Hartree Equation Research Articles

Detection of a Planar Tetracoordinate Hydrogen within the Indium Framework by Quantum Dynamics Theory.

In this work, we consider the question how to detect the planar tetracoordination hydrogen geometry which was recently proposed by electronic structure calculations on the In4H+ system (Angew. Chem. Int. Ed. 2024, 63, e202317312; e202400927; e202403214). Keeping the C4v symmetry, a two-dimensional model of In4H+ is designed to build the nonadiabatic Hamiltonian operator with the lowest-lying singlet and triplet states coupled with spin-orbit coupling. The electronic energies in fitting the potential energy matrix are computed at either the MRCI or CCSD (T) level. Having constructed Hamiltonian, the multiconfigurational time-dependent Hartree product method predicts vibrational eigenstates for spectrum and recrossing probability of the proton. These quantum dynamics calculations predict a period of ∼60 fs for the proton recrossing the In4 moiety and further indicate that experimental observation of the D4h geometry for the triplet state is highly probable at room temperature even though the present MRCI and previous CASPT2 calculations (Angew. Chem. Int. Ed. 2024, 63, e202400927) both predict the C4v symmetry. To measure the C4v geometry, a low temperature of ≲30 K could be adopted. Despite the low temperature, the experiment might still miss the C4v geometry due to the nonzero recrossing probability of H at the low kinetic energy region. Further, a new type of hydrogen bonding in the D4h geometry is proposed to explain the interaction between the C4v geometries.

Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to $$D^{1,p}$$-critical quasi-linear static Schrödinger–Hartree equation involving p-Laplacian $$-\Delta _{p}$$

Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to $$D^{1,p}$$-critical quasi-linear static Schrödinger–Hartree equation involving p-Laplacian $$-\Delta _{p}$$

Coulomb Interaction-Driven Entanglement of Electrons on Helium

The generation and evolution of entanglement in many-body systems is an active area of research that spans multiple fields, from quantum information science to the simulation of quantum many-body systems encountered in condensed matter, subatomic physics, and quantum chemistry. Motivated by recent experiments exploring quantum information processing systems with electrons trapped above the surface of cryogenic noble gas substrates, we theoretically investigate the generation of entanglement between two electrons via their unscreened Coulomb interaction. The model system consists of two electrons confined in separate electrostatic traps that establish microwave-frequency quantized states of their motion. We compute the motional energy spectra of the electrons, as well as their entanglement, by diagonalizing the model Hamiltonian with respect to a single-particle Hartree product basis. We also compare our results with the predictions of an effective Hamiltonian. The computational procedure outlined here can be employed for device design and guidance of experimental implementations. In particular, the theoretical tools developed here can be used for fine-tuning and optimization of control parameters in future experiments with electrons trapped above the surface of superfluid helium or solid neon. Published by the American Physical Society 2024

Ground states for mass critical two coupled semi-relativistic Hartree equations with attractive interactions

Ground states for mass critical two coupled semi-relativistic Hartree equations with attractive interactions

The modified scattering of two dimensional semi-relativistic Hartree equations

The modified scattering of two dimensional semi-relativistic Hartree equations

Perturbation limiting behaviors of normalized ground states to focusing mass-critical Hartree equations with Local repulsion

Perturbation limiting behaviors of normalized ground states to focusing mass-critical Hartree equations with Local repulsion

Energy scattering for a 3D Hartree equation with inverse square potential

This work studies the focusing inhomogeneous nonlinear equation of Hartree type \mathrm{i} \partial_{t} u-\mathcal{K}_{\lambda}u +|x|^{-\tau}|u|^{p-2}(\mathcal{J}_{\alpha} *|\cdot|^{-\tau}|u|^{p})u=0, \quad u(t,x):\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{C}. Here, the linear Schrödinger operator reads \mathcal{K}_{\lambda}:= -\Delta+\frac{\lambda}{|x|^{2}} for some \lambda>-\frac{1}{4} . The Riesz potential is \mathcal{J}_{\alpha}(x)=C_{\alpha}|x|^{-(3-\alpha)} for certain 0<\alpha<3 . The singular decaying term |x|^{-\tau} for some \tau>0 gives an inhomogeneous non-linearity. One considers the inter-critical regime, namely, 1+\frac{2(1-\tau)+\alpha}{3}<p<1+{2(1-\tau)+\alpha} . Moreover, one assumes that p\geq2 in order to avoid a singular term |u|^{p-2} . Furthermore, one restricts \lambda>0 because there is no dispersive estimate L^{1}\to L^{\infty} for \lambda<0 . Contrarily to the homogeneous case \tau=0 , for \lambda>0 , there is a ground state which minimizes the associated Gagliardo–Nirenberg-type estimate. The purpose is to investigate the energy scattering of global solutions under the ground state threshold. One uses the method of Dodson–Murphy based on Tao’s scattering criteria and Morawetz estimates. The decay of the inhomogeneous term |x|^{-\tau} avoids any spherically symmetric assumption.

