spheres with radius a/N, moving in the three-dimensional unit torus Λ. Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit N→∞. The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose–Einstein condensate and describing correlations on large scales

A new second-order upper bound for the ground state energy of dilute Bose gases

We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee–Huang–Yang formula. This result was first established in [20] by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported $V \in L^3 (\mathbb {R}^3)$, gives better rates and, in our opinion, is substantially simpler

Low energy behavior in few-particle quantum systems: Efimov effect and zero-range interactions.

We investigate the emergence of a universal behavior in certain few-particle quantum sys- tems at low-energy. First we consider a system composed by two identical fermions of mass one and a dif- ferent particle of mass m in dimension three. Under the assumption that the two-particle Hamiltonians composed by one of the fermions and the third particle have a resonance at zero-energy, and for m less than a mass threshold m∗, we prove the occurrence of the Efimov effect, i. e., the existence of an infinite number of three-body bound states accumulating at zero. Then we study three-particle systems with zero-range interactions. In dimension one we give a rigorous definition of the Hamiltonian for three identical bosons and we prove that it is the limit of suitably rescaled regular Hamiltonians. In dimension three we write the expression of the quadratic form associated to the STM extension for a generic three-particle system. Then we focus on a system of three identical bosons proving stability outside the s-wave subspace. As a third example of universal behavior in few-particle system we con- sider a quantum Lorentz gas in dimension three: a particle moving through N obstacles whose positions are independently chosen according to a given common probability density. We assume that the particle interact with each obstacle via a Gross Pitaevskii potential. We prove the convergence, as N → ∞, to a Hamiltonian depending on the common distribution density of the obstacles and such that the only dependence on the interaction potential is through its scattering length

Universal low-energy behavior in a quantum Lorentz gas with Gross-Pitaevskii potentials

We consider a quantum particle interacting with $N$ obstacles, whose positions are independently chosen according to a given probability density, through a two-body potential of the form $N^2 V(Nx)$ (Gross-Pitaevskii potential). We show convergence of the $N$ dependent one-particle Hamiltonian to a limiting Hamiltonian where the quantum particle experiences an effective potential depending only on the scattering length of the unscaled potential and the density of the obstacles. In this sense our Lorentz gas model exhibits a universal behavior for $N$ large. Moreover we explicitely characterize the fluctuations around the limit operator. Our model can be considered as a simplified model for scattering of slow neutrons from condensed matter

The three-body problem in dimension one: From short-range to contact interactions

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 Hamiltonian converges to one with zero-range (also called delta or point) interactions. The convergence is understood in norm resolvent sense. The two-body rescaled potentials are of the form $v^{\varepsilon}_{\sigma}(x_{\sigma})= \varepsilon^{-1} v_{\sigma}(\varepsilon^{-1}x_\sigma )$, where $\sigma = 23, 12, 31$ is an index that runs over all the possible pairings of the three particles, $x_{\sigma}$ is the relative coordinate between two particles, and $\varepsilon$ is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials $v_\sigma$ with $\alpha_\sigma \delta_\sigma$, where $\delta_\sigma$ is the Dirac delta-distribution centered on the coincidence hyperplane $x_\sigma=0$ and $\alpha_\sigma = \int_{\mathbb{R}} v_\sigma dx_\sigma$. To prove the convergence of the resolvents we make use of Faddeev's equations.Comment: 21 page

Ground state energy of a Bose gas in the Gross–Pitaevskii regime

We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere

Ground state energy of a Bose gas in the Gross-Pitaevskii regime

We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson's upper bound, which provides the correct leading order asymptotics for hard spheres. Afterwards, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross-Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross-Pitaeavskii limit. Here we only sketch the main ideas, details will appear elsewhere.Comment: Postprint version. To appear in JM