Relativistic Schrödinger Theory and the Hartree–Fock Approach

Within the framework of Relativistic Schrödinger Theory (RST), the scalar two-particle systems with electromagnetic interactions are treated on the basis of a non-Abelian gauge group U(2) which is broken down to the Abelian subgroup U(1)×U(1). In order that the RST dynamics be consistent with the (non-Abelian) Maxwell equations, there arises a compatibility condition which yields cross relationships for the links between the field strengths and currents of both particles such that self-interactions are eliminated. In the non-relativistic limit, the RST dynamics becomes identical to the well-known Hartree–Fock equations (for spinless particles). Consequently the original RST field equations may be considered as the relativistic generalization of the Hartree–Fock equations, and the exchange interactions of the conventional theory (induced by the anti-symmetrization postulate) do reappear here as ordinary gauge interactions due to a broken symmetry.     

A Statistical Approach to the Asymptotic Behavior of a Class of Generalized Nonlinear Schrödinger Equations

A statistical relaxation phenomenon is studied for a general class of dispersive wave equations of nonlinear Schrödinger-type which govern non-integrable, non-singular dynamics. In a bounded domain the solutions of these equations have been shown numerically to tend in the long-time limit toward a Gibbsian statistical equilibrium state consisting of a ground-state solitary wave on the large scales and Gaussian fluctuations on the small scales. The main result of the paper is a large deviation principle that expresses this concentration phenomenon precisely in the relevant continuum limit. The large deviation principle pertains to a process governed by a Gibbs ensemble that is canonical in energy and microcanonical in particle number. Some supporting Monte-Carlo simulations of these ensembles are also included to show the dependence of the concentration phenomenon on the properties of the dispersive wave equation, especially the high frequency growth of the dispersion relation. The large deviation principle for the process governed by the Gibbs ensemble is based on a large deviation principle for Gaussian processes, for which two independent proofs are given.This research was supported in part by grants from the Department of Energy (DE-FG02-99ER25376) and from the National Science Foundation (NSF-DMS-0202309)This research was partially supported by a Mathematical Sciences Postdoctoral Research Fellowship from the National Science Foundation.This research was supported in part by grants from the Department of Energy (DE-FG02-99ER25376) and from the National Science Foundation (NSF-DMS-0207064).     

A New Approach to the Exact Solutions of the Effective Mass Schrödinger Equation

Effective mass Schrödinger equation is solved exactly for a given potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrödinger equation is also solved for the Morse potential transforming to the constant mass Schrödinger equation for a potential. One can also get solution of the effective mass Schrödinger equation starting from the constant mass Schrödinger equation.     

The Hamiltonian Structure of the Nonlinear Schrödinger Equation and the Asymptotic Stability of its Ground States

In this paper we prove that ground states of the NLS which satisfy the sufficient conditions for orbital stability of M. Weinstein, are also asymptotically stable, for seemingly generic equations. The key issue is to prove that a certain coefficient is non-negative because is a square power. We assume that the NLS has a smooth short range nonlinearity. We assume also the presence of a very short range and smooth linear potential, to avoid translation invariance. The basic idea is to perform a Birkhoff normal form argument on the hamiltonian, as in a paper by Bambusi and Cuccagna on the stability of the 0 solution for NLKG. But in our case, the natural coordinates arising from the linearization are not canonical. So we need also to apply the Darboux Theorem. With some care though, in order not to destroy some nice features of the initial hamiltonian.     

Asymptotic Study to the N-Dimensional Radial Schrdinger Equation for the Quark-Antiquark System

Here an asymptotic study to the N-dimensional radial Schrdinger equation for the quark-antiquark interaction potential employing asymptotic iteration method via an ansatz to the wavefunction is carried out. The complete energy spectra of the consigned system is obtained by computing and adding energy eigenvalues for ground state, for large " r" and for small " r". From this analysis, the mass spectra of heavy quarkonia is derived in three dimensions. Our analytical and numerical results are in good correspondence with other experimental and theoretical studies.     

A New Approach to the Relativistic Schrödinger Equation with Central Potential: Ansatz Method

Applying an ansatz to the eigenfunction, we obtain the exact closed-form solutions of the relativistic Schrödinger equation with the potential V(r) = –a/r + b/r^1/2 both in three dimensions and in two dimensions. The restrictions on the parameters of the given potential and the angular momentum quantum number are also presented.     

On the computation of ground state and dynamics of Schrödinger–Poisson–Slater system

In this paper, we deal with the computation of ground state and dynamics of the Schrödinger–Poisson–Slater (SPS) system. To this end, backward Euler and time-splitting pseudospectral methods are proposed for the nonlinear Schrödinger equation with the nonlocal Hartree potential approximated by solving a Poisson equation. The approximation approaches for the Hartree potential include fast convolution algorithms, which are accelerated by using FFT in 1D and fast multipole method (FMM) in 2D and 3D, and sine/Fourier pseudospectral methods. The inconsistency in 0-mode in Fourier pseudospectral approach is pointed out, which results in a significant loss of high-order of accuracy as expected for spectral methods. Numerical comparisons show that in 1D the fast convolution and sine pseudospectral approaches are compatible. While, in 3D the fast convolution approach based on FMM is second-order accurate and the Fourier pseudospectral approach is better than it from both efficiency and accuracy point of view. Among all these approaches, the sine pseudospectral one is the best candidate in the numerics of the SPS system. Finally, we apply the backward Euler and time-splitting sine pseudospectral methods to study the ground state and dynamics of 3D SPS system in different setups.     

