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arXiv:1804.06747

Zero range (contact) interactions conspire to produce Efimov trimers and quadrimers

Gianfausto Dell’Antonio

Published Apr 17, 2018• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1804.06747

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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 H0 H_0H0​ restricted to functions with support away from the \emph{contact manifold} Γ≡∪Γi,j      Γi,j≡{xi=xj  i≠j}  ,  xi∈R3Γ\equiv \cup Γ_{i,j} \;\;\; Γ_{i,j}\equiv \{x_i = x_j \; i \not= j \} \;,\; x_i \in R^3Γ≡∪Γi,j​Γi,j​≡{xi​=xj​i=j},xi​∈R3. These extensions are defined by boundary conditions at Γ ΓΓ. 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→0 ε\to 0ε→0 of N-body hamiltonians Hε=H0+∑i,j1ε3Vi,j(∣xi−xj∣ε),  Vi,j∈L1H_ε= H_0+ \sum_{i,j} \frac {1}{ε^3}V_{i,j}( \frac {|x_i-x_j|}ε) ,\; V_{i,j} \in L^1Hε​=H0​+∑i,j​ε31​Vi,j​(ε∣xi​−xj​∣​),Vi,j​∈L1. 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.
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.
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.
In Classical Mechanics constraints describe forces restricting the motion of two systems when they are in contact. In Quantum Mechanics in the Schrodinger representation the state of the system is described by a (probability) wave and the Schrodinger equation is dispersive (it does not preserve locality of the wave function): at any time the wave function is spread out over all space. It is difficult to define "contact". To avoid this difficulty it is convenient to use the Heisenberg representation and describe the system by means of self-adjoint operators on some function space. Each self-adjoint operator operator has a domain of definition. We consider first in some detail the dynamics in R^3 and later consider the case of dimension two and dimension one. In three dimension contact interactions gives the Efimov spectrum of trimers and quadrimers in low energy physics and the Bose-Einstein condensate, both in low density and in high density. The case of dimension one is particularly interesting because the system may represent three particles on a Y-shaped graph with contact interaction at the vertex. We consider both the case of Bose particles which satisfy the Schrodinger equation and the case of spin 1\2 fermions which satisfy the Pauli equation. They form respectively Bose crystals and Fermi crystals (not necessary periodic). In both cases there is only one bound state at each vertex. In an extended crystal, due to Fermi-Dirac statistics, all bound states may be occupied to form the Fermi sea. The particles on the surface have a Dirac spectrum. We prove that in the semiclassical limit the dynamics of particles on the surface of the Fermi sea is the (classical) motion studied in detail by Novikov and Maltsev. In this semiclassical description the Fermi surface can be deformed by a magnetic field and one may have topological resonances.
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 re-investigate the entire family of many center point interaction Hamiltonians. Under the assumption of exchange symmetry with respect to the point positions, we show that a large sub-family of point interaction Hamiltonian operators does not become either singular or trivial when the positions of two or more scattering centers tend to coincide. In this sense, they appear to be renormalized by default as opposed to the point interaction Hamiltonians usually considered in the literature. Functions in their domains satisfy regularized boundary conditions which turn out to be very similar to the ones proposed recently in many-body quantum mechanics to define three-particle system Hamiltonians with contact interactions bounded from below. In the two-center case, we study the behavior of the negative eigenvalues as a function of the center distance. The result is used to analyze a formal Born–Oppenheimer approximation of a three-particle system with two heavy bosons and one light particle. We demonstrate that this simplified model describes a stable system (no ‘fall to the center’ problem is present). Furthermore, in the unitary limit, the energy spectrum is characterized by an infinite sequence of negative energy eigenvalues accumulating at zero according to the geometrical Efimov law.
We study the Born–Oppenheimer approximation of a mass-imbalanced three-body system made of two heavy particles of mass M and one light particle of mass m for arbitrary angular momentum. In this system, heavy-light pairs interact via a zero-range force. We construct the light-particle Hamiltonian using self-adjoint extensions of the two-center point interaction and show that the corresponding effective potential is regular at the coincidence point of the heavy particles. Consequently, this model presents an alternative method to finite-range, cutoff, or short-distance heavy-heavy regularizations: the necessary three-body input is encoded in the self-adjoint realization of the light-particle Hamiltonian, while the heavy-light interactions remain point-like. In the unitary limit, after fixing the characteristic length scale, we derive an explicit Efimov spectrum. Our results recover the zero-angular-momentum case of [] and provide a sufficient condition ensuring the absence of non-Efimov bound states. Away from unitarity, we show that the spatial size of the shallowest trimer near the threshold is approximately 2.8 times the heavy-light scattering length, in contrast with the common assumption that these two length scales coincide. We also derive a Bargmann-type bound on the number of three-body bound states and obtain an estimate sharper than previous results. Finally, we illustrate the method with numerical results for selected alkali mixtures.