Narrow Results

In 1983, Berestycki and Lions [Nonlinear scalar field equations I. Existence of a ground state, Arch. Ration. Mech. Anal.82 (1983) 313–346] studied the following elliptic problem:

In this note, we complete the study made in [The elliptic Kirchhoff equation in ℝ^N perturbed by a local nonlinearity, Differential Integral Equations 25 (2012) 543–554] on a Kirchhoff type equation with a Berestycki–Lions nonlinearity. We also correct Theorem 0.6 inside.

We consider a class of $L^2$-supercritical inhomogeneous nonlinear Schrödinger equations in two dimensions

We study the behavior of Hardy-weights for a class of variational quasilinear elliptic operators of $p$-Laplacian type. In particular, we obtain necessary sharp decay conditions at infinity on the Hardy-weights in terms of their integrability with respect to certain integral weights. Some of the results are extended also to nonsymmetric linear elliptic operators. Applications to various examples are discussed as well.

An otherwise free classical particle moving through an extended spatially homogeneous medium with which it may exchange energy and momentum will undergo a frictional drag force in the direction opposite to its velocity with a magnitude which is typically proportional to a power of its speed. We study here the quantum equivalent of a classical Hamiltonian model for this friction phenomenon that was proposed in [11]. More precisely, we study the spectral properties of the quantum Hamiltonian and compare the quantum and classical situations. Under suitable conditions on the infrared behavior of the model, we prove that the Hamiltonian at fixed total momentum has no ground state except when the total momentum vanishes, and that its spectrum is otherwise absolutely continuous.

We consider a model of quantum particles coupled to a massless quantum scalar field, called the massless Nelson model, in a non-Fock representation of the time-zero fields which satisfy the canonical commutation relations. We show that the model has a ground state for all values of the coupling constant even in the case where no infrared cutoff is made. The non-Fock representation used is inequivalent to the Fock one if no infrared cutoff is made.

In this paper, we study dynamics of the ground state and central vortex states in Bose–Einstein condensation (BEC) analytically and numerically. We show how to define the energy of the Thomas–Fermi (TF) approximation, prove that the ground state is a global minimizer of the energy functional over the unit sphere and all excited states are saddle points in linear case, derive a second-order ordinary differential equation (ODE) which shows that time-evolution of the condensate width is a periodic function with/without a perturbation by using the variance identity, prove that the angular momentum expectation is conserved in two dimensions (2D) with a radial symmetric trap and 3D with a cylindrical symmetric trap for any initial data, and study numerically stability of central vortex states as well as interaction between a few central vortices with winding numbers ±1 by a fourth-order time-splitting sine-pseudospectral (TSSP) method. The merit of the numerical method is that it is explicit, unconditionally stable, time reversible and time transverse invariant. Moreover, it conserves the position density, performs spectral accuracy for spatial derivatives and fourth-order accuracy for time derivative, and possesses "optimal" spatial/temporal resolution in the semiclassical regime. Finally we find numerically the critical angular frequency for single vortex cycling from the ground state under a far-blue detuned Gaussian laser stirrer in strong repulsive interaction regime and compare our numerical results with those in the literatures.

Graphene samples are identified as minimizers of configurational energies featuring both two- and three-body atomic-interaction terms. This variational viewpoint allows for a detailed description of ground-state geometries as connected subsets of a regular hexagonal lattice. We investigate here how these geometries evolve as the number $n$ of carbon atoms in the graphene sample increases. By means of an equivalent characterization of minimality via a discrete isoperimetric inequality, we prove that ground states converge to the ideal hexagonal Wulff shape as $n\to \infty$. Precisely, ground states deviate from such hexagonal Wulff shape by at most ${Kn}^{3/4}+\mathrm{\text{o}}(n^{3/4})$ atoms, where both the constant $K$ and the rate $n^{3/4}$ are sharp.

We study the dynamics for the focusing nonlinear Klein–Gordon equation, $u_{tt}-\mathrm{\Delta }u+m^{2}u=V(x)|u{|}^{p-1}u,$ with positive radial potential $V$ and initial data in the energy space. Under suitable assumption on the potential, we establish the existence and uniqueness of the ground state solution. This enables us to define a threshold size for the initial data that separates global existence and blow-up. An appropriate Gagliardo–Nirenberg inequality gives a critical exponent depending on $V$. For subcritical exponent and subcritical energy global existence vs blow-up conditions are determined by a comparison between the nonlinear term of the energy solution and the nonlinear term of the ground state energy. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary domains.

We consider a model of a quantum mechanical system coupled to a (massless) Bose field, called the generalized spin-boson model (A. Arai and M. Hirokawa, J. Funct. Anal. 151 (1997), 455–503), without infrared regularity condition. We define a regularized Hamiltonian H (ν) with a parameter ν ≥ 0 such that H = H (0) is the Hamiltonian of the original model. We clarify a relation between ground states of H (ν) and those of H by formulating sufficient conditions under which weak limits, as ν → 0, of the ground states of H (ν)'s are those of H. We also establish existence theorems on ground states of H (ν) and H under weaker conditions than in the previous paper mentioned above.

