Ground State Threshold Research Articles

Well-posedness and scattering for a 2D inhomogeneous NLS with Aharonov-Bohm magnetic potential

We consider the magnetic nonlinear inhomogeneous Schrödinger equationi∂tu−(−i∇+α|x|2(−x2,x1))2u=±|x|−ϱ|u|p−1u,(t,x)∈R×R2, where α∈R∖Z,ϱ>0,p>1. We establish a dichotomy between global existence and scattering versus finite-time blow-up for energy solutions below the ground state threshold in the inter-critical regime. The scattering result is obtained by employing the novel approach developed by Dodson and Murphy (A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, Proc. Am. Math. Soc. (2017)). We employ Tao's scattering criteria and Morawetz estimates as the foundation of this method. The novelty of our approach lies in two aspects: firstly, we explore the case of nonzero ϱα, and secondly, we consider general energy initial data that need not be radially symmetric. The specific instance of α=0, known as INLS, has garnered significant attention in recent years. Additionally, X. Gao and C. Xu recently investigated the case of ϱ=0 in the homogeneous regime, establishing scattering theory for spherically symmetric data in their work (Scattering theory for NLS with inverse-square potential in 2D, J. Math. Anal. Appl. (2020)). In the radial framework, the aforementioned problem can be translated to the INLS with an inverse square potential, which has been extensively studied in higher-dimensional spaces. The Hardy inequality ‖|x|−1f‖L2(RN)≤2N−2‖∇f‖L2(RN), which establishes the norm equivalence ‖f‖H1≃‖f‖H1+‖|x|−1f‖L2, does not hold in two space dimensions. Consequently, it remains unclear how to handle the NLS with an inverse square potential in the H1 framework for two-dimensional space. This article stands out as the first one to address the NLS with Aharonov-Bohm magnetic potential within the inhomogeneous regime, specifically when ϱ≠0.

Scattering of solutions with group invariance for the fourth-order nonlinear Schrödinger equation

In this paper, we consider the focusing, L 2-supercritical and H˙2 -subcritical fourth-order nonlinear Schrödinger equations. We show the scattering of group-invariant solutions below the ground state threshold, under the hypothesis that the threshold for group-invariant solutions is less than a certain value.

Long‐time dynamics for the radial focusing fractional INLS

We consider the following fractional NLS with focusing inhomogeneous power‐type nonlinearity: where , , , and . We prove the ground state threshold of global existence and scattering versus finite time blowup of energy solutions in the inter‐critical regime with spherically symmetric initial data. The scattering is proved by the new approach of Dodson–Murphy. This method is based on Tao's scattering criteria and Morawetz estimates. We describe the threshold using some non‐conserved quantities in the spirit of the recent paper by Dinh. The radial assumption avoids a loss of regularity in Strichartz estimates. The challenge here is to overcome two main difficulties. The first one is the presence of a non‐local fractional Laplacian operator. The second one is the presence of a singular weight in the nonlinearity. The greater part of this paper is devoted to the scattering of global solutions in . The Lorentz spaces and the Strichartz estimates play crucial roles in our approach.

Scattering threshold for a focusing inhomogeneous non-linear Schrödinger equation with inverse square potential

This work studies the three space dimensional focusing inhomogeneous Schrödinger equation with inverse square potential i∂tu−(−Δ+λ|x|2)u+|x|−2τ|u|2(q−1)u=0,u(t,x):R×R3→C.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ i\\partial _{t} u-\\biggl(-\\Delta +\\frac{\\lambda}{ \\vert x \\vert ^{2}}\\biggr)u + \\vert x \\vert ^{-2\ au} \\vert u \\vert ^{2(q-1)}u=0 , \\qquad u(t,x):\\mathbb{R}\ imes \\mathbb{R}^{3}\ o \\mathbb{C}. $$\\end{document} The purpose is to investigate the energy scattering of global inter-critical solutions below the ground state threshold. The scattering is obtained by using the new approach of Dodson-Murphy, based on Tao’s scattering criteria and Morawetz estimates. This work naturally extends the recent paper by J. An et al. (Discrete Contin. Dyn. Syst., Ser. B 28(2): 1046–1067 2023). The threshold is expressed in terms the non-conserved potential energy. As a consequence, it can be given with a classical way with the conserved mass and energy. The inhomogeneous term |x|^{-2tau} for tau >0 guarantees the existence of ground states for lambda geq 0, contrarily to the homogeneous case tau =0. Moreover, the decay of the inhomogeneous term enables to avoid any radial assumption on the datum. Since there is no dispersive estimate of L^{1}to L^{infty} for the free Schrödinger equation with inverse square potential for lambda <0, one restricts this work to the case lambda geq 0.

