Existence and asymptotics of ground states to the nonlinear Dirac equation with Coulomb potential

In this paper we study the Dirac equation with Coulomb potential − i α · ∇ u + a β u − μ | x | u = f ( x , | u | ) u , x ∈ R 3 where a is a positive constant, μ is a positive parameter, α = ( α 1 , α 2 , α 3 ), α i and β are 4 × 4 Pauli–Dirac matrices. The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. Under some suitable conditions, we prove that the problem possesses a ground state solution which is exponentially decay, and the least energy has continuous dependence about μ. Moreover, we are able to obtain the asymptotic property of ground state solution as μ → 0 + , this result can characterize some relationship of the above problem between μ > 0 and μ = 0.

Ground state solutions of the non-autonomous Schrödinger–Bopp–Podolsky system

In this paper, we consider the following non-autonomous Schrödinger–Bopp–Podolsky system $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + V(x) u + q^2\phi u = f(u)\\ -\Delta \phi + a^2 \Delta ^2 \phi = 4\pi u^2 \end{array}\right. } \hbox { in }{\mathbb {R}}^3. \end{aligned}$$ - Δ u + V ( x ) u + q 2 ϕ u = f ( u ) - Δ ϕ + a 2 Δ 2 ϕ = 4 π u 2 in R 3 . By using some original analytic techniques and new estimates of the ground state energy, we prove that this system admits a ground state solution under mild assumptions on V and f. In the final part of this paper, we give a min-max characterization of the ground state energy.

Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

Fully differentiable optimization protocols for non-equilibrium steady states

In the case of quantum systems interacting with multiple environments, the time-evolution of the reduced density matrix is described by the Liouvillian. For a variety of physical observables, the long-time limit or steady state solution is needed for the computation of desired physical observables. For inverse design or optimal control of such systems, the common approaches are based on brute-force search strategies. Here, we present a novel methodology, based on automatic differentiation, capable of differentiating the steady state solution with respect to any parameter of the Liouvillian. Our approach has a low memory cost, and is agnostic to the exact algorithm for computing the steady state. We illustrate the advantage of this method by inverse designing the parameters of a quantum heat transfer device that maximizes the heat current and the rectification coefficient. Additionally, we optimize the parameters of various Lindblad operators used in the simulation of energy transfer under natural incoherent light. We also present a sensitivity analysis of the steady state for energy transfer under natural incoherent light as a function of the incoherent- light pumping rate.

Saladin, The Latin Crusaders, and Palestine: David Eldridge's Holy Warriors and the ‘Clash of Civilisations’ Propaganda

This article explores representations of the Third Crusade in David Eldridge's play Holy Warriors: A Fantasia on the Third Crusade and History of Violent Struggle in the Holy Lands (2014). It argues that Eldridge tries in some instances to present the Israeli-Palestinian conflict as a legacy of European imperialism in the Middle East and warns against contemporary Western involvement in the region. However, on other occasions, he suggests that Islamic cultures are incompatible with Western values of secular democracy and therefore the two-state solution is more applicable a solution that the one-state settlement. Ultimately, Eldridge shares some of the ideas behind Huntington's theory of the ‘Clash of Civilisations’ and supports Western military action in Muslim-majority countries.