Decay estimates for defocusing energy-critical Hartree equation

Abstract In this paper, we are devoted to establishing the point-wise decay estimates for solution to the 5D defocusing energy-critical Hartree equation with an initial data in H 2 ( R 5 ) ∩ L 1 ( R 5 ) ${H}^{2}\left({\mathbb{R}}^{5}\right)\cap {L}^{1}\left({\mathbb{R}}^{5}\right)$ . We show that the nonlinear solution has the same time decay rate as the linear one. The main new ingredient is that we used the theories of Lorentz spaces to overcome the low power of nonlinearity.

Effects of multiplicative noise on the fractional Hartree equation

This paper is dedicated to radial solutions to the Cauchy problem for the fractional Hartree equation with multiplicative noise. First, we establish a stochastic Strichartz estimate related to the fractional Schrödinger propagator. Local well-posedness for the Cauchy problem is proved by using stochastic and radial deterministic Strichartz estimates. Then, based on Itô’s formula and stopping time arguments, the existence of a global solution is studied. Finally, we investigate the blow-up phenomenon and give a criterion via localized virial estimates.

Scattering threshold for the focusing energy-critical generalized Hartree equation

Abstract This work investigates the asymptotic behavior of energy solutions to the focusing nonlinear Schrödinger equation of Choquard type i ∂ t u + Δ u + ∣ u ∣ p − 2 ( I α * ∣ u ∣ p ) u = 0 , p = 1 + 2 + α N − 2 , N ≥ 3 . i{\partial }_{t}u+\Delta u+{| u| }^{p-2}\left({I}_{\alpha }* {| u| }^{p})u=0,\hspace{1.0em}p=1+\frac{2+\alpha }{N-2},\hspace{1.0em}N\ge 3. Indeed, in the energy-critical spherically symmetric regime, one proves a global existence and scattering versus finite time blow-up dichotomy. Precisely, if the data have an energy less than the ground state one, two cases are possible. If the kinetic energy of the radial data is less than the ground state one, then the solution is global and scatters. Otherwise, if the data have a finite variance or is spherically symmetric and have a finite mass, then the solution is nonglobal. The main difficulty is to deal with the nonlocal source term. The argument is the concentration-compactness-rigidity method introduced by Kenig and Merle (Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645–675). This note naturally complements the work by Saanouni (Scattering theory for a class of defocusing energy-critical Choquard equations, J. Evol. Equ. 21 (2021), 1551–1571), where the scattering of the defocusing energy-critical generalized Hartree equation was obtained.

Pickl’s proof of the quantum mean-field limit and quantum Klimontovich solutions

This paper discusses the mean-field limit for the quantum dynamics of N identical bosons in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{R}}^3$$\\end{document} interacting via a binary potential with Coulomb-type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in Golse and Paul (Commun Math Phys 369:1021–1053, 2019) . Our first main result is a definition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials slightly less general than those considered in Kato (Trans Am Math Soc 70:195–211, 1951). Our second main result is a new operator inequality satisfied by the quantum Klimontovich solution in the case of an interaction potential with Coulomb-type singularity. When evaluated on an initial bosonic pure state, this operator inequality reduces to a Gronwall inequality for a functional introduced in Pickl (Lett Math Phys 97:151-164, 2011), resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation.

