Quadratic Confinement Research Articles

Nonlocal anisotropic interactions of Coulomb type

In this paper, we review some recent results on nonlocal interaction problems. The focus is on interaction kernels that are anisotropic variants of the classical Coulomb kernel. In other words, while preserving the same singularity at zero of the Coulomb kernel, they present preferred directions of interaction. For kernels of this kind and general confinement we will prove existence and uniqueness of minimizers of the corresponding energy. In the case of a quadratic confinement we will review a recent result by Carrillo and Shu about the explicit characterization of minimizers, and present a new proof, which has the advantage of being extendable to higher dimensions. In light of this result, we will re-examine some previous works motivated by applications to dislocation theory in materials science. Finally, we will discuss some related results and open questions.

Existence of Radial Global Smooth Solutions to the Pressureless Euler–Poisson Equations with Quadratic Confinement

We consider the pressureless Euler–Poisson equations with quadratic confinement. For spatial dimension dge 2,,dne 4, we give a necessary and sufficient condition for the existence of radial global smooth solutions, which is formulated explicitly in terms of the initial data. This condition appears to be much more restrictive than the critical-threshold conditions commonly seen in the study of Euler-type equations. To obtain our results, the key observation is that every characteristic satisfies a periodic ODE system, and the existence of a global smooth solution requires the period of every characteristic to be identical.

Exotic states with configuration

In the framework of a non-relativistic quark model, we have calculated the masses of the (Q = c, b, s and ) tetraquark states. Using the linear and quadratic confinement potentials inside a Cornell-type potential along with all possible spin and color configurations, the Schrödinger equation masses of these tetraquark states have been calculated. Based on our numerical analysis, linear confinement and quadratic confinement produce acceptable results. Models that use linear confinement estimate charmonium-like tetraquark mass close to experimental data, as well as bottomonium-like tetraquark mass below the threshold of their meson–meson channels.

Aufbau principle and singlet‐triplet gap in spherical Hooke atoms

AbstractSinglet and triplet spin state energies for three‐dimensional Hooke atoms, that is, electrons in a quadratic confinement, with even number of electrons (2, 4, 6, 8, 10) is discussed using Full‐CI and CASSCF type wavefunctions with a variety of basis sets and considering perturbative corrections up to second order. The effect of the screening of the electron–electron interaction is also discussed by using a Yukawa‐type potential with different values of the Yukawa screening parameter (λee = 0.2, 0.4, 0.6, 0.8, 1.0). Our results show that the singlet state is the ground state for two and eight electron Hooke atoms, whereas the triplet is the ground spin state for 4‐, 6‐, and 10‐electron systems. This suggests the following Aufbau structure 1s < 1p < 1d with singlet ground spin states for systems in which the generation of the triplet implies an inter‐shell one‐electron promotion, and triplet ground states in cases when there is a partial filling of electrons of a given shell. It is also observed that the screening of electron–electron interactions has a sizable quantitative effect on the relative energies of both spin states, specially in the case of two‐ and eight‐electron systems, favoring the singlet state over the triplet. However, the screening of the electron–electron interaction does not provoke a change in the nature of the ground spin state of these systems. By analyzing the different components of the energy, we have gained a deeper understanding of the effects of the kinetic, confinement and electron–electron interaction components of the energy.

Newtonian repulsion and radial confinement: Convergence toward steady state

We investigate the large time behavior of multi-dimensional aggregation equations driven by Newtonian repulsion, and balanced by radial attraction and confinement. In case of Newton repulsion with radial confinement we quantify the algebraic convergence decay rate toward the unique steady state. To this end, we identify a one-parameter family of radial steady states, and prove dimension-dependent decay rate in energy and 2-Wassertein distance, using a comparison with properly selected radial steady states. We also study Newtonian repulsion and radial attraction. When the attraction potential is quadratic it is known to coincide with quadratic confinement. Here, we study the case of perturbed radial quadratic attraction, proving that it still leads to one-parameter family of unique steady states. It is expected that this family to serve for a corresponding comparison argument which yields algebraic convergence toward steady repulsive-attractive solutions.

The equilibrium measure for an anisotropic nonlocal energy

In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies I_alpha defined on probability measures in {mathbb {R}}^n, with nge 3. The energy I_alpha consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for alpha =0 and is anisotropic otherwise, and a quadratic confinement. The two-dimensional case arises in the study of defects in metals and has been solved by the authors by means of complex-analysis techniques. We prove that for alpha in (-1, n-2], the minimiser of I_alpha is unique and is the (normalised) characteristic function of a spheroid. This result is a paradigmatic example of the role of the anisotropy of the kernel on the shape of minimisers. In particular, the phenomenon of loss of dimensionality, observed in dimension n=2, does not occur in higher dimension at the value alpha =n-2 corresponding to the sign change of the Fourier transform of the interaction potential.

