Semiclassical Scaling Research Articles

Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential

Recently the leading order of the correlation energy of a Fermi gas in a coupled mean-field and semiclassical scaling regime has been derived, under the assumption of an interaction potential with a small norm and with compact support in Fourier space. We generalize this result to large interaction potentials, requiring only |cdot | hat{V} in ell ^1 (mathbb {Z}^3). Our proof is based on approximate, collective bosonization in three dimensions. Significant improvements compared to recent work include stronger bounds on non-bosonizable terms and more efficient control on the bosonization of the kinetic energy.

Effective Dynamics of Extended Fermi Gases in the High-Density Regime

We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus on the high-density regime, in the semiclassical scaling, and we consider a class of initial data describing zero-temperature states. In the non-relativistic case we prove that, as the density goes to infinity, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation, for short macroscopic times. In the case of relativistic dispersion, we show convergence of the many-body evolution to the relativistic Hartree equation for all macroscopic times. With respect to previous work, the rate of convergence does not depend on the total number of particles, but only on the density: in particular, our result allows us to study the quantum dynamics of extensive many-body Fermi gases.

Scaling asymptotics of Wigner distributions of harmonic oscillator orbital coherent states

The main result of this article gives scaling asymptotics of the Wigner distributions of isotropic harmonic oscillator orbital coherent states concentrating along Hamiltonian orbits γ in shrinking tubes around γ in phase space. In particular, these Wigner distributions exhibit a hybrid semi-classical scaling. That is, simultaneously, we have an Airy scaling when the tube has radius normal to the energy surface Σ E , and a Gaussian scaling when the tube has radius tangent to Σ E .

Bosonization of Fermionic Many-Body Dynamics

We consider the quantum many-body evolution of a homogeneous Fermi gas in three dimensions in the coupled semiclassical and mean-field scaling regime. We study a class of initial data describing collective particle-hole pair excitations on the Fermi ball. Using a rigorous version of approximate bosonization, we prove that the many-body evolution can be approximated in Fock space norm by a quasifree bosonic evolution of the collective particle-hole excitations.

Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation

We consider the dispersive logarithmic Schr{\"o}dinger equation in a semi-classical scaling. We extend the results about the large time behaviour of the solution (dispersion faster than usual with an additional logarithmic factor, convergence of the rescaled modulus of the solution to a universal Gaussian profile) to the case with semi-classical constant. We also provide a sharp convergence rate to the Gaussian profile in Kantorovich-Rubinstein metric through a detailed analysis of the Fokker-Planck equation satisfied by this modulus. Moreover, we perform the semiclassical limit of this equation thanks to the Wigner Transform in order to get a (Wigner) measure. We show that those two features are compatible and the density of a Wigner Measure has the same large time behaviour as the modulus of the solution of the logarithmic Schr{\"o}dinger equation. Lastly, we discuss about the related kinetic equation (which is the Kinetic Isothermal Euler System) and its formal properties, enlightened by the previous results and a new class of explicit solutions.

Combined Mean-Field and Semiclassical Limits of Large Fermionic Systems

We study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale hbar = N^{-1/3} in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.

The role of collapsed matter in the decay of black holes

We try to shed some light on the role of matter in the final stages of black hole evaporation from the fundamental frameworks of classicalization and the black-to-white hole bouncing scenario. Despite being based on very different grounds, these two approaches attempt at going beyond the background field method and treat black holes as fully quantum systems rather than considering quantum field theory on the corresponding classical manifolds. They also lead to the common prediction that the semiclassical description of black hole evaporation should break down and the system be disrupted by internal quantum pressure, but they both arrive at this conclusion neglecting the matter that formed the black hole. We instead estimate this pressure from the bootstrapped description of black holes, which allows us to express the total Arnowitt–Deser–Misner mass in terms of the baryonic mass still present inside the black hole. We conclude that, although these two scenarios provide qualitatively similar predictions for the final stages, the corpuscular model does not seem to suggest any sizeable deviation from the semiclassical time scale at which the disruption should occur, unlike the black-to-white hole bouncing scenario. This, in turn, makes the phenomenology of corpuscular black holes more subtle from an astrophysical perspective.