Limit behaviors of pseudo-relativistic Hartree equation with power-type perturbations

We consider the following pseudo-relativistic Hartree equations with power-type perturbation,i∂tψ=−Δ+m2ψ−(1|x|⁎|ψ|2)ψ+ε|ψ|p−2ψ, with (t,x)∈R×R3 where 2<p<3, ε>0 and m>0, p=83 can be viewed as a Slater modification. We mainly focus on the normalized ground state solitary waves φε, where ‖φε‖22=N. Firstly, we prove the existence and nonexistence of normalized ground states under L2-subcritical, L2-critical (p=83) and L2-supercritical perturbations. Secondly, we classify perturbation limit behaviors of ground states when ε→0+, and obtain two different blow-up profiles for N=Nc and N>Nc, where Nc be regard as “Chandrasekhar limiting mass”. We prove that 〈φε,−Δφε〉∼ε−23p−4 for N=Nc and 2<p<3, while 〈φε,−Δφε〉∼ε−23p−8 for N>Nc and 83<p<3. Finally, we study the asymptotic behavior for ε→+∞, and obtain an energy limit limε→+∞⁡eε(N)=12mN and a vanishing rate ∫R3|φε|pdx≲ε−1 when N>Nc and 83<p<3.

Orthonormal Strichartz estimate for dispersive equations with potentials

In this paper we prove the orthonormal Strichartz estimates for the higher order and fractional Schrödinger, wave, Klein-Gordon and Dirac equations with potentials. As in the case of the Schrödinger operator, the proofs are based on the smooth perturbation theory by T. Kato. However, for the Klein-Gordon and Dirac equations, we also use a method of the microlocal analysis in order to prove the estimates for wider range of admissible pairs. As applications we prove the global existence of a solution to the higher order or fractional Hartree equation with potentials which describes the dynamics of infinitely many particles. We also give a local existence result for the semi-relativistic Hartree equation with electromagnetic potentials. As another application, the refined Strichartz estimates are proved for higher order and fractional Schrödinger, wave and Klein-Gordon equations.

Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities

Abstract In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation ( − Δ ) s u + μ u = ( I α * F ( u ) ) F ′ ( u ) in R N , ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ &gt; 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ &gt; 0 fixed and the case ∫ R N u 2 = m &gt; 0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ &gt;}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m &gt; 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1-regularity.

Standing waves and global well-posedness for the 2d Hartree equation with a point interaction

We study a class of two-dimensional nonlinear Schrödinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces that are necessary for the study of ground states and evolution flow. We include in our treatment both mass sub-critical and mass critical Hartree non-linearities. Our analysis is two-fold: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy space. The first goal, unlike other treatments emerging in parallel with the present work, is achieved by a nontrivial adaptation of the standard properties of Schwartz symmetrization for the modified Weinstein functional. This produces, among others, modified Gagliardo-Nirenberg type inequalities that allow to efficiently control the non-linearity and obtain well-posedness by energy methods. The evolution flow is proved to be global in time in the defocusing case, and in the focusing and mass sub-critical case. It is also global in the focusing and mass critical case, for initial data that are suitably small in terms of the best Gagliardo-Nirenberg constant.

The Hunt for Perpendicular Magnetic Field Measurements in Plasma

The one consistent technique for remotely estimating the magnetic field in plasma has been Faraday rotation. It is only sensitive to the portion of the vector parallel to the propagation path. We show how to remotely detect the portion of the vector that is perpendicular using a modified measurement. Isolating this electromagnetic propagation wave mode to measure the magnetic field enables us to (i) study more about the magnetic field vector in plasma, (ii) reduce error in total electron content measurements, and (iii) discover new magnetic field information from archived data sets. The Appleton–Hartree equation is used to verify a new approach to calculating the phase change to an electromagnetic wave propagating through a plasma at frequencies larger than the gyrofrequency, the cyclotron frequency, and the upper hybrid frequency. Focusing on the perpendicular propagation modes, the simplified equation for the integrated path effect from a perpendicular magnetic field is calculated. The direction of the perpendicular component is unknown, because the magnetic field is squared. Isolating the magnetic term in the equation with dual frequency waves is shown. We also show how to eliminate the magnetic field contribution to total electron content measurements with a similar approach. In combination with Faraday rotation, the degeneracy of the magnetic field vector direction is reduced to a cone configuration.

A note on the energy critical inhomogeneous Hartree equation

A note on the energy critical inhomogeneous Hartree equation

Existence and Stability of Standing Wavesfor the Fourth-Order Hartree Equation withMixed Dispersion Terms

Existence and Stability of Standing Wavesfor the Fourth-Order Hartree Equation withMixed Dispersion Terms

On the blowup for the generalized Hartree equations

On the blowup for the generalized Hartree equations

Scattering for the Non-Radial Defocusing Nonlinear Inhomogeneous Hartree Equation

Scattering for the Non-Radial Defocusing Nonlinear Inhomogeneous Hartree Equation