N-body Hamiltonians Research Articles

Topologically nontrivial three-body contact interaction in one dimension

Abstract It is known that three-body contact interactions in one-dimensional (1D) n(≥3)-body problems of nonidentical particles can be topologically nontrivial: they are all classified by unitary irreducible representations of the pure twin group PTn. It was, however, unknown how such interactions are described in the Hamiltonian formalism. In this paper, we study topologically nontrivial three-body contact interactions from the viewpoint of the path integral. Focusing on spinless particles, we construct an n(n − 1)(n − 2)/3!-parameter family of n-body Hamiltonians that corresponds to one particular 1D unitary representation of PTn. These Hamiltonians are written in terms of background Abelian gauge fields that describe infinitely thin magnetic fluxes in the n-body configuration space.

Scattering amplitudes and N-body post-Minkowskian Hamiltonians in general relativity and beyond

We present a general framework for calculating post-Minskowskian, classical, conservative Hamiltonians for N non-spinning bodies in general relativity from relativistic scattering amplitudes. Novel features for N > 2 are described including the subtraction of tree-like iteration contributions and the calculation of non-trivial many-body Fourier transform integrals needed to construct position space potentials. A new approach to calculating these integrals as an expansion in the hierarchical limit is described based on the method of regions. As an explicit example, we present the mathcal{O} (G2) 3-body momentum space potential in general relativity as well as for charged bodies in Einstein-Maxwell. The result is shown to be in perfect agreement with previous post-Newtonian calculations in general relativity up to mathcal{O} (G2v4). Furthermore, in appropriate limits the result is shown to agree perfectly with relativistic probe scattering in multi-center extremal black hole backgrounds and with the scattering of slowly-moving extremal black holes in the moduli space approximation.

A regularity result for the bound states of N-body Schrödinger operators: blow-ups and Lie manifolds

We prove regularity estimates in weighted Sobolev spaces for the L^2-eigenfunctions of Schrödinger-type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual N-body Hamiltonians with Coulomb-type singular potentials are covered by our result: in that case, the weight is , where is the usual Euclidean distance to the union of the set of collision planes {mathcal {F}}. The proof is based on blow-ups of manifolds with corners and Lie manifolds. More precisely, we start with the radial compactification {overline{X}} of the underlying space X and we first blow up the spheres {mathbb {S}}_Y subset {mathbb {S}}_X at infinity of the collision planes Y in {mathcal {F}} to obtain the Georgescu–Vasy compactification. Then, we blow up the collision planes {mathcal {F}}. We carefully investigate how the Lie manifold structure and the associated data (metric, Sobolev spaces, differential operators) change with each blow-up. Our method applies also to higher-order differential operators, to certain classes of pseudodifferential operators, and to matrices of scalar operators.

On the structure of the field C⁎-algebra of a symplectic space and spectral analysis of the operators affiliated to it

We show that the C⁎-algebra generated by the field operators associated to a symplectic space Ξ is graded by the semilattice of all finite dimensional subspaces of Ξ. If Ξ is finite dimensional we give a simple intrinsic description of the components of the grading, we show that the self-adjoint operators affiliated to the algebra have a many channel structure similar to that of N-body Hamiltonians, in particular their essential spectrum is described by a kind of HVZ theorem, and we point out a large class of operators affiliated to the algebra.

A Comparison of the Georgescu and Vasy Spaces Associated to the N-Body Problems and Applications

We provide new insight into the analysis of N-body problems by studying a compactification $$M_N$$ of $${\mathbb {R}}^{3N}$$ that is compatible with the analytic properties of the N-body Hamiltonian $$H_N$$ . We show that our compactification coincides with a compactification introduced by Vasy using blow-ups in order to study the scattering theory of N-body Hamiltonians and with a compactification introduced by Georgescu using $$C^*$$ -algebras. In particular, the compactifications introduced by Georgescu and by Vasy coincide (up to a homeomorphism that is the identity on $${\mathbb {R}}^{3N}$$ ). Our result has applications to the spectral theory of N-body problems and to some related approximation properties. For instance, results about the essential spectrum, the resolvents, and the scattering matrices of $$H_N$$ (when they exist) may be related to the behavior near $$M_N{\setminus } {\mathbb {R}}^{3N}$$ (i.e., “at infinity”) of their distribution kernels, which can be efficiently studied using our methods. The compactification $$M_N$$ is compatible with the action of the permutation group $$S_N$$ , which allows to implement bosonic and fermionic (anti-)symmetry relations. We also indicate how our results lead to a regularity result for the eigenfunctions of $$H_N$$ .

