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In this work, we explore advanced machine learning techniques for minimizing Gibbs free energy in full 3D micromagnetic simulations. Building on Brown's bounds for magnetostatic self-energy, we revisit their application in the context of variational formulations of the transmission problems for the scalar and vector potential. To overcome the computational challenges posed by whole-space integrals, we reformulate these bounds on a finite domain, making the method more efficient and scalable for numerical simulation. Our approach utilizes an alternating optimization scheme for joint minimization of Brown's energy bounds and the Gibbs free energy. The Cayley transform is employed to rigorously enforce the unit norm constraint, while R-functions are used to impose essential boundary conditions in the computation of magnetostatic fields. Our results highlight the potential of mesh-free Physics-Informed Neural Networks (PINNs) and Extreme Learning Machines (ELMs) when integrated with hard constraints, providing highly accurate approximations. These methods exhibit competitive performance compared to traditional numerical approaches, showing significant promise in computing magnetostatic fields and the application for energy minimization, such as the computation of hysteresis curves. This work opens the path for future directions of research on more complex geometries, such as grain structure models, and the application to large scale problem settings which are intractable with traditional numerical methods., Comment: 39 pages, 9 figures

We consider the cubic nonlinear Schr\"odinger equation with a spatially rough potential, a key equation in the mathematical setup for nonlinear Anderson localization. Our study comprises two main parts: new optimal results on the well-posedness analysis on the PDE level, and subsequently a new efficient numerical method, its convergence analysis and simulations that illustrate our analytical results. In the analysis part, our results focus on understanding how the regularity of the solution is influenced by the regularity of the potential, where we provide quantitative and explicit characterizations. Ill-posedness results are also established to demonstrate the sharpness of the obtained regularity characterizations and to indicate the minimum regularity required from the potential for the NLS to be solvable. Building upon the obtained regularity results, we design an appropriate numerical discretization for the model and establish its convergence with an optimal error bound. The numerical experiments in the end not only verify the theoretical regularity results, but also confirm the established convergence rate of the proposed scheme. Additionally, a comparison with other existing schemes is conducted to demonstrate the better accuracy of our new scheme in the case of a rough potential., Comment: 54 pages, 8 figures

The self-consistent Pauli-Poisswell equation for 2-spinors is the first order in $1/c$ semi-relativistic approximation of the Dirac-Maxwell equation for 4-spinors coupled to the self-consistent electromagnetic fields generated by the density and current density of a fast moving electric charge. It consists of a vector-valued magnetic Schr\"odinger equation with an extra term coupling spin and magnetic field via the Pauli matrices coupled to 1+3 Poisson type equations as the magnetostatic approximation of Maxwell's equations. The Pauli-Poisswell equation is a consistent $O(1/c)$ model that keeps both relativistic effects magnetism and spin which are both absent in the non-relativistic Schr\"odinger-Poisson equation and inconsistent in the magnetic Schr\"odinger-Maxwell equation. We present the mathematically rigorous semiclassical limit $\hbar \rightarrow 0$ of the Pauli-Poisswell equation towards the magnetic Euler-Poisswell equation. We use WKB analysis which is valid locally in time only. A key step is to obtain an a priori energy estimate for which we have to take into account the Poisson equations for the magnetic potential with the current as source term. Additionally we obtain the weak convergence of the monokinetic Wigner transform and strong convergence of the density and the current density. We also prove local wellposedness of the Euler-Poisswell equation which is global unless a finite time blow-up occurs.

