On the spectral analysis of many-body systems

We describe the essential spectrum and prove the Mourre estimate for quantum particle systems interacting through k-body forces and creation-annihilation processes which do not preserve the number of particles. For this we compute the “Hamiltonian algebra” of the system, i.e. the C^∗-algebra C generated by the Hamiltonians we want to study, and show that, as in the N-body case, it is graded by a semilattice. Hilbert C^∗-modules graded by semilattices are involved in the construction of C. For example, if we start with an N-body system whose Hamiltonian algebra is CN and then we add field type couplings between subsystems, then the many-body Hamiltonian algebra C is the imprimitivity algebra of a graded Hilbert CN-module.     

On the spectral analysis of quantum field Hamiltonians

We define C^∗-algebras on a Fock space such that the Hamiltonians of quantum field models with positive mass are affiliated to them. We describe the quotient of such algebras with respect to the ideal of compact operators and deduce consequences in the spectral theory of these Hamiltonians: we compute their essential spectrum and give a systematic procedure for proving the Mourre estimate.     

Gaudin models with irregular singularities

We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from B. Feigin et al. (1994) [17] to the case of non-highest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P¹ with regular as well as irregular singularities at finitely many points. We construct eigenvectors of these Hamiltonians, using Wakimoto modules of critical level, and show that their spectra on finite-dimensional representations are given by opers with trivial monodromy. We also comment on the connection between the generalized Gaudin models and the geometric Langlands correspondence with ramification.     

Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type

Let (M,g) be a globally symmetric space of noncompact type, of arbitrary rank, and Δ its Laplacian. We introduce a new method to analyze Δ and the resolvent (Δ-σ)^-1; this has origins in quantum N-body scattering, but is independent of the ‘classical’ theory of spherical functions, and is analytically much more robust. We expect that, suitably modified, it will generalize to locally symmetric spaces of arbitrary rank. As an illustration of this method, we prove the existence of a meromorphic continuation of the resolvent across the continuous spectrum to a Riemann surface multiply covering the plane. We also show how this continuation may be deduced using the theory of spherical functions. In summary, this paper establishes a long-suspected connection between the analysis on symmetric spaces and N-body scattering.     

Mean field dynamics of fermions and the time-dependent Hartree–Fock equation

The time-dependent Hartree–Fock equations are derived from the N-body linear Schrödinger equation with the mean-field scaling in the limit N→+∞ and for initial data that are close to Slater determinants. Only the case of bounded, symmetric binary interaction potentials is treated in this work. We prove that, as N→+∞, the first partial trace of the N-body density operator approaches the solution of the time-dependent Hartree–Fock equations (in operator form) in the sense of the trace norm.     

Spectral theory of time-periodic many-body systems

We study the spectrum of the monodromy operator for an N-body quantum system in a time-periodic external field with time-mean equal to zero. This includes AC-Stark and circularly polarized fields, and pair potentials with a local singularity up to (and including) the Coulomb singularity. In the framework of Floquet theory we prove a local commutator estimate and use it to prove a Limiting Absorption Principle for the Floquet Hamiltonian as well as exponential decay estimates on non-threshold eigenfunctions. These two results are then used to obtain a second-order perturbation theory for embedded eigenvalues. The principal tool is a new extended Mourre theory.     

On the existence of the N-body Efimov effect

In this paper, we prove the existence of the Efimov effect for N-body quantum systems with N?4. Under the conditions that the bottom of the essential spectrum, E₀, of the N-body operator is attained by the spectra of a unique three-cluster Subhamiltonian and its three associated two-cluster Subhamiltonians, and that at least two of these two-cluster Subhamiltonians have a resonance at the threshold E₀, we give a lower bound of the form C₀|log(E₀−λ)| for the number of eigenvalues on the left of , where C₀ is a positive constant depending only on the reduced masses in the three-cluster decomposition. We also obtain a lower bound on the number of discrete eigenvalues in coupling constant perturbation.     

Minimizing configurations and Hamilton-Jacobi equations of homogeneous N-body problems

For N-body problems with homogeneous potentials we define a special class of central configurations related with the reduction of homotheties in the study of homogeneous weak KAM solutions. For potentials in 1/r ^α with α ∈ (0, 2) we prove the existence of homogeneous weak KAM solutions. We show that such solutions are related to viscosity solutions of another Hamilton-Jacobi equation in the sphere of normal configurations. As an application we prove for the Newtonian three-body problem that there are no smooth homogeneous solutions to the critical Hamilton-Jacobi equation.     

Isometries, Fock spaces, and spectral analysis of Schrödinger operators on trees

We construct conjugate operators for the real part of a completely non-unitary isometry and we give applications to the spectral and scattering theory of a class of operators on (complete) Fock spaces, natural generalizations of the Schrödinger operators on trees. We consider C^*-algebras generated by such Hamiltonians with certain types of anisotropy at infinity, we compute their quotient with respect to the ideal of compact operators, and give formulas for the essential spectrum of these Hamiltonians.     

Vlasov limit and discreteness effects in cosmological N-body simulations

We present the problematic of controlling the discreteness effects in cosmological N-body simulations. We describe a perturbative treatment which gives an approximation describing the evolution under self-gravity of a lattice perturbed from its equilibrium, which allows to trace the evolution of the fully discrete distribution until the time when particles approach one another (“shell-crossing”). Perturbed lattices are typical initial conditions for cosmological N-body simulations and thus we can describe precisely the early time evolution of these simulations. A quantitative comparison with fluid Lagrangian theory permits to study discreteness effects in the linear regime of the simulations. We show finally some work in progress about quantifying discreteness effects in the non-perturbative (highly non-linear) regime of cosmological N-body simulations by evolving different discretizations of the same continuous density field.     