The energy levels of excited states of ground state, β- and γ-bands of the lanthanide and actinide even–even nuclei have been studied within the Davydov–Chaban model (for three different types of the potential energy of the β-deformations) and approximations for a small and free triaxiality. It is shown that the approximation with a free triaxiality better describes the spectrum of the excited collective states for considered nuclei.

Let $\mathrm{\Omega }$ be a $C^1$ open bounded domain in ${ℝ}^N$ ($N\ge 3$) with $0\in \partial \mathrm{\Omega }$. Suppose that $\partial \mathrm{\Omega }$ is $C^2$ at $0$ and the mean curvature of $\partial \mathrm{\Omega }$ at $0$ is negative. Consider the following perturbed PDE involving two Hardy–Sobolev critical exponents:

We study the Cauchy problem of an antiferromagnetic spin-1 Bose–Einstein condensate under Ioffe–Pritchard magnetic field $B$. We then address the existence of ground state solutions and characterize the orbit of standing waves.

Finding or estimating the lowest eigenstate of quantum system Hamiltonians is an important problem for quantum computing, quantum physics, quantum chemistry, and material science. Several quantum computing approaches have been developed to address this problem. The most frequently used method is variational quantum eigensolver (VQE). Many quantum systems, and especially nanomaterials, are described using tight-binding Hamiltonians, but until now no quantum computation method has been developed to find the lowest eigenvalue of these specific, but very important, Hamiltonians. We address the problem of finding the lowest eigenstate of tight-binding Hamiltonians using quantum walks. Quantum walks is a universal model of quantum computation equivalent to the quantum gate model. Furthermore, quantum walks can be mapped to quantum circuits comprising qubits, quantum registers, and quantum gates and, consequently, executed on quantum computers. In our approach, probability distributions, derived from wave function probability amplitudes, enter our quantum algorithm as potential distributions in the space where the quantum walk evolves. Our results showed the quantum walker localization in the case of the lowest eigenvalue is distinctive and characteristic of this state. Our approach will be a valuable computation tool for studying quantum systems described by tight-binding Hamiltonians.

We consider two kinds of stability (under a perturbation) of the ground state of a self-adjoint operator: the one is concerned with the sector to which the ground state belongs and the other is about the uniqueness of the ground state. As an application to the Wigner–Weisskopf model which describes one mode fermion coupled to a quantum scalar field, we prove in the massive case the following: (a) For a value of the coupling constant, the Wigner–Weisskopf model has degenerate ground states; (b) for a value of the coupling constant, the Wigner–Weisskopf model has a first excited state with energy level below the bottom of the essential spectrum. These phenomena are nonperturbative.

We prove in dimension n = 3 an asymptotic stability result for ground states of the Nonlinear Schrödinger Equation which contain one internal mode.

The operator-theoretic renormalization group (RG) methods are powerful analytic tools to explore spectral properties of field-theoretical models such as quantum electrodynamics (QED) with non-relativistic matter. In this paper, these methods are extended and simplified. In a companion paper, our variant of operator-theoretic RG methods is applied to establishing the limiting absorption principle in non-relativistic QED near the ground state energy.

The half-lives of deformed nuclei are reported. We consider an angle-dependent potential which yields multipole approximations. We see that the results are in better agreement with the experimental data when the multipole approximation is solely considered for the daughter nuclei.

The second (unphysical) critical charge in the three-body quantum Coulomb system of a nucleus of positive charge $Z$ and mass $m_p$, and two electrons, predicted by Stillinger has been calculated to be equal to $Z_B^{\infty }=0.904854$ and $Z_B^{m_p}=0.905138$ for infinite and finite (proton) mass $m_p$, respectively. It is shown that in both cases, the ground state energy $E(Z)$ (analytically continued beyond the first critical charge $Z_c$, for which the ionization energy vanishes, to $\mathrm{\text{Re}}Z$$<$$Z_c$) has a square-root branch point with exponent 3/2 at $Z=Z_B$ in the complex $Z$-plane. Based on analytic continuation, the second, excited, spin-singlet bound state of negative hydrogen ion H${}^{-}$ is predicted to be at −0.51554 a.u. (−0.51531 a.u. for the finite proton mass $m_p$). The first critical charge $Z_c$ is found accurately for a finite proton mass $m_p$ in the Lagrange mesh method, $Z_c^{m_p}=0.911069724655$.

We consider a two-component dark matter halo (DMH) of a galaxy containing ultra-light axions (ULA) of different mass. The DMH is described as a Bose–Einstein condensate (BEC) in its ground state. In the mean-field (MF) limit, we have derived the integro-differential equations for the spherically symmetrical wave functions of the two DMH components. We studied, numerically, the radial distribution of the mass density of ULA and constructed the parameters which could be used to distinguish between the two- and one-component DMH. We also discuss an interesting connection between the BEC ground state of a one-component DMH and Black Hole temperature and entropy, and Unruh temperature.