We present the self-consistent Pauli equation, a semi-relativistic model for charged spin-$1/2$-particles with self-interaction with the electromagnetic field. The Pauli equation arises as the $O(1/c)$ approximation of the relativistic Dirac equation. The fully relativistic self-consistent model is the Dirac-Maxwell equation where the description of spin and the magnetic field arises naturally. In the non-relativistic setting the correct self-consistent equation is the Schr\"odinger-Poisson equation which does not describe spin and the magnetic field and where the self-interaction is with the electric field only. The Schr\"odinger-Poisson equation also arises as the mean field limit of the $N$-body Schr\"odinger equation with Coulomb interaction. We propose that the Pauli-Poisson equation arises as the mean field limit $N \rightarrow \infty$ of the linear $N$-body Pauli equation with Coulomb interaction where one has to pay extra attention to the fermionic nature of the Pauli equation. We present the semiclassical limit of the Pauli-Poisson equation by the Wigner method to the Vlasov equation with Lorentz force coupled to the Poisson equation which is also consistent with the hierarchy in $1/c$ of the self-consistent Vlasov equation. This is a non-trivial extension of the groundbreaking works by Lions & Paul and Markowich & Mauser, where we need methods like magnetic Lieb-Thirring estimates.

We consider the expansion of wave packets governed by the free Schr\"odinger equation. This seemingly simple task plays an important role in simulations of various quantum experiments and in particular in the field of matter-wave interferometry. The initial tight confinement results in a very fast expansion of the wave function at later times which significantly complicates an efficient and precise numerical evaluation. In many practical cases the expansion time is too short for the validity of the stationary phase approximation and too long for an efficient application of Fourier collocation-based methods. We develop an alternative method based on a discretization of the free-particle propagator. This simple approach yields highly accurate results which readily follows from the exceptionally fast convergence of the trapezoidal rule approximation of integrals involving smooth and rapidly decaying functions. We discuss and analyze our approach in detail and demonstrate how to estimate the numerical error in the one-dimensional setting. Furthermore, we show that by exploiting the separability of the Green's function, the numerical effort of the multi-dimensional approximation is considerably reduced. Our method is very fast, highly accurate, and easy to implement on modern hardware.

We study the full 3d static micromagnetic equations via a physics-informed neural network (PINN) ansatz for the continuous magnetization configuration. PINNs are inherently mesh-free and unsupervised learning models. In our approach we can learn to minimize the total Gibbs free energy with additional conditional parameters, such as the exchange length, by a single low-parametric neural network model. In contrast, traditional numerical methods would require the computation and storage of a large number of solutions to interpolate the continuous spectrum of quasi-optimal magnetization configurations. We also consider the important and computationally expensive stray field problem via PINNs, where we use a basically linear learning ansatz, called extreme learning machines (ELM) within a splitting method for the scalar potential. This reduces the stray field training to a linear least squares problem with precomputable solution operator. We validate the stray field method by means of numerical example comparisons from literature and illustrate the full micromagnetic approach via the NIST $\mu$MAG standard problem $\#3$., Comment: 17 pages, 10 figures

"Generalized Hydrodynamics" (GHD) stands for a model that describes one-dimensional \textit{integrable} systems in quantum physics, such as ultra-cold atoms or spin chains. Mathematically, GHD corresponds to nonlinear equations of kinetic type, where the main unknown, a statistical distribution function $f(t,z,\theta)$, lives in a phase space which is constituted by a one-dimensional position variable $z$, and a one-dimensional "kinetic" variable $\theta$, actually a wave-vector, called "rapidity". Two key features of GHD equations are first a non-local and nonlinear coupling in the advection term, and second an infinite set of conserved quantities, which prevent the system from thermalizing. To go beyond this, we consider the dissipative GHD equations, which are obtained by supplementing the right-hand side of the GHD equations with a non-local and nonlinear diffusion operator or a Boltzmann-type collision integral. In this paper, we deal with new high-order numerical methods to efficiently solve these kinetic equations. In particular, we devise novel backward semi-Lagrangian methods for solving the advective part (the so-called Vlasov equation) by using a high-order time-Taylor series expansion for the advection fields, whose successive time derivatives are obtained by a recursive procedure. This high-order temporal approximation of the advection fields are used to design new implicit/explicit Runge-Kutta semi-Lagrangian methods, which are compared to Adams-Moulton semi-Lagrangian schemes. For solving the source terms, constituted by the diffusion and collision operators, we use and compare different numerical methods of the literature., Comment: 35 pages, 8 figures, 1 table