A stable spin-structure found in a 3-body system with spin-3 cold atoms and its role in N-body condensates

We have found a stable spin-structure of N = 3 system in which three spin-3 atoms are trapped and coupled to total spin S = 3. We have proved that a pair of this structure is nearly an exact solution for N = 6 systems in a very broad district in the parameter-space. Comparing with the well-known singlet pairs, this pair is a more promising candidate to serve as a building block for large N systems with spin-3 atoms. This is because the spin-structure of the latter can be modified depending on the interactions to reduce the energy while the singlet pairs can not. In fact, we have proved that, for a specific set of strengths (a point in the parameter-space) the product state based on this pair is an exact solution of the N-body Hamiltonian. Thus, in the neighborhood of this point, the product state will appear as an approximate solution. However, how broad this neighborhood would be remains to be clarified.

Improving the accuracy of simulated chaotic N-body orbits using smoothness

ABSTRACT Symplectic integrators are a foundation to the study of dynamical N-body phenomena, at scales ranging from planetary to cosmological. These integrators preserve the Poincaré invariants of Hamiltonian dynamics. The N-body Hamiltonian has another, perhaps overlooked, symmetry: it is smooth, or, in other words, it has infinite differentiability class order (DCO) for particle separations greater than 0. Popular symplectic integrators, such as hybrid methods or block adaptive stepping methods do not come from smooth Hamiltonians and it is perhaps unclear whether they should. We investigate the importance of this symmetry by considering hybrid integrators, whose DCO can be tuned easily. Hybrid methods are smooth, except at a finite number of phase space points. We study chaotic planetary orbits in a test considered by Wisdom. We find that increasing smoothness, at negligible extra computational cost in particular tests, improves the Jacobi constant error of the orbits by about 5 orders of magnitude in long-term simulations. The results from this work suggest that smoothness of the N-body Hamiltonian is a property worth preserving in simulations.

Exhaustive families of representations of C⁎-algebras associated with N-body Hamiltonians with asymptotically homogeneous interactions

We continue the analysis of algebras introduced by Georgescu, Nistor, and their coauthors, in order to study N-body type Hamiltonians with interactions. More precisely, let Y⊂X be a linear subspace of a finite-dimensional Euclidean space X, and vY be a continuous function on X/Y that has uniform homogeneous radial limits at infinity. We consider, in this paper, Hamiltonians of the form H=−Δ+∑Y∈SvY, where the subspaces Y⊂X belong to some given family S of subspaces. Georgescu and Nistor have considered the case when S consists of all subspaces Y⊂X, and Nistor and coauthors considered the case when S is a finite semilattice and Georgescu generalized these results to any family. In this paper, we develop new techniques to prove their results on the spectral theory of the Hamiltonian to the case where S is any family of subspaces also, and extend those results to other operators affiliated to a larger algebra of pseudodifferential operators associated with the action of X introduced by Connes. In addition, we exhibit Fredholm conditions for such elliptic operators. We also note that the algebras we consider answer a question of Melrose and Singer.

PoMiN: A Post-Minkowskian N-body Solver

Abstract In this paper, we introduce , a lightweight N-body code based on the post-Minkowskian N-body Hamiltonian of Ledvinka et al., which includes general relativistic effects up to first order in Newton’s constant G, and all orders in the speed of light c. is written in and uses a fourth-order Runge–Kutta integration scheme. has also been written to handle an arbitrary number of particles (both massive and massless), with a computational complexity that scales as O(N 2). We describe the methods we used to simplify and organize the Hamiltonian, and the tests we performed (convergence, conservation, and analytical comparison tests) to validate the code.