Scattering for a class of inhomogeneous generalized Hartree equations

ABSTRACT This paper studies the asymptotic behavior of energy solutions to a non-linear generalized Hartree equation. Indeed, in the inter-critical regime, one revisits the scattering versus finite time blow-up of energy solutions with non-necessarily spherically symmetric datum. Here, one uses the new approach due to Dodson and Murphy. The novelty in this work is to express the scattering threshold in terms of some non-conserved quantities. The main result of this note seems to be stronger than the classical scattering versus finite time blow-up dichotomy given in terms of the conserved mass and energy by Holmer and Roudenko. Indeed, as an application, one investigates the scattering in three different regimes: under, at and beyond the ground state threshold. The main result given here, which can be seen as a criteria of scattering versus finite time blow-up of energy solutions, enables us to give a unified approach to deal with the above generalised Hartree equation in different regimes. This paper follows some new ideas presented recently in the classical Schrödinger equation with a local source term, by V.D. Dinh.

Finite time/Infinite time blow-up behaviors for the inhomogeneous nonlinear Schrödinger equation

In this work, we consider the following focusing inhomogeneous nonlinear Schrödinger equation i∂tu+Δu+|x|−b|u|pu=0,(t,x)∈R×RNwith 0<b<min{2,N} and 4−2bN<p<4−2bN−2. Assume that u0∈H1(RN) and beyond the ground state threshold, then we prove the following two statements, (1) when 4−2bN<p<min{4N,4−2bN−2}, or p=4N when b∈(0,4N), then the corresponding solution blows up in finite time; (2) when 4N<p<4−2bN−2, we prove the finite or infinite time blow-up. Moreover, we can further obtain a precise lower bound of infinite time blow-up rate, that is supt∈[0,T]‖∇u(t)‖L2≳Tκ,for someκ>0.To our knowledge, the statement (1) establishes the first finite time blow-up result for this equation in the intercritical case when the initial data u0 does not have finite variance and is non-radial. The statement (2) gives the first result for the infinite time blow-up rate for this equation.

Temperature Evolution of Two-State Lasing in Microdisk Lasers with InAs/InGaAs Quantum Dots.

One-state and two-state lasing is investigated experimentally and through numerical simulation as a function of temperature in microdisk lasers with Stranski-Krastanow InAs/InGaAs/GaAs quantum dots. Near room temperature, the temperature-induced increment of the ground-state threshold current density is relatively weak and can be described by a characteristic temperature of about 150 K. At elevated temperatures, a faster (super-exponential) increase in the threshold current density is observed. Meanwhile, the current density corresponding to the onset of two-state lasing was found to decrease with increasing temperature, so that the interval of current density of pure one-state lasing becomes narrower with the temperature increase. Above a certain critical temperature, ground-state lasing completely disappears. This critical temperature drops from 107 to 37 °C as the microdisk diameter decreases from 28 to 20 μm. In microdisks with a diameter of 9 μm, a temperature-induced jump in the lasing wavelength from the first excited-state to second excited-state optical transition is observed. A model describing the system of rate equations and free carrier absorption dependent on the reservoir population provides a satisfactory agreement with experimental results. The temperature and threshold current corresponding to the quenching of ground-state lasing can be well approximated by linear functions of saturated gain and output loss.

Threshold solutions in the focusing 3D cubic NLS equation outside a strictly convex obstacle

We study the dynamics of the focusing 3d cubic nonlinear Schrödinger equation in the exterior of a strictly convex obstacle at the mass-energy threshold, namely, when EΩ[u0]MΩ[u0]=ER3[Q]MR3[Q] and ‖∇u0‖L2(Ω)‖u0‖L2(Ω)<‖∇Q‖L2(R3)‖Q‖L2(R3), where u0∈H01(Ω) is the initial data, Q is the ground state on the Euclidean space, E is the energy and M is the mass. In the whole Euclidean space Duyckaerts and Roudenko (following the work of Duyckaerts and Merle on the energy-critical problem) have proved the existence of a specific global solution that scatters for negative times and converges to the soliton in positive times. We prove that these heteroclinic orbits do not exist for the problem in the exterior domain and that all solutions at the threshold are globally defined and scatter. This is the first step in the study of the global dynamics of the equation above the ground-state threshold. The main difficulty is to control the position of the center of mass of the solution for large time without the momentum conservation law and the Galilean transformation which are not available for this equation.