Spin waves are ideal candidates for wave-based computing, but the construction of magnetic circuits is blocked by a lack of an efficient mechanism to excite long-running exchange spin waves with normalised amplitudes. Here, we solve the challenge by exploiting the deeply nonlinear phenomena of forward-volume spin waves in 200 nm wide nanoscale waveguides and validate our concept with microfocused Brillouin light scattering spectroscopy. An unprecedented nonlinear frequency shift of >2 GHz is achieved, corresponding to a magnetisation precession angle of 55{\deg} and enabling the excitation of exchange spin waves with a wavelength of down to ten nanometres with an efficiency of >80%. The amplitude of the excited spin waves is constant and independent of the input microwave power due to the self-locking nonlinear shift, enabling robust adjustment of the spin wave amplitudes in future on-chip magnonic integrated circuits., Comment: 21 pages, 5 figures

Generalized Hydrodynamics (GHD) has recently been devised as a method to solve the dynamics of integrable quantum many-body systems beyond the mean-field approximation. In its original form, a major limitation is the inability to predict equal-time correlations. Here we present a new method to treat thermal fluctuations of a 1D bosonic degenerate gas within the GHD framework. We show how the standard results using the thermodynmaic Bethe ansatz can be obtained through sampling of collective bosonic excitations, revealing the connection or duality between GHD and effective field theories such as the standard hydrodynamic equations. As an example, we study the damping of a coherently excited density wave and show how equal-time phase correlation functions can be extracted from the GHD evolution. Our results present a conceptually new way of treating fluctuations beyond the linearized regime of GHD., Comment: 10 pages, 7 figures

This work introduces a latent space method to calculate the demagnetization reversal process of multigrain permanent magnets. The algorithm consists of two deep learning models based on neural networks. The embedded Stoner-Wohlfarth method is used as a reduced order model for computing samples to train and test the machine learning approach. The work is a proof of concept of the method since the used microstructures are simple and the only varying parameters are the magnetic anisotropy axes of the hard magnetic grains. A convolutional autoencoder is used for nonlinear dimensionality reduction, in order to reduce the required amount of training samples for predicting the hysteresis loops. We enriched the loss function with physical information about the underlying problem which increases the accuracy of the machine learning approach. A deep learning regressor is operating in the latent space of the autoencoder. The predictor takes a series of previously encoded magnetization states and outputs the next magnetization state along the hysteresis curve. To predict the complete demagnetization loop, we apply a recursive learning scheme.

Recently, Kim & Wilkening (Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation, Quart. Appl. Math., 76, 383-405, 2018) proposed two novel predictor-corrector methods for the Landau-Lifshitz-Gilbert equation (LLG) in micromagnetics, which models the dynamics of the magnetization in ferromagnetic materials. Both integrators are based on the so-called Landau-Lifshitz form of LLG, use mass-lumped variational formulations discretized by first-order finite elements, and only require the solution of linear systems, despite the nonlinearity of LLG. The first(-order in time) method combines a linear update with an explicit projection of an intermediate approximation onto the unit sphere in order to fulfill the LLG-inherent unit-length constraint at the discrete level. In the second(-order in time) integrator, the projection step is replaced by a linear constraint-preserving variational formulation. In this paper, we extend the analysis of the integrators by proving unconditional well-posedness and by establishing a close connection of the methods with other approaches available in the literature. Moreover, the new analysis also provides a well-posed integrator for the Schr\"odinger map equation (which is the limit case of LLG for vanishing damping). Finally, we design an implicit-explicit strategy for the treatment of the lower-order field contributions, which significantly reduces the computational cost of the schemes, while preserving their theoretical properties., Comment: 29 pages, 6 figures