The equally spaced energy levels of the quantum harmonic oscillator revisited: a back-to-front reconstruction of an n-body Hamiltonian

The ‘back-to-front’ derivation of the properties of the quantum harmonic oscillator (QHO), starting with its equally spaced energy levels, is re-examined. A new derivation that exploits the natural rotational symmetry of the QHO is proposed. The new approach allows the ‘back-to-front’ idea to be extended further by showing that it is possible to derive the Hamiltonian of a system of particles from the starting point that the population is represented by a natural number. This involves the symmetry properties of phasors and Schwinger's theory of angular momentum. The analysis is also extended to multi-mode bosonic systems and fermionic systems. It is suggested that these results offer an alternative way to formulate physics, based on discreteness.

The tenfold way redux: Fermionic systems with N-body interactions

We provide a systematic treatment of the tenfold way of classifying fermionic systems that naturally allows for the study of those with arbitrary N-body interactions. We identify four types of symmetries such systems can possess, which consist of one ordinary type (usual unitary symmetries), and three non-ordinary symmetries (such as time reversal, charge conjugation and sublattice). Focusing on systems that possess no non-trivial ordinary symmetries, we demonstrate that the non-ordinary symmetries are strongly constrained. This approach not only leads very naturally to the tenfold classes, but also obtains the canonical representations of these symmetries in each of the ten classes. We also provide a group cohomological perspective of our results in terms of projective representations. We then use the canonical representations of the symmetries to obtain the structure of Hamiltonians with arbitrary N-body interactions in each of the ten classes. We show that the space of N-body Hamiltonians has an affine subspace (of a vector space) structure in classes which have either or both charge conjugation and sublattice symmetries. Our results can help address many open questions including the topological classification of interacting fermionic systems.

On the essential spectrum of $N$-body Hamiltonians with asymptotically homogeneous interactions

We determine the essential spectrum of N-body Hamiltonians with 2-body (or, more generally, k-body) potentials that have radial limits at infinity. The classical N-body Hamiltonians appearing in the well known HVZ-theorem are a particular case of this type of potentials corresponding to zero limits at infinity. Our result thus extends the HVZ-theorem that describes the essential spectrum of the usual N-body Hamiltonians. More precisely, if the configuration space of the system is a finite dimensional real vector space X, then let e(X) be the C *-algebra of functions on X generated by the algebras C(X/Y), where Y runs over the set of all linear subspaces of X and C(X/Y) is the space of continuous functions on X/Y that have radial limits at infinity. This is the algebra used to define the potentials in our case, while in the classical case the C(X/Y) are replaced by C0 (X/Y). The proof of our main results is based on the study of the structure of the algebra e(X), in particular, we determine its character space and the structure of its cross-product e(X) := e(X) ⋊ X by the natural action τ of X on e(X). Our techniques apply also to more general classes of Hamiltonians that have a many-body type structure. We allow, in particular, potentials with local singularities and more general behaviours at infinity. We also develop general techniques that may be useful for other operators and other types of questions, such as the approximation of eigenvalues.

A Doob h-Transform of the Gross–Pitaevskii Hamiltonian

We perform a Doob h-transformation of the N-body Hamiltonian for N trapped interacting Bose particles starting from the unique real strictly positive ground state. This identifies the unitarily equivalent Markov generator describing the N interacting diffusion processes. By applying the convergence results coming from the Gross–Pitaevskii (GP) scaling limit to the energy form associated with this Markov generator we obtain the asymptotic generator uniquely corresponding to the GP Hamiltonian, which has been rigorously derived from the N-body Hamiltonian by performing the same scaling limit. Our asymptotic generator is a non linear diffusion generator with a killing rate governed by the GP order parameter. The associated stochastic process properly describes the single particle motion in the Bose–Einstein condensate including the self-interaction which acts here as a probability density-dependent killing rate. Finally, in a one-dimensional setting, we discuss under which conditions the process conditioned to survival converges to a quasi-stationary distribution having density with respect to the speed measure given by the principal eigenfunction of the diffusion generator with killing.