Asymptotic Properties of Solutions to 3-Particle Schrödinger Equations

We construct a generalized Fourier transformation ℱ(λ) associated with the 3-body Schr?dinger operator H=−Δ+Σ ^a V ^a (x ^a ) and characterize all solutions of (H−λ)u= 0 in the Agmon–H?rmander space ℬ^* as the image of ℱ(λ)^*. These stationary solutions admit asymptotic expansions in ℬ^* in terms of spherical waves associated with scattering channels. Received: 20 September 2000 / Accepted: 20 May 2001     

A Vector Fokas-Lenells System from the Coupled Nonlinear Schrödinger Equations

With the aid of the spectral gradient method of Fuchssteiner, the compatible pair of Hamiltonian operators for the coupled NLS hierarchy is rediscovered. This result enables us to construct a hierarchy, which contains a vector generalization of Fokas-Lenells system. The vector Fokas-Lenells system is shown to be bi-Hamiltonian and to possess a Lax pair.     

On the Asymptotic Stability of {N}-Soliton Solutions of the Defocusing Nonlinear Schrödinger Equation

We consider the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation for finite density type initial data. Using the \({\overline{\partial}}\) generalization of the nonlinear steepest descent method of Deift and Zhou, we derive the leading order approximation to the solution of NLS for large times in the solitonic region of space–time, \({|x| < 2 t}\), and we provide bounds for the error which decay as \({ t \rightarrow \infty}\) for a general class of initial data whose difference from the non vanishing background possesses a fixed number of finite moments and derivatives. Using properties of the scattering map of NLS we derive, as a corollary, an asymptotic stability result for initial data that are sufficiently close to the N-dark soliton solutions of NLS.     

Solitary waves in the resonant nonlinear Schrödinger equation: Stability and dynamical properties

The stability and dynamical properties of the so-called resonant nonlinear Schrödinger (RNLS) equation, are considered. The RNLS is a variant of the nonlinear Schrödinger (NLS) equation with the addition of a perturbation used to describe wave propagation in cold collisionless plasmas. We first examine the modulational stability of plane waves in the RNLS model, identifying the modifications of the associated conditions from the NLS case. We then move to the study of solitary waves with vanishing and nonzero boundary conditions. Interestingly the RNLS, much like the usual NLS, exhibits both dark and bright soliton solutions depending on the relative signs of dispersion and nonlinearity. The corresponding existence, stability and dynamics of these solutions are studied systematically in this work.     

A General Approach for the Exact Solution of the Schrödinger Equation

The Schrödinger equation is solved exactly for some well known potentials. Solutions are obtained reducing the Schrödinger equation into a second order differential equation by using an appropriate coordinate transformation. The Nikiforov-Uvarov method is used in the calculations to get energy eigenvalues and the corresponding wave functions.     

Asymptotic Stability, Concentration, and Oscillation in Harmonic Map Heat-Flow, Landau-Lifshitz, and Schrödinger Maps on {\mathbb R^2}

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schrödinger flow as special cases) for degree m equivariant maps from ${\mathbb {R}^2}We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schr?dinger flow as special cases) for degree m equivariant maps from \mathbb R²{\mathbb {R}^2} to \mathbb S²{\mathbb {S}^2} . If m ≥ 3, we prove that near-minimal energy solutions converge to a harmonic map as t → ∞ (asymptotic stability), extending previous work (Gustafson et al., Duke Math J 145(3), 537–583, 2008) down to degree m = 3. Due to slow spatial decay of the harmonic map components, a new approach is needed for m = 3, involving (among other tools) a “normal form” for the parameter dynamics, and the 2D radial double-endpoint Strichartz estimate for Schr?dinger operators with sufficiently repulsive potentials (which may be of some independent interest). When m = 2 this asymptotic stability may fail: in the case of heat-flow with a further symmetry restriction, we show that more exotic asymptotics are possible, including infinite-time concentration (blow-up), and even “eternal oscillation”.     

On Asymptotic Stability in Energy Space of Ground States for Nonlinear Schrödinger Equations

We consider nonlinear Schrödinger equations $iu_t +\Delta u +\beta (|u|^2)u=0\, ,\, \text{for} (t,x)\in \mathbb{R}\times \mathbb{R}^d,$ where d ≥ 3 and β is smooth. We prove that symmetric finite energy solutions close to orbitally stable ground states converge to a sum of a ground state and a dispersive wave as t → ∞ assuming the so called the Fermi Golden Rule (FGR) hypothesis. We improve the “sign condition” required in a recent paper by Gang Zhou and I.M.Sigal.     

Thermodynamic Gravity and the Schrödinger Equation

We adopt a formulation of the Mach principle that the rest mass of a particle is a measure of it’s long-range collective interactions with all other particles inside the horizon. As a consequence, all particles in the universe form a ‘gravitationally entangled’ statistical ensemble and one can apply the approach of classical statistical mechanics to it. It is shown that both the Schrödinger equation and the Planck constant can be derived within this Machian model of the universe. The appearance of probabilities, complex wave functions, and quantization conditions is related to the discreetness and finiteness of the Machian ensemble.