Machine learning (ML) entered the field of computational micromagnetics only recently. The main objective of these new approaches is the automatization of solutions of parameter-dependent problems in micromagnetism such as fast response curve estimation modeled by the Landau-Lifschitz-Gilbert (LLG) equation. Data-driven models for the solution of time- and parameter-dependent partial differential equations require high dimensional training data-structures. ML in this case is by no means a straight-forward trivial task, it needs algorithmic and mathematical innovation. Our work introduces theoretical and computational conceptions of certain kernel and neural network based dimensionality reduction approaches for efficient prediction of solutions via the notion of low-dimensional feature space integration. We introduce efficient treatment of kernel ridge regression and kernel principal component analysis via low-rank approximation. A second line follows neural network (NN) autoencoders as nonlinear data-dependent dimensional reduction for the training data with focus on accurate latent space variable description suitable for a feature space integration scheme. We verify and compare numerically by means of a NIST standard problem. The low-rank kernel method approach is fast and surprisingly accurate, while the NN scheme can even exceed this level of accuracy at the expense of significantly higher costs.

We present a detailed analysis of the relaxation dynamics in an extended bosonic Josephson junction. We show that stochastic classical field simulations using Gross-Pitaevskii equations in three spatial dimensions reproduce the main experimental findings of M. Pigneur et al., Phys. Rev. Lett. 120, 173601 (2018). We give an analytic solution describing the short time evolution through multimode dephasing. For longer times, the observed relaxation to a phase locked state is caused by nonlinear dynamics beyond the sine-Gordon model, induced by the longitudinal confinement potential and persisting even at zero temperature. Finally, we analyze different experimentally relevant trapping geometries to mitigate these effects. Our results provide the basis for future experimental implementations aiming to study nonlinear and quantum effects of the relaxation in extended bosonic Josephson junctions., Comment: 12 pages, 9 figures, restructured and extended article format

We establish a machine learning model for the prediction of the magnetization dynamics as function of the external field described by the Landau-Lifschitz-Gilbert equation, the partial differential equation of motion in micromagnetism. The model allows for fast and accurate determination of the response to an external field which is illustrated by a thin-film standard problem. The data-driven method internally reduces the dimensionality of the problem by means of nonlinear model reduction for unsupervised learning. This not only makes accurate prediction of the time steps possible, but also decisively reduces complexity in the learning process where magnetization states from simulated micromagnetic dynamics associated with different external fields are used as input data. We use a truncated representation of kernel principal components to describe the states between time predictions. The method is capable of handling large training sample sets owing to a low-rank approximation of the kernel matrix and an associated low-rank extension of kernel principal component analysis and kernel ridge regression. The approach entirely shifts computations into a reduced dimensional setting breaking down the problem dimension from the thousands to the tens.

We analyze a numerical method to solve the time-dependent linear Pauli equation in three space dimensions. The Pauli equation is a semi-relativistic generalization of the Schr\"odinger equation for 2-spinors which accounts both for magnetic fields and for spin, with the latter missing in preceding numerical work on the linear magnetic Schr\"odinger equation. We use a four operator splitting in time, prove stability and convergence of the method and derive error estimates as well as meshing strategies for the case of given time-independent electromagnetic potentials, thus providing a generalization of previous results for the magnetic Schr\"odinger equation., Comment: 15 pages, 3 figures

Dipolar Bose-Einstein condensates have recently attracted much attention in the world of quantum many body experiments. While the theoretical principles behind these experiments are typically supported by numerical simulations, the application of optimal control algorithms could potentially open up entirely new possibilities. As a proof of concept, we demonstrate that the formation process of a single dipolar droplet state could be dramatically accelerated using advanced concepts of optimal control. More specifically, our optimization is based on a multilevel B-spline method reducing the number of required cost function evaluations and hence significantly reducing the numerical effort. Moreover, our strategy allows to consider box constraints on the control inputs in a concise and efficient way. To further improve the overall efficiency, we show how to evaluate the dipolar interaction potential in the generalized Gross-Pitaevskii equation without sacrificing the spectral convergence rate of the underlying time-splitting spectral method.