The essential spectrum of N-body systems with asymptotically homogeneous order-zero interactions

We study the essential spectrum of N-body Hamiltonians with potentials defined by functions that have radial limits at infinity. The results extend the HVZ theorem which describes the essential spectrum of usual N-body Hamiltonians. The proof is based on a careful study of algebras generated by potentials and their cross-products. We also describe the topology on the spectrum of these algebras, thus extending to our setting a result of A. Mageira. Our techniques apply to more general classes of potentials associated with translation invariant algebras of bounded uniformly continuous functions on a finite-dimensional vector space X.

Approximate solutions for N-body Hamiltonians with identical particles in D dimensions

A method based on the envelope theory is presented to compute approximate solutions for N-body Hamiltonians with identical particles in D dimensions (D⩾2). In some favorable cases, the approximate eigenvalues can be analytically determined and can be lower or upper bounds. The accuracy of the method is tested with several examples, and an application to a N-body system with a minimal length is studied. Finally, a semiclassical interpretation is given for the generic formula of the eigenvalues.

Boundary conditions for star matter and other periodic fermionic systems

Bulk fermionic matter, as it can be notably found in supernova matter and neutrons stars, is subject to correlations of infinite range due to the antisymmetrisation of the N-body wave function, which cannot be explicitly accounted for in a practical simulation. This problem is usually addressed in condensed matter physics by means of the so-called Twist Averaged Boundary Condition method. A different ansatz based on the localized Wannier representation has been proposed in the context of antisymmetrized molecular dynamics. In this paper we work out the formal relation between the two approaches. We show that, while the two coincide when working with exact eigenstates of the N-body Hamiltonian, differences appear in the case of variational approaches, which are currently used for the description of stellar matter. Some model applications with Fermionic Molecular Dynamics are shown.

The Quantum N-Body Problem and the Auxiliary Field Method: New Developments

Approximate analytical energy formulas for N-body semirelativistic Hamiltonians with one- and two-body interactions are obtained within the framework of the auxiliary field method. We first review the method in the case of nonrelativistic two-body problems. A general procedure is then given for N-body systems and a connection is presented between the method and the generalized virial theorem. The procedure is applied to the case of baryons in the large-N c limit.

Analytic harmonic approach to the N-body problem

We consider an analytic way to make the interacting N-body problem tractable by using harmonic oscillators in place of the relevant two-body interactions. The two-body terms of the N-body Hamiltonian are approximated by considering the energy spectrum and radius of the relevant two-body problem which gives frequency, centre position, and zero point energy of the corresponding harmonic oscillator. Adding external harmonic one-body terms, we proceed to solve the full quantum mechanical N-body problem analytically for arbitrary masses. Energy eigenvalues, eigenmodes, and correlation functions like density matrices can then be computed analytically. As a first application of our formalism, we consider the N-boson problem in two and three dimensions where we fit the two-body interactions to agree with the well-known zero-range model for two particles in a harmonic trap. Subsequently, condensate fractions, spectra, radii, and eigenmodes are discussed as a function of dimension, boson number N, and scattering length obtained in the zero-range model. We find that energies, radii, and condensate fraction increase with scattering length as well as boson number, while radii decrease with increasing boson number. Our formalism is completely general and can also be applied to fermions, Bose–Fermi mixtures, and to more exotic geometries.

The Quantum N-Body Problem and the Auxiliary Field Method

Approximate analytical energy formulas for N-body semirelativistic Hamiltonians with one- and two-body interactions are obtained within the framework of the auxiliary field method. We first review the method in the case of nonrelativistic two-body problems. A general procedure is then given for N-body systems and applied to the case of baryons in the large-N c limit.

The quantum N-body problem and the auxiliary field method

Approximate analytical energy formulas for N-body semirelativistic Hamiltonians with one- and two-body interactions are obtained within the framework of the auxiliary field method. This method has already been proven to be a powerful technique in the case of two-body problems. A general procedure is given and applied to various Hamiltonians of interest, in atomic and hadronic physics in particular. A test of formulas is performed for baryons described as a three-quark system.