Non-radial scattering theory for nonlinear Schrödinger equations with potential

In this paper, we study a class of nonlinear Schrödinger equations (NLS) with potential $$\begin{aligned} i\partial _t u +\Delta u - Vu = \pm |u|^\alpha u, \quad (t,x) \in \mathbb R\times \mathbb R^3, \end{aligned}$$ where $$\frac{4}{3}<\alpha <4$$ and V is a Kato-type potential including the genuine Yukawa potential as a special case. By using variational analysis and interaction Morawetz estimates, we establish a scattering criterion for the equation with non-radial initial data. As a consequence, we prove the energy scattering for the focusing problem with data below the ground state threshold. Our result extends the recent works of Hong (Commun Pure Appl Anal 15(5):1571–1601, 2016) and Hamano and Ikeda (J Evolut Equ 20:1131–1172, 2020). As a by product of the scattering criterion and the concentration-compactness lemma à la P. L. Lions, we study long time dynamics of global solutions to the focusing problem with data at the ground state threshold. Our result is robust and can be applicable to show the energy scattering for the focusing NLS with Coulomb potential.

Scattering for the non-radial energy-critical inhomogeneous NLS

We prove scattering below the ground state threshold for an energy-critical inhomogeneous nonlinear Schrödinger equation in three space dimensions. In particular, we extend results of Cho, Hong, and Lee [8] , [9] from the radial to the non-radial setting.

Global dynamics above the ground state energy for the combined power-type nonlinear Schrödinger equations with energy-critical growth at low frequencies

We consider the combined power-type nonlinear Schrödinger equations with energy-critical growth, and study the solutions slightly above the ground state threshold at low frequencies, so that we obtain a so-called nine-set theory developed by Nakanishi and Schlag.

Blow-up and scattering for the 1D NLS with point nonlinearity above the mass–energy threshold

In this paper, we study the nonlinear Schrödinger equation with focusing point nonlinearity in dimension one. First, we establish a scattering criterion for the equation based on Kenig–Merle’s compactness-rigidity argument. Then we prove the energy scattering below and above the mass-energy threshold. We also describe the dynamics of solutions with data at the ground state threshold. Finally, we prove a blow-up criteria for the equation with initial data with arbitrarily large energy.

Global wellposedness for the energy-critical Zakharov system below the ground state

The Cauchy problem for the Zakharov system in the energy-critical dimension d=4 is considered. We prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a Strichartz estimate for the Schrödinger equation with a potential. More precisely, a Strichartz estimate is proved to hold uniformly for any potential solving the free wave equation with mass below the ground state constraint. The key new ingredient is a bilinear (adjoint) Fourier restriction estimate for solutions of the inhomogeneous Schrödinger equation with forcing in dual endpoint Strichartz spaces.

Dynamics of radial solutions for the focusing fourth-order nonlinear Schrödinger equations

We consider the following class of focusing L 2-supercritical fourth-order nonlinear Schrödinger equations where N ⩾ 2, μ ⩾ 0, and with if N ⩾ 5 and α* = ∞ if N ⩽ 4. By using the localized Morawetz estimates and radial Sobolev embedding, we establish the energy scattering below the ground state threshold for the equation with radially symmetric initial data. We also address the existence of finite time blow-up radial solutions to the equation. In particular, we show a sharp threshold for scattering and blow-up for the equation with radial data. Our scattering result not only extends the one proved by Guo (2016 Commun. PDE 41 185–207), where the scattering was proven for μ = 0, but also provides an alternative simple proof that completely avoids the use of the concentration/compactness and rigidity argument. In the case μ > 0, our blow-up result extends an earlier result proved by Boulenger–Lenzmann (2017 Ann. Sci. Éc. Norm. Supér. 50 503–544), where the finite time blow-up was shown for initial data with negative energy.

The energy-critical nonlinear wave equation with an inverse-square potential

We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocussing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.

Global existence and blowup for a class of the focusing nonlinear Schrödinger equation with inverse-square potential

We consider a class of the focusing nonlinear Schrödinger equation with inverse-square potentiali∂tu+Δu−c|x|−2u=−|u|αu,u(0)=u0∈H1,(t,x)∈R×Rd, where d≥3, 4d≤α≤4d−2 and c≠0 satisfies c>−λ(d):=−(d−22)2. In the mass-critical case α=4d, we prove the global existence and blowup below ground states for the equation with d≥3 and c>−λ(d). In the mass and energy intercritical case 4d<α<4d−2, we prove the global existence and blowup below the ground state threshold for the equation. This extends similar results of [18] and [22] to any dimensions d≥3 and a full range c>−λ(d). We finally prove the blowup below ground states for the equation in the energy-critical case α=4d−2 with d≥3 and c>−d2+4d(d+2)2λ(d).