Coulomb Gases Under Constraint: Some Theoretical and Numerical Results

We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a large deviations technique known as the Gibbs conditioning principle. The technical part amounts to establishing that the conditioning ensemble is an I-continuity set of the energy. This leads to characterizing the conditioned equilibrium measure as the solution of a modified variational problem. For simplicity, we focus on linear statistics and on quadratic statistics constraints. Finally, we numerically illustrate our predictions and explore cases in which no explicit solution is known. For this, we use a Generalized Hybrid Monte Carlo algorithm for sampling from the conditioned distribution for a finite but large system.

Explicit minimizers of some non-local anisotropic energies: a short proof

In this paper we consider non-local energies defined on probability measures in the plane, given by a convolution interaction term plus a quadratic confinement. The interaction kernel is , , with . This kernel is anisotropic except for the Coulomb case . We present a short compact proof of the known surprising fact that the unique minimizer of the energy is the normalized characteristic function of the domain enclosed by an ellipse with horizontal semi-axis and vertical semi-axis . Letting , we find that the semicircle law on the vertical axis is the unique minimizer of the corresponding energy, a result related to interacting dislocations, and previously obtained by some of the authors. We devote the first sections of this paper to presenting some well-known background material in the simplest way possible, so that readers unfamiliar with the subject find the proofs accessible.

Pressureless Euler with nonlocal interactions as a singular limit of degenerate Navier-Stokes system

We show that weak solutions of degenerate Navier-Stokes equations converge to the strong solutions of the pressureless Euler system with linear drag term, Newtonian repulsion and quadratic confinement. The proof is based on the relative entropy method using the artificial velocity formulation for the one-dimensional Navier-Stokes system.

D̄0D0* (D0D̄0*) system in QCD-improved many body potential**BM and FA acknowledge the support of PU research (D/605/Est.I Sr. 20 Project 2014-15, D/34/Est.1 Sr. 109 Project 2013-14), SG is thankful to the Higher Education Commission (HEC) of Pakistan for its financial support through (17-5-4(Ps3-128) HEC/Sch/2006)

For a system of current interest (composed of charm, anticharm and a pair of light quarks), we show trends in phenomenological implications of QCD-based improvements to a simple quark model treatment. We employ a resonating group method to render this difficult four-body problem manageable. We use a quadratic confinement so as to be able to improve beyond the Born approximation. We report the position of the pole corresponding to the D̄0D0* molecule for the best fit of a model parameter to the relevant QCD simulations. We point out the interesting possibility that the pole can be shifted to 3872 MeV by introducing another parameter I0 that changes the strength of the interaction in this one component of X(3872). The revised value of this second parameter can guide future trends in modeling of the full exotic meson X(3872). We also report the changes with I0 in the S-wave spin averaged cross sections for D̄0D0* ⟶ ωJ/ψ and D̄0D0* ⟶ ρJ/ψ. These cross sections are important regarding the study of QGP (quark gluon plasma).

On the pressureless damped Euler–Poisson equations with quadratic confinement: Critical thresholds and large-time behavior

We analyze the one-dimensional pressureless Euler–Poisson equations with linear damping and nonlocal interaction forces. These equations are relevant for modeling collective behavior in mathematical biology. We provide a sharp threshold between the supercritical region with finite-time breakdown and the subcritical region with global-in-time existence of the classical solution. We derive an explicit form of solution in Lagrangian coordinates which enables us to study the time-asymptotic behavior of classical solutions with the initial data in the subcritical region.

An analytical solution for the graphene electronic spectrum in the presence of external fields and confinement potential

An algebraic solution of [Formula: see text]-dimensional Dirac equation describing the massless Dirac fermions in the presence of both magnetic field perpendicular to the graphene plane and in-plane homogeneous electric field together with a quadratic confinement potential is investigated. The associated energy eigenvalues are obtained within the framework of unitary transformation scheme. It is shown that the decreasing quadratic confinement potential strength leads to the enhancement of the Landau levels.

The Pion Renormalized Light-Cone Wave Function

An approximate light-cone wave function for the pion effective quark-antiquark Fock sector corresponding to a small value of the renormalization group parameter is presented. The approximate wave function is motivated by the LF-holography and the quadratic confinement potential in the front form of Hamiltonian dynamics, which is in harmony with the linear confining potential in the instant form. The pion radius, decay constant and form-factor are also presented.

Scattering for Nonlinear Schrödinger Equation Under Partial Harmonic Confinement

We consider the nonlinear Schrodinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which we infer the existence of wave operators thanks to suitable vector-fields. Conversely, given an initial Cauchy datum, the solution is global in time and asymptotically free, provided that confinement affects one spatial direction only. This stems from anisotropic Morawetz estimates, involving a marginal of the position density.