We establish a time-stepping learning algorithm and apply it to predict the solution of the partial differential equation of motion in micromagnetism as a dynamical system depending on the external field as parameter. The data-driven approach is based on nonlinear model order reduction by use of kernel methods for unsupervised learning, yielding a predictor for the magnetization dynamics without any need for field evaluations after a data generation and training phase as precomputation. Magnetization states from simulated micromagnetic dynamics associated with different external fields are used as training data to learn a low-dimensional representation in so-called feature space and a map that predicts the time-evolution in reduced space. Remarkably, only two degrees of freedom in feature space were enough to describe the nonlinear dynamics of a thin-film element. The approach has no restrictions on the spatial discretization and might be useful for fast determination of the response to an external field.

We present our open-source Python module Commics for the study of the magnetization dynamics in ferromagnetic materials via micromagnetic simulations. It implements state-of-the-art unconditionally convergent finite element methods for the numerical integration of the Landau-Lifshitz-Gilbert equation. The implementation is based on the multiphysics finite element software Netgen/NGSolve. The simulation scripts are written in Python, which leads to very readable code and direct access to extensive post-processing. Together with documentation and example scripts, the code is freely available on GitLab.

We consider laser-pumped one-dimensional two-component bosons in a parabolic trap embedded in a high-finesse optical cavity. Above a threshold pump power, the photons that populate the cavity modify the effective atom trap and mediate a coupling between the two components of the Bose-Einstein condensate. We calculate the ground state of the laser-pumped system and find different stages of self-organization depending on the power of the laser. The modified potential and the laser-mediated coupling between the atomic components give rise to rich many-body physics: an increase of the pump power triggers a self-organization of the atoms while an even larger pump power causes correlations between the self-organized atoms -- the BEC becomes fragmented and the reduced density matrix acquires multiple macroscopic eigenvalues. In this fragmented superradiant state, the atoms can no longer be described as two-level systems and the mapping of the system to the Dicke model breaks down., Comment: 8 pages, 3 figures, software available at http://ultracold.org

Longitudinal relaxation is the process by which an excited spin ensemble decays into its thermal equilibrium with the environment. In solid-state spin systems relaxation into the phonon bath usually dominates over the coupling to the electromagnetic vacuum. In the quantum limit the spin lifetime is determined by phononic vacuum fluctuations. However, this limit was not observed in previous studies due to thermal phonon contributions or phonon-bottleneck processes. Here we use a dispersive detection scheme based on cavity quantum electrodynamics (cQED) to observe this quantum limit of spin relaxation of the negatively charged nitrogen vacancy ($\mathrm{NV}^-$) centre in diamond. Diamond possesses high thermal conductivity even at low temperatures, which eliminates phonon-bottleneck processes. We observe exceptionally long longitudinal relaxation times $T_1$ of up to 8h. To understand the fundamental mechanism of spin-phonon coupling in this system we develop a theoretical model and calculate the relaxation time ab initio. The calculations confirm that the low phononic density of states at the $\mathrm{NV}^-$ transition frequency enables the spin polarization to survive over macroscopic timescales.

A practical and efficient scheme for the higher order integration of the Landau-Lifschitz-Gilbert (LLG) equation is presented. The method is based on extrapolation of the two-step explicit midpoint rule and incorporates adaptive time step and order selection. We make use of a piecewise time-linear stray field approximation to reduce the necessary work per time step. The approximation to the interpolated operator is embedded into the extrapolation process to keep in step with the hierarchic order structure of the scheme. We verify the approach by means of numerical experiments on a standardized NIST problem and compare with a higher order embedded Runge-Kutta formula. The efficiency of the presented approach increases when the stray field computation takes a larger portion of the costs for the effective field evaluation.

