Ground states in non-relativistic quantum electrodynamics

The excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay, but such a particle should have a true ground state – one that minimizes the energy and satisfies the Schr?dinger equation. We prove quite generally that this state exists for all values of the fine-structure constant and the ultraviolet cutoff. We also show the same thing for a many-particle system under physically natural conditions. Oblatum 21-IX-2000 & 8-IV-2001?Published online: 18 June 2001     

Gibbs states of the Hopfield model in the regime of perfect memory

Summary We study the thermodynamic properties of the Hopfield model of an autoassociative memory. IfN denotes the number of neurons andM (N) the number of stored patterns, we prove the following results: IfM/N 0 asN , then there exists an infinite number of infinite volume Gibbs measures for all temperaturesT<1 concentrated on spin configurations that have overlap with exactly one specific pattern. Moreover, the measures induced on the overlap parameters are Dirac measures concentrated on a single point and the Gibbs measures on spin configurations are products of Bernoulli measures. IfM/N , asN for small enough, we show that for temperaturesT smaller than someT()<1, the induced measures can have support only on a disjoint union of balls around the previous points, but we cannot construct the infinite volume measures through convergent sequences of measures.Work partially supported by the Commission of the European Communities under contract No. SC1-CT91-0695     

Ground states of a prescribed mean curvature equation

We study the existence of radial ground state solutions for the problem

     

Ground states of one-dimensional antiferromagnetic models with long-range interaction

Theoretical and Mathematical Physics -     

具外部位势分数阶Choquard方程的基态解

研究具外部位势非自治分数阶Choquard方程:{(-?)~su+mu+V(x)u=(1+a(x))(I_α*|u|p)|u|~(p-2)u,x∈R~N u(x)→0,当|x|→∞时,基态解的存在性.利用Nehari流形技巧、集中紧性原理和山路引理得到了基态解的存在性.     

Hierarchical-matrix method for a class of diffusion-dominated partial integro-differential equations

A hierarchical matrix approach for solving diffusion-dominated partial integro-differential problems is presented. The corresponding diffusion-dominated differential operator is discretized by a second-order accurate finite-volume scheme, while the Fredholm integral term is approximated by the trapezoidal rule. The hierarchical matrix approach is used to approximate the resulting algebraic problem and includes the implementation of an efficient preconditioned generalized minimum residue (GMRes) solver. This approach extends previous work on integral forms of boundary element methods by taking into account inherent characteristics of the diffusion-dominated differential operator in the resultant algebraic problem. Numerical analysis estimates of the accuracy and stability of the finite-volume and the trapezoidal rule approximation are presented and combined with estimates of the hierarchical-matrix approximation and with the accuracy of the GMRes iterates. Results of numerical experiments are reported that successfully validate the theoretical accuracy and convergence estimates, and demonstrate the almost optimal computational complexity of the proposed solution procedure.     

Ground states for nonlinear Kirchhoff equations with critical growth

This paper is concerned with the existence, multiplicity and concentration behavior of positive solutions for the critical Kirchhoff-type problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\left(\varepsilon ^2a+\varepsilon b\int _{\mathbb{R }^{3}}|\nabla u|^2\right)\Delta u+V(x)u=u^{2^*-1}+\lambda f(u)&\text{ in}~{\mathbb{R }^{3}},\\ u\in H^1({\mathbb{R }^{3}}), ~u(x)>0&\text{ in}~{\mathbb{R }^{3}}, \end{array}\right. \end{aligned}$$ where $\varepsilon $ and $\lambda $ are positive parameters, and $a,b>0$ are constants, $2^*(=6)$ is the critical Sobolev exponent in dimension three, $V$ is a positive continuous potential satisfying some conditions, and $f$ is a subcritical nonlinear term. We use the variational methods to relate the number of solutions with the topology of the set where $V$ attains its minimum, for all sufficiently large $\lambda $ and small $\varepsilon $ .     