Boundary conditions of Sturm-Liouville operators with mixed spectra

We study bounds on averages of spectral functions corresponding to Sturm-Liouville operators on the half line for different boundary conditions. As a consequence constraints are obtained which imply existence of singular spectrum embedded in a.c. spectrum for sets of boundary conditions with positive measure and potentials vanishing in an interval [0,N]. These constraints are related to estimates on the measure of sets where the spectral density is positive.     

Periodic solutions of the N-body problem

This paper proves the existence of six new classes of periodic solutions to the N-body problem by small parameter methods. Three different methods of introducing a small parameter are considered and an appropriate method of scaling the Hamiltonian is given for each method. The small parameter is either one of the masses, the distance between a pair of particles or the reciprocal of the distances between one particle and the center of mass of the remaining particles. For each case symmetric and non-symmetric periodic solutions are established. For every relative equilibrium solution of the (N ? 1)-body problem each of the six results gives periodic solutions of the N-body problem. Under additional mild non-resonance conditions the results are roughly as follows. Any non-degenerate periodic solutions of the restricted N-body problem can be continued into the full N-body problem. There exist periodic solutions of the N-body problem, where N ? 2 particles and the center of mass of the remaining pair move approximately on a solution of relative equilibrium and the pair move approximately on a small circular orbit of the two-body problems around their center of mass. There exist periodic solutions of the N-body problem, where one small particle and the center of mass of the remaining N ? 1 particles move approximately on a large circular orbit of the two body problems and the remaining N ? 1 bodies move approximately on a solution of relative equilibrium about their center of mass. There are three similar results on the existence of symmetric periodic solutions.     

Eigenvalue Asymptotics for a Schrödinger Operator with Non-Constant Magnetic Field Along One Direction

We consider the discrete spectrum of the two-dimensional Hamiltonian H = H ₀ + V, where H ₀ is a Schrödinger operator with a non-constant magnetic field B that depends only on one of the spatial variables, and V is an electric potential that decays at infinity. We study the accumulation rate of the eigenvalues of H in the gaps of its essential spectrum. First, under certain general conditions on B and V, we introduce effective Hamiltonians that govern the main asymptotic term of the eigenvalue counting function. Further, we use the effective Hamiltonians to find the asymptotic behavior of the eigenvalues in the case where the potential V is a power-like decaying function and in the case where it is a compactly supported function, showing a semiclassical behavior of the eigenvalues in the first case and a non-semiclassical behavior in the second one. We also provide a criterion for the finiteness of the number of eigenvalues in the gaps of the essential spectrum of H.     

Weak interaction limit for nuclear matter and the time-dependent Hartree-Fock equation

We consider an effective model of nuclear matter including spin and isospin degrees of freedom, described by an N-body Hamiltonian with suitably renormalized two-body and three-body interaction potentials. We show that the corresponding mean-field theory (the time-dependent Hartree-Fock approximation) is “exact” as N tends to infinity.     

Perturbation of resonances in quantum mechanics

Schrödinger operators on L²($\text{R}$³) of the form ?Δ + V_λ with potentials V_λ real-analytic in λ are discussed. The analytic structure in V_λ and k (with k² the energy variable) of the resolvent kernel, the eigenvalues and resonances is exhibited and we obtain in particular convergent perturbation expansions for the resonances and the corresponding resonance functions. The lower order expansion coefficients are computed explicitly. The resonances and the corresponding functions are also computed for a particle moving under the action of n point interactions. This gives asymptotic low energy information about Schrödinger Hamiltonians with short range potentials. The perturbation theory of resonances, eigenvalues and of the corresponding functions for Hamiltonians describing n point interactions perturbed by a potential is also given.     

Spectra of algebras of holomorphic functions of bounded type

We prove that if U is a balanced H_b (U)-domain of holomorphy in Tsirelson's space then the spectrum of H_b (U) is identified with U. We derive theorems of Banach Stone type for algebras of holomorphic functions and algebras of holomorphic germs.     

Many-Particle Models with Two Constants of Interaction

We discuss integrable models of quantum field theory and statistical mechanics. The dynamics and kinematics of these models are defined by Hamiltonians with symmetries determined by Lie algebras. The paper is devoted to the characterization of models such that the root vectors of their symmetry algebras run two orbits under the action of the Weyl group. Such properties possess root systems of the type B _N , C _N , and G ₂. The main focus is on models with the symmetry of algebras B _N . In this case the main characteristics of the process are obtained from the system of Yang-Baxter equation and the reflection equations.We consider the Calogero-Moser and Calogero-Sutherland models and also the formalisms of the Lax and Dunkl operators. The connection between these formalisms and method of describing these models in terms of the generalized Knizhnik-Zamolodchikov equations with the system of roots of the type B _N by the example of the Gaudin model with reflection are discussed. Examples of many-particle systems that interact with each other with reflections are presented.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ?S_n and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon’s multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B.     

On algebras of holomorphic functions of a given type

We show that several spaces of holomorphic functions on a Riemann domain over a Banach space, including the nuclear and Hilbert–Schmidt bounded type, are locally m-convex Fréchet algebras. We prove that the spectrum of these algebras has a natural analytic structure, which we use to characterize the envelope of holomorphy. We also show a Cartan–Thullen type theorem.     

Curved Koszul duality of algebras over unital versions of binary operads

We develop a curved Koszul duality theory for algebras presented by quadratic-linear-constant relations over unital versions of binary quadratic operads. As an application, we study Poisson n-algebras given by polynomial functions on a standard shifted symplectic space. We compute explicit resolutions of these algebras using curved Koszul duality. We use these resolutions to compute derived enveloping algebras and factorization homology on parallelized simply connected closed manifolds with coefficients in these Poisson n-algebras.