The Hilbert space for three fermions in six orbitals, lately dubbed the "Borland-Dennis setting," is a proving ground for insights into electronic structure. Borland and Dennis discovered that, when referred to coordinate systems defined in terms of its natural orbitals, a wave function in the Borland-Dennis setting has the same structure as a 3-qubit state. By dint of the Borland-Dennis Theorem, canonical forms for 3-qubit states have analogs in the Borland-Dennis setting. One of these canonical forms is based upon "max-overlap Slater determinant approximations." Any max-overlap Slater determinant approximation of a given wave function is the leading term in a 5-term configuration interaction (CI) expansion of that wave function. Our main result is that "max-overlap CIS approximations" also lead to 5-term CI expansions, distinct from those based on max-overlap Slater determinant approximations, though of the same symmetric shape. We also prove the analog of this result for 3-qubit setting.

We study the optimal Slater-determinant approximation of an $N$-fermion wave function analytically. That is, we seek the Slater-determinant (constructed out of $N$ orthonormal single-particle orbitals) wave function having largest overlap with a given $N$-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the Greenberger-Horne-Zeilinger state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the optimal Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the optimal Slater approximation of some special target wave functions., Comment: To appear in PRA

This article is a brief review of "nonfreeness" and related measures of "correlation" for many-fermion systems. The many-fermion states we deem "uncorrelated" are the gauge-invariant quasi-free states. Uncorrelated states of systems of finitely many fermions we call simply "free" states. Slater determinant states are free; all other free states are "substates" of Slater determinant states or limits of such. The nonfreeness of a many-fermion state equals the minimum of its entropy relative to all free states. Correlation functionals closely related to nonfreeness can be defined in terms of R\'enyi entropies; nonfreeness is the one that uses Shannon entropy. These correlation functionals all share desirable additivity and monotonicity properties, but nonfreeness has some additional attractive properties.

We introduce an accurate and efficient method for a class of nonlocal potential evaluations with free boundary condition, including the 3D/2D Coulomb, 2D Poisson and 3D dipolar potentials. Our method is based on a Gaussian-sum approximation of the singular convolution kernel and Taylor expansion of the density. Starting from the convolution formulation, for smooth and fast decaying densities, we make a full use of the Fourier pseudospectral (plane wave) approximation of the density and a separable Gaussian-sum approximation of the kernel in an interval where the singularity (the origin) is excluded. Hence, the potential is separated into a regular integral and a near-field singular correction integral, where the first integral is computed with the Fourier pseudospectral method and the latter singular one can be well resolved utilizing a low-order Taylor expansion of the density. Both evaluations can be accelerated by fast Fourier transforms (FFT). The new method is accurate (14-16 digits), efficient ($O(N \log N)$ complexity), low in storage, easily adaptable to other different kernels, applicable for anisotropic densities and highly parallelable.

In the context of many-fermion systems, "correlation" refers to the inadequacy of an independent-particle model. Using "free" states as archetypes of our independent-particle model, we have proposed a measure of correlation that we called "nonfreeness" [Int. J. Quant. Inf. 5, 815 (2007)]. The nonfreeness of a many-fermion state was defined to be its entropy relative to the unique free state with the same 1-matrix. In this article, we prove that the nonfreeness of a state is the minimum of its entropy relative to all free states. We also extend the definition of nonfreeness to many-boson states and discuss a couple of examples.

We present a steepest descent energy minimization scheme for micromagnetics. The method searches on a curve that lies on the sphere which keeps the magnitude of the magnetization vector constant. The step size is selected according to a modified Barzilai-Borwein method. Standard linear tetrahedral finite elements are used for space discretization. For the computation of static hysteresis loops the steepest descent minimizer is faster than a Landau-Lifshitz micromagnetic solver by more than a factor of two. The speed up on a graphic processor is 4.8 as compared to the fastest single-core CPU implementation.

In this paper, we present the Stroboscopic Averaging Method (SAM), recently introduced in [7,8,10,12], which aims at numerically solving highly-oscillatory differential equations. More specifically, we first apply SAM to the Schr\"odinger equation on the 1-dimensional torus and on the real line with harmonic potential, with the aim of assessing its efficiency: as compared to the well-established standard splitting schemes, the stiffer the problem is, the larger the speed-up grows (up to a factor 100 in our tests). The geometric properties of SAM are also explored: on very long time intervals, symmetric implementations of the method show a very good preservation of the mass invariant and of the energy. In a second series of experiments on 2-dimensional equations, we demonstrate the ability of SAM to capture qualitatively the long-time evolution of the solution (without spurring high oscillations).