Ballistic front dynamics after joining two semi-infinite quantum Ising chains

We consider two semi-infinite quantum Ising chains initially at thermal equilibrium at two different temperatures and subsequently joined by an interaction between their end points. Transport properties such as the heat current are determined by the dynamics of the left- and right-moving fermionic quasiparticles which characterize the ensuing unitary dynamics. Within the so-called semiclassical space-time scaling limit we extend known results by determining the full space and time dependence of the density and current of energy and of fermionic quasiparticles. Upon approaching the edge of the propagating front, these quantities as well as the two-point correlation function display qualitatively different behaviors depending on the transverse field of the chain being critical or not. While in the latter case corrections to the leading behavior are described, as expected, by the Airy kernel, in the former a novel scaling form emerges with universal features.

Mean Field Evolution of Fermions with Coulomb Interaction

We study the many body Schr\"odinger evolution of weakly coupled fermions interacting through a Coulomb potential. We are interested in a joint mean field and semiclassical scaling, that emerges naturally for initially confined particles. For initial data describing approximate Slater determinants, we prove convergence of the many-body evolution towards Hartree-Fock dynamics. Our result holds under a condition on the solution of the Hartree-Fock equation, that we can only show in a very special situation (translation invariant data, whose Hartree-Fock evolution is trivial), but that we expect to hold more generally.

DParFit: A computer program for fitting diatomic molecule spectral data to parameterized level energy expressions

This paper describes Fortran program dParFit, which performs least-squares fits of diatomic molecule spectroscopic data involving one or more electronic states and one or more isotopologues, to parameterized expressions for the level energies. The data may consist of any combination of microwave, infrared or electronic vibrotational bands, fluorescence series or binding energies (from photo-association spectroscopy). The level energies for each electronic state may be described by one of: (i) band constants {Gv,Bv,Dv,…} for each vibrational level, (ii) generalized Dunham expansions, (iii) pure near-dissociation expansions (NDEs), (iv) mixed Dunham/NDE expressions, or (v) individual term values for each distinct level of each isotopologue. Different representations may be used for different electronic states and/or for different types of constants in a given fit (e.g., Gv and Bv may be represented one way and centrifugal distortion constants another). The effect of Λ-doubling or 2Σ splittings may be represented either by band constants (qvB or γvB, qvD or γvD, etc.) for each vibrational level of each isotopologue, or by using power series expansions in (v+12) to represent those constants. Fits to Dunham or NDE expressions automatically incorporate normal first-order semiclassical mass scaling to allow combined analyses of multi-isotopologue data. In addition, dParFit may fit to determine atomic-mass-dependent terms required to account for breakdown of the Born–Oppenheimer and first-order semiclassical approximations. In any of these types of fits, one or more subsets of these parameters for one or more of the electronic states may be held fixed, while a limited parameter set is varied. The program can also use a set of read-in constants to make predictions and calculate deviations [ycalc−yobs] for any chosen input data set, or to generate predictions of arbitrary data sets.

On semi-classical limit of nonlinear quantum scattering

We consider the nonlinear Schr{\"o}dinger equation with a short-range external potential, in a semi-classical scaling. We show that for fixed Planck constant, a com-plete scattering theory is available, showing that both the potential and the nonlinearity are asymptotically negligible for large time. Then, for data under the form of coherent state, we show that a scattering theory is also available for the approximate envelope of the propagated coherent state, which is given by a nonlinear equation. In the semi-classical limit, these two scattering operators can be compared in terms of classical scattering the-ory, thanks to a uniform in time error estimate. Finally, we infer a large time decoupling phenomenon in the case of finitely many initial coherent states.

Semiclassics for Particles with Spin via a Wigner–Weyl-Type Calculus

We show how to relate the full quantum dynamics of a spin-½ particle on \({\mathbb{R}^d}\) to a classical Hamiltonian dynamics on the enlarged phase space \({\mathbb{R}^{2d} \times \mathbb{S}^{2}}\) up to errors of second order in the semiclassical parameter. This is done via an Egorov-type theorem for normal Wigner–Weyl calculus for \({\mathbb{R}^d}\) (Folland, Harmonic Analysis on Phase Space, 1989; Lein, Weyl Quantization and Semiclassics, 2010) combined with the Stratonovich–Weyl calculus for SU(2) (Varilly and Gracia-Bondia, Ann Phys 190:107–148, 1989). For a specific class of Hamiltonians, including the Rabi- and Jaynes–Cummings model, we prove an Egorov theorem for times much longer than the semiclassical time scale. We illustrate the approach for a simple model of the Stern–Gerlach experiment.