On the 4D nonlinear Schrödinger equation with combined terms under the energy threshold

In this paper, we consider the longtime dynamics of the solutions to focusing energy-critical Schr\"odinger equation with a defocusing energy-subcritical perturbation term under a ground state energy threshold in four spatial dimension. This extends the results in Miao et al. (Commun Math Phys 318(3):767-808, 2013, The dynamics of the NLS with the combined terms in five and higher dimensions. Some topics in harmonic analysis and applications, advanced lectures in mathematics, ALM34, Higher Education Press, Beijing, pp 265-298, 2015) to four dimension without radial assumption and the proof of scattering is based on the interaction Morawetz estimates developed in Dodson (Global well-posedness and scattering for the focusing, energy-critical nonlinear Schr\"oinger problem in dimension $d=4$ for initial data below a ground state threshold, arXiv:1409.1950), the main ingredients of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.

Theoretical study of laser-mode competition in quantum-dot semiconductor lasers using a self-consistent electro-opto-thermal model

We present a new model for quantum-dot (QD) semiconductor lasers which includes the self-consistent treatment of electrical, optical, and thermal interactions. The approach is developed from the energy balance equation to incorporate the carrier temperature and a three-level rate equation to describe the carrier and photon dynamics within the QD active region operating simultaneously on two longitudinal modes: via ground-state (GS) and first-excited-state (ES) transitions. Using the presented model, the steady-state, small-signal modulation response and transient device characteristics are calculated and analyzed. Similar to the situation of mode competition of laser modes, it is found that the light output power of the GS mode enters the roll-off regime, while the power of the ES mode increases. The model also predicts the negative temperature dependency of the power threshold for the ES mode, which can be understood from the negative temperature-dependent behavior of incomplete clamping of the ES population above the GS threshold. The simulated results are in good agreement with experimental data from the devices reported earlier.

Absolute single-photoionization cross sections ofSe2+: Experiment and theory

Absolute single-photoionization cross-section measurements for ${\mathrm{Se}}^{2+}$ ions were performed at the Advanced Light Source at Lawrence Berkeley National Laboratory using the merged-beams photo-ion technique. Measurements were made at a photon energy resolution of $24\ifmmode\pm\else\textpm\fi{}3$ meV in the photon energy range 23.5--42.5 eV, spanning the ground state and low-lying metastable state ionization thresholds. To clearly resolve the resonant structure near the ground-state threshold, high-resolution measurements were made from 30.0 to 31.9 eV at a photon energy resolution of $6.7\ifmmode\pm\else\textpm\fi{}0.7$ meV. Numerous resonance features observed in the experimental spectra are assigned and their energies and quantum defects tabulated. The high-resolution cross-section measurements are compared with large-scale, state-of-the-art theoretical cross-section calculations obtained from the Dirac Coulomb $R$-matrix method. Suitable agreement is obtained over the entire photon energy range investigated. These results are an experimental determination of the absolute photoionization cross section of doubly ionized selenium and include a detailed analysis of the photoionization resonance spectrum of this ion.

Barrierless proton transfer across weak CH⋯O hydrogen bonds in dimethyl ether dimer.

We present a combined computational and threshold photoelectron photoion coincidence study of two isotopologues of dimethyl ether, (DME - h6)n and (DME - d6)n n = 1 and 2, in the 9-14 eV photon energy range. Multiple isomers of neutral dimethyl ether dimer were considered, all of which may be present, and exhibited varying C-H⋯O interactions. Results from electronic structure calculations predict that all of them undergo barrierless proton transfer upon photoionization to the ground electronic state of the cation. In fact, all neutral isomers were found to relax to the same radical cation structure. The lowest energy dissociative photoionization channel of the dimer leads to CH3OHCH3 (+) by the loss of CH2OCH3 with a 0 K appearance energy of 9.71 ± 0.03 eV and 9.73 ± 0.03 eV for (DME - h6)2 and deuterated (DME - d6)2, respectively. The ground state threshold photoelectron spectrum band of the dimethyl ether dimer is broad and exhibits no vibrational structure. Dimerization results in a 350 meV decrease of the valence band appearance energy, a 140 meV decrease of the band maximum, thus an almost twofold increase in the ground state band width, compared with DME - d6 monomer.