We derive a Kronecker product approximation for the micromagnetic long range interactions in a collocation framework by means of separable sinc quadrature. Evaluation of this operator for structured tensors (Canonical format, Tucker format, Tensor Trains) scales below linear in the volume size. Based on efficient usage of FFT for structured tensors, we are able to accelerate computations to quasi linear complexity in the number of collocation points used in one dimension. Quadratic convergence of the underlying collocation scheme as well as exponential convergence in the separation rank of the approximations is proved. Numerical experiments on accuracy and complexity confirm the theoretical results., Comment: 4 figures

We analyze the emergence of Shapiro resonances in tunnel-coupled Bose-Einstein condensates, realizing a bosonic Josephson junction. Our analysis is based on an experimentally relevant implementation using magnetic double well potentials on an atomchip. In this configuration the potential bias (implementing the junction voltage) and the potential barrier (realizing the Josephson link) are intrinsically coupled. We show that the dynamically driven system exhibits significantly enhanced Shapiro resonances which will facilitate experimental observation. To describe the systems response to the dynamic drive we compare a single-mode Gross-Pitaevskii (GP) description, an improved two-mode (TM) model and the self-consistent multi-configurational time dependent Hartree for Bosons (MCTDHB) method. We show that in the case of significant atom-atom interactions or strong driving, the spatial dynamics of the involved modes have to be taken into account, and only the MCTDHB method allows reliable predictions., Comment: 16 pages, 4 figures

The multiconfiguration methods are widely used by quantum physicists and chemists for numerical approximation of the many electron Schr\"odinger equation. Recently, first mathematically rigorous results were obtained on the time-dependent models, e.g. short-in-time well-posedness in the Sobolev space $H^2$ for bounded interactions (C. Lubichand O. Koch} with initial data in $H^2$, in the energy space for Coulomb interactions with initial data in the same space (Trabelsi, Bardos et al.}, as well as global well-posedness under a sufficient condition on the energy of the initial data (Bardos et al.). The present contribution extends the analysis by setting an $L^2$ theory for the MCTDHF for general interactions including the Coulomb case. This kind of results is also the theoretical foundation of ad-hoc methods used in numerical calculation when modification ("regularization") of the density matrix destroys the conservation of energy property, but keeps invariant the mass., Comment: This work was supported by the Viennese Science Foundation (WWTF) via the project "TDDFT" (MA-45), the Austrian Science Foundation (FWF) via the Wissenschaftkolleg "Differential equations" (W17) and the START Project (Y-137-TEC) and the EU funded Marie Curie Early Stage Training Site DEASE (MEST-CT-2005-021122)

Smoothed Wigner transforms have been used in signal processing, as a regularized version of the Wigner transform, and have been proposed as an alternative to it in the homogenization and / or semiclassical limits of wave equations. We derive explicit, closed formulations for the coarse-scale representation of the action of pseudodifferential operators. The resulting ``smoothed operators'' are in general of infinite order. The formulation of an appropriate framework, resembling the Gelfand-Shilov spaces, is necessary. Similarly we treat the ``smoothed Wigner calculus''. In particular this allows us to reformulate any linear equation, as well as certain nonlinear ones (e.g. Hartree and cubic non-linear Schr\"odinger), as coarse-scale phase-space equations (e.g. smoothed Vlasov), with spatial and spectral resolutions controlled by two free parameters. Finally, it is seen that the smoothed Wigner calculus can be approximated, uniformly on phase-space, by differential operators in the semiclassical regime. This improves the respective weak-topology approximation result for the Wigner calculus., Comment: 58 pages, plain Tex