Ground states of a nonlinear curl-curl problem in cylindrically symmetric media

We consider the nonlinear curl-curl problem \({\nabla\times\nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U}\) in \({\mathbb{R}^3}\) related to the Kerr nonlinear Maxwell equations for fully localized monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of \({H({\rm curl};\mathbb{R}^3)}\). Under a cylindrical symmetry assumption corresponding to a photonic fiber geometry on the functions V and \({\Gamma}\) the variational problem can be posed in a symmetric subspace of \({H({\rm curl};\mathbb{R}^3)}\). For a defocusing case \({{\rm sup} \Gamma < 0}\) with large negative values of \({\Gamma}\) at infinity we obtain ground states by the direct minimization method. For the focusing case \({{\rm inf} \Gamma > 0}\) the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator \({\nabla \times \nabla \times +V(x)}\). Examples of cylindrically symmetric functions V are provided for which this holds.     

Ground states for the higher-order dispersion managed NLS equation in the absence of average dispersion

The problem of existence of ground states in higher-order dispersion managed NLS equation is considered. The ground states are stationary solutions to dispersive equations with nonlocal nonlinearity which arise as averaging approximations in the context of strong dispersion management in optical communications. The main result of this note states that the averaged equation possesses ground state solutions in the practically and conceptually important case of the vanishing residual dispersions.     

p-Laplacian Keller–Segel equation: Fair competition and diffusion-dominated cases

This work deals with the aggregation diffusion equation${?}_t\rho ={\mathrm{\Delta }}_p\rho +\lambda \mathrm{div}((K_a?\rho )\rho ),$ where $K_a(x)=\frac{x}{|x{|}^a}$ is an attraction kernel and ${\mathrm{\Delta }}_p$ is the so called p-Laplacian. We show that the domain $a<p(d+1)?2d$ is subcritical with respect to the competition between the aggregation and diffusion by proving the existence of a solution unconditionally with respect to the mass. In the critical case, we show the existence of a solution in a small mass regime for an $L\ln ?L$ initial condition.     

Ground states for nonlinear fractional Choquard equations with general nonlinearities

We study the existence of ground states for the nonlinear Choquard equation driven by fractional Laplacian: where the nonlinearity satisfies the general Berestycki–Lions‐type assumptions. Copyright © 2016 John Wiley & Sons, Ltd.     

Ground states and spectrum of quantum electrodynamics of nonrelativistic particles

A system consisting of finitely many nonrelativistic particles bound on an external potential and minimally coupled to a massless quantized radiation field without the dipole approximation is considered. An ultraviolet cut-off is imposed on the quantized radiation field. The Hamiltonian of the system is defined as a self-adjoint operator in a Hilbert space. The existence of the ground states of the Hamiltonian is established. It is shown that there exist asymptotic annihilation and creation operators. Hence the location of the absolutely continuous spectrum of the Hamiltonian is specified.

     

Ground states for Schrödinger-type equations with nonlocal nonlinearity

The problem of the existence of stable solitary wave solutions for nonlinear Schrödinger-type equations with a generalized cubic nonlinearity is considered. These types of equations have recently arisen in the context of optical communications as averaging approximations to nonlinear dispersive equations with widely separated time scales. In this paper, it is shown that under general conditions on the kernel of the nonlocal term, stable standing wave solutions exist for these equations.     

Ground states for translationally invariant Pauli-Fierz models at zero momentum

We consider the translationally invariant Pauli-Fierz model describing a charged particle interacting with the electromagnetic field. We show under natural assumptions that the fiber Hamiltonian at zero momentum has a ground state.     

Ground states of nonlinear Schrödinger equations with potentials

In this paper we study the nonlinear Schrödinger equation:

We give general conditions which assure the existence of ground state solutions. Under a Nehari type condition, we show that the standard Ambrosetti–Rabinowitz super-linear condition can be replaced by a more natural super-quadratic condition.     

Ground states for Kirchhoff‐type equations with critical or supercritical growth

In this paper, we study the following Kirchhoff‐type equation with critical or supercritical growth where a>0, b>0, λ>0, p≥6 and f is a continuous superlinear but subcritical nonlinearity. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for small λ>0 by Nehari method. Moreover, we regard b as a parameter and obtain a convergence property of the ground state solution as b↘0. Our main contribution is related to the fact that we are able to deal with the case p>6.