A Note on the Second Order Universality at the Edge of Coulomb Gases on the Plane

We consider in this note a class of two-dimensional determinantal Coulomb gases confined by a radial external field. As the number of particles tends to infinity, their empirical distribution tends to a probability measure supported in a centered ring of the complex plane. A quadratic confinement corresponds to the complex Ginibre Ensemble. In this case, it is also already known that the asymptotic fluctuation of the radial edge follows a Gumbel law. We establish in this note the universality of this edge behavior, beyond the quadratic case. The approach, inspired by earlier works of Kostlan and Rider, boils down to identities in law and to an instance of the Laplace method.

On Nonlinear Schrödinger-Type Equations with Nonlinear Damping

odinger-type augmented by nonlinear damping terms. We show that nonlinear damping prevents finite time blow-up in several situations, which we describe. We also prove that the presence of a quadratic confinement in all spatial directions drives the solution of our model to zero for large time. In the case without external potential we prove that the solution may not go to zero for large time due to (nontrivial) scattering.

Dynamical effects of QCD in q2 $ \bar{{q}}^{{2}}_{}$ systems

We study the coupling of a tetraquark system to an exchanged meson-meson channel, using a pure gluonic theory based four-quark potential {\em matrix} model which is known to fit well a large number of data points for lattice simulations of different geometries of a four-quark system. We find that if this minimal-area-based potential matrix replaces the earlier used simple Gaussian form for the gluon field overlap factor $f$ in its off-diagonal terms, the resulting $T$-matrix and phase shifts develop an angle dependence whose partial wave analysis reveals $D$ wave and higher angular momentum components in it. In addition to the obvious implications of this result for the meson-meson scattering, this new feature indicates the possibility of orbital excitations influencing properties of meson-meson molecules through a polarization potential. We have used a formalism of the resonating group method, treated kinetic energy and overlap matrices on model of the potential matrix, but decoupled the resulting complicated integral equations through the Born approximation. In this exploratory study we have used a quadratic confinement and not included the spin-dependence; we also used the approximation of equal constituent quark masses.

Classical phase-space approach for coherent matter waves

We investigate a classical phase-space approach of matter-wave propagation based on the truncated Wigner equation (TWE). We show that such a description is suitable for ideal matter waves in quadratic time-dependent confinement as well as for harmonically trapped Bose-Einstein condensates in the Thomas-Fermi regime. In arbitrary interacting regimes, the TWE combined with the moment method yields the low-energy spectrum of a condensate as predicted by independent variational methods. TWE also gives the right breathing-mode frequency for long-ranged interactions decaying as $1/{r}^{2}$ in three dimensions and for a contact potential in two dimensions. Quantum signatures, beyond the TWE, may only be found in the condensate dynamics beyond the regimes of classical phase-space propagation identified here.

Two-body Problems with Confining Potentials

A formalism is presented that allows an asymptotically exact solution of non-relativistic and semi-relativistic two-body problems with infinitely rising confining potentials. We consider both linear and quadratic confinement. The additional short-range terms are expanded in a Coulomb-Sturmian basis. Such kinds of Hamiltonians are frequently used in atomic, nuclear, and particle physics.

Quark models of dibaryon resonances in nucleon-nucleon scattering

We look for $\ensuremath{\Delta}\ensuremath{\Delta}$ and $N\ensuremath{\Delta}$ resonances by calculating $\mathit{NN}$ scattering phase shifts of two interacting baryon clusters of quarks with explicit coupling to these dibaryon channels. Two phenomenological nonrelativistic chiral quark models giving similar low-energy $\mathit{NN}$ properties are found to give significantly different dibaryon resonance structures. In the chiral quark model (ChQM), the dibaryon system does not resonate in the $\mathit{NNS}$ waves, in agreement with the experimental SP07 $\mathit{NN}$ partial-wave scattering amplitudes. In the quark delocalization and color screening model (QDCSM), the $S$-wave $\mathit{NN}$ resonances disappear when the nucleon size $b$ falls below 0.53 fm. Both quark models give an ${\mathit{IJ}}^{P}={03}^{+}\ensuremath{\Delta}\ensuremath{\Delta}$ resonance. At $b=0.52$ fm, the value favored by the baryon spectrum, the resonance mass is 2390 (2420) MeV for the ChQM with quadratic (linear) confinement, and 2360 MeV for the QDCSM. Accessible from the ${}^{3}{D}_{3}^{\mathit{NN}}$ channel, this resonance is a promising candidate for the known isoscalar ABC structure seen more clearly in the $\mathit{pn}\ensuremath{\rightarrow}d\ensuremath{\pi}\ensuremath{\pi}$ production cross section at 2410 MeV in the recent preliminary data reported by the CELSIUS-WASA Collaboration. In the isovector dibaryon sector, our quark models give a bound or almost bound ${}^{5}{S}_{2}^{\ensuremath{\Delta}\ensuremath{\Delta}}$ state that can give rise to a ${}^{1}{D}_{2}^{\mathit{NN}}$ resonance. None of the quark models used have bound $N\ensuremath{\Delta}P$ states that might generate odd-parity resonances.