"Nonfreeness" is the (negative of the) difference between the von Neumann entropies of a given many-fermion state and the free state that has the same 1-particle statistics. It also equals the relative entropy of the two states in question, i.e., it is the entropy of the given state relative to the corresponding free state. The nonfreeness of a pure state is the same as its "particle-hole symmetric correlation entropy", a variant of an established measure of electron correlation. But nonfreeness is also defined for mixed states, and this allows one to compare the nonfreeness of subsystems to the nonfreeness of the whole. Nonfreeness of a part does not exceed that in the whole; nonfreeness is additive over independent subsystems; and nonfreeness is superadditive over subsystems that are independent on the 1-particle level., Comment: 20 pages. Submitted to Phys. Rev. B

We propose to quantify the "correlation" inherent in a many-electron (or many-fermion) wavefunction by comparing it to the unique uncorrelated state that has the same single-particle density operator as it does., Comment: Final version to appear in PRL

This article concerns the time-dependent Hartree-Fock (TDHF) approximation of single-particle dynamics in systems of interacting fermions. We find that the TDHF approximation is accurate when there are sufficiently many particles and the initial many-particle state is any Gibbs equilibrium state for noninteracting fermions (with Slater determinants as a special example). Assuming a bounded two-particle interaction, we obtain a bound on the error of the TDHF approximation, valid for short times. We further show that the error of the TDHF approximation vanishes at all times in the mean field limit., Comment: 18 pages. To appear in Journal of Statistical Physics

This article examines the time-dependent Hartree-Fock (TDHF) approximation of single-particle dynamics in systems of interacting fermions. We find the TDHF approximation to be accurate when there are sufficiently many particles and the initial many-particle state is a Slater determinant, or any Gibbs equilibrium state for noninteracting fermions. Assuming a bounded two-particle interaction, we obtain a bound on the error of the TDHF approximation, valid for short times. We further show that the error of the the TDHF approximation vanishes at all times in the mean field limit., Comment: 20 pages

We deal with the ``nonrelativistic limit'', i.e. the limit c to infinity, where c is the speed of light, of the nonlinear PDE system obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the ``moving charge'' of the spinor. This limit, sometimes also called ``Post-Newtonian'' limit, yields a Schroedinger-Poisson system, where the spin and the magnetic field no longer appear. However, our splitting of the 4-spinor into two 2-spinors preserves the symmetry of "electrons'' and "positrons''; the latter obeying a Schroedinger equation with ``negative mass'' in the limit. We rigorously prove that in the nonrelativistic limit solutions of the Dirac-Maxwell system converge in the energy space $C([0,T];H^{1})$ to solutions of a Schroedinger-Poisson system, under appropriate (convergence) conditions on the initial data. We also prove that the time interval of existence of local solutions of Dirac-Maxwell is bounded from below by log(c). In fact, for this result we only require uniform $H^{1}$ bounds on the initial data, not convergence. Our key technique is "null form estimates'', extending the work of Klainerman and Machedon and our previous work on the nonrelativistic limit of the Klein-Gordon-Maxwell system.

The time-dependent Hartree-Fock equations are derived from the N-particle Schr\"odinger equation with mean-field scaling in the infinite particle limit, for initial data that are like Slater determinants. Only the case of bounded interaction potentials is treated in this work. We prove that, in the infnite particle limit, the first partial trace of the N-particle density operator approaches the solution of the time-dependent Hartree-Fock equations in the trace norm., Comment: 23 pages. This is the final version, to appear in Journal de Mathematiques Pures et Appliquees

We consider the Cauchy-problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of high-frequency asymptotics of such models is reviewed,in particular we highlight the difficulties in crossing caustics when using (time-dependent) WKB-methods. Using Wigner measures we present an alternative approach to such asymptotic problems. We first discuss the connection of the naive WKB solutions to transport equations of Liouville type (with mono-kinetic solutions) in the prebreaking regime. Further we show that the Wigner measure approach can be used to analyze high-frequency limits in the post-breaking regime, in comparison with the traditional Fourier integral operator method. Finally we present some illustrating examples., Comment: 38 pages