Local smoothing for the Schrödinger equation with a prescribed loss

We consider a family of surfaces of revolution, each with a single periodic geodesic which is degenerately unstable. We prove a local smoothing estimate for solutions to the linear Schrödinger equation with a loss that depends on the degeneracy, and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale. As a byproduct of our proof, we obtain a cutoff resolvent estimate with a sharp polynomial loss.

Mathematical theory and numerical methods for Bose-Einstein condensation

In this paper, we mainly review recent results on mathematical theory andnumerical methods for Bose-Einstein condensation (BEC),based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are thencarried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models forspin-1 BEC, Bogoliubov excitation and BEC at finite temperature.

A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds

We prove a bilinear $L^2(\R^d) \times L^2(\R^d) \to L^2(\R^{d+1})$ estimate for a pair of oscillatory integral operators with different asymptotic parameters and phase functions satisfying a transversality condition. This is then used to prove a bilinear refinement to Strichartz estimates on closed manifolds, similar to that on $\R^d$, but at a relevant semi-classical scale. These estimates will be employed elsewhere to prove global well-posedness below $H^1$ for the cubic nonlinear Schr\odinger equation on closed surfaces.

Positive ions of SiH4: Jahn-Teller-like distortions for $${{\rm SiH}_{4}^{+}}$$ and eventually molecular collapse on further ionisation.

We have recently studied by quantum-chemical calculations some fifteen tetrahedral and octahedral molecules. These seemingly disparate numerical tabulations were, it was demonstrated, pulled together by comparison with model semiclassical scaling laws for (i) nuclear–nuclear repulsion energy at the equilibrium geometry and (ii) total energies likewise. Here, we again appeal to such model scaling predictions, but now for positive ions of SiH4. We then report Hartree-Fock equilibrium geometries and MP2 corrections. For \({{\rm SiH}_{4}^{+}}\), we assumed the symmetry to be C2v , by analogy with \({{\rm CH}_{4}^{+}}\) for which experimental confirmation of this symmetry is available. Larger distortions, but still for C2v symmetry, are found from our quantum-chemical studies in the case of \({{\rm SiH}_{4}^{2+}}\). But for \({{\rm SiH}_{4}^{3+}}\) there is a marked tendency to return to a configuration quite close to tetrahedral symmetry. But non-convergence is found for \({{\rm SiH}_{4}^{4+}}\). Finally relations to the admittedly simplistic semiclassical geometry scaling predictions of Lawes and March are conjectured.

Nonlinear dynamics of semiclassical coherent states in periodic potentials

We consider nonlinear Schrödinger equations with either local or nonlocal nonlinearities. In addition, we include periodic potentials as used, for example, in matter wave experiments in optical lattices. By considering the corresponding semiclassical scaling regime, we construct asymptotic solutions, which are concentrated both in space and in frequency around the effective semiclassical phase-space flow induced by Bloch’s spectral problem. The dynamics of these generalized coherent states is governed by a nonlinear Schrödinger model with effective mass. In the case of nonlocal nonlinearities, we establish a novel averaging-type result in the critical case.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Mathematical and computational methods for semiclassical Schrödinger equations

We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical regime.

The top resonances of the cubic oscillator**In memory of Pierre Duclos.

We study the top resonance states of the cubic anharmonic oscillator for β on the complex plane cut on the negative semiaxis. In particular, by the semiclassical scaling and semiclassical methods, we prove that the top resonance states do not belong to .

Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials

This work concerns superharmonic perturbations of a Gaussian measure given by a special class of positive weights in the complex plane of the form $w(z) = \exp(-|z|^2 + U^{\mu}(z))$, where $U^{\mu}(z)$ is the logarithmic potential of a compactly supported positive measure $\mu$. The equilibrium measure of the corresponding weighted energy problem is shown to be supported on subharmonic generalized quadrature domains for a large class of perturbing potentials $U^{\mu}(z)$. It is also shown that the $2\times 2$ matrix d-bar problem for orthogonal polynomials with respect to such weights is well-defined and has a unique solution given explicitly by Cauchy transforms. Numerical evidence is presented supporting a conjectured relation between the asymptotic distribution of the zeroes of the orthogonal polynomials in a semi-classical scaling limit and the Schwarz function of the curve bounding the support of the equilibrium measure, extending the previously studied case of harmonic polynomial perturbations with weights $w(z)$ supported on a compact domain.