Construction of the ground state in nonrelativistic QED by continuous flows

For a nonrelativistic hydrogen atom minimally coupled to the quantized radiation field we construct the ground state projection Pgs by a continuous approximation scheme as an alternative to the iteration scheme recently used by Fröhlich, Pizzo, and the first author [V. Bach, J. Fröhlich, A. Pizzo, Infrared-finite algorithms in QED: The groundstate of an atom interacting with the quantized radiation field, Comm. Math. Phys. (2006), doi: 10.1007/s00220-005-1478-3]. That is, we construct Pgs=limt→∞Pt as the limit of a continuously differentiable family (Pt)_t?0 of ground state projections of infrared regularized Hamiltonians Ht. Using the ODE solved by this family of projections, we show that the norm of their derivative is integrable in t which in turn yields the convergence of Pt by the fundamental theorem of calculus.     

An invariant set in energy space for supercritical NLS in 1D

We consider radial solutions of a mass supercritical monic NLS and we prove the existence of a set, which looks like a hypersurface, in the space of finite energy functions, invariant for the flow and formed by solutions which converge to ground states.     

Infrared renormalization in non-relativistic QED and scaling criticality

We consider a spin- electron in the framework of non-relativistic Quantum Electrodynamics (QED). Let denote the fiber Hamiltonian corresponding to the conserved total momentum of the electron and the photon field, regularized by a fixed ultraviolet cutoff in the interaction term, and an infrared regularization parametrized by 0<σ?1. Ultimately, our goal is to remove the latter by taking σ↘0. We prove that there exists a constant 0<α₀?1 independent of σ>0 such that for all and all values of the finestructure constant 0<α<α₀, there exists a ground state eigenvalue of multiplicity two at the bottom of the essential spectrum. Moreover, we prove that the renormalized electron mass satisfies , uniformly in σ?0, in units where the bare mass has the value 1, and we prove the existence of the renormalized mass in the limit σ↘0. Our analysis uses the isospectral renormalization group method of Bach, Fröhlich, Sigal introduced in [V. Bach, J. Fröhlich, I.M. Sigal, Quantum electrodynamics of confined non-relativistic particles, Adv. Math. 137 (2) (1998) 299-395; V. Bach, J. Fröhlich, I.M. Sigal, Renormalization group analysis of spectral problems in quantum field theory, Adv. Math. 137 (1998) 205-298] and further developed in [V. Bach, T. Chen, J. Fröhlich, I.M. Sigal, Smooth Feshbach map and operator-theoretic renormalization group methods, J. Funct. Anal. 203 (1) (2003) 44-92; V. Bach, T. Chen, J. Fröhlich, I.M. Sigal, The renormalized electron mass in non-relativistic QED, J. Funct. Anal. 243 (2) (2007) 426-535]. The limit σ↘0 determines a scaling-critical (or endpoint type) renormalization group problem, in which the interaction is strictly marginal (of scale-independent size). A main result of this paper is the development of a method that provides rigorous control of the renormalization of a strictly marginal quantum field theory characterized by a non-trivial scaling limit. The key ingredients entering this analysis include a hierarchy of exact algebraic cancelation identities exploiting the spatial and gauge symmetries of the model, and a combination of the isospectral renormalization group method with the strong induction principle.     

Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions

In [T. Duyckaerts, F. Merle, Dynamic of threshold solutions for energy-critical NLS, preprint, arXiv:0710.5915 [math.AP]], T. Duyckaerts and F. Merle studied the variational structure near the ground state solution W of the energy critical NLS and classified the solutions with the threshold energy E(W) in dimensions d=3,4,5 under the radial assumption. In this paper, we extend the results to all dimensions d?6. The main issue in high dimensions is the non-Lipschitz continuity of the nonlinearity which we get around by making full use of the decay property of W.     

Nonindependent splittings and Gibbs states

We discuss a nonindependent (beam) splitting for which the related thinning leaves the class of equilibrium states for a one mode electromagnetic field invariant. The thinning affects only the parameters of the state, showing a nonlinear loss of energy. After the splitting, the energy values of both split parts are independent. This independence is a characteristic property of the geometric distribution, the distribution of energy values in the equilibrium state. Also, we observe that the class of states where the full states of the split parts are independent is formed by the so-called phase states.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 598–605, October, 1998.This research was partially supported by the International Science Foundation under grant No. 96-0698     

Multiplicity of ground states in quantum field models: applications of asymptotic fields

Ground states of Hamiltonian H of quantum field models are investigated. The infimum of the spectrum of H is in the edge of its essential spectrum. By means of the asymptotic field theory, we give a necessary and sufficient condition for that the expectation value of the number operator of ground states is finite, from which we give an upper bound of the multiplicity of ground states of H. Typical examples are massless GSB models and the Pauli-Fierz model with spin 1/2.     

Minimum energy with infinite horizon: From stationary to non-stationary states

We study a non-standard infinite horizon, infinite dimensional linear–quadratic control problem arising in the physics of non-stationary states (see e.g. Bertini et al. (2004, 2005)): finding the minimum energy to drive a given stationary state $\bar{x}=0$ (at time $t=-\infty$) into an arbitrary non-stationary state $x$ (at time $t=0$). This is the opposite to what is commonly studied in the literature on null controllability (where one drives a generic state $x$ into the equilibrium state $\bar{x}=0$). Consequently, the Algebraic Riccati Equation (ARE) associated with this problem is non-standard since the sign of the linear part is opposite to the usual one and since its solution is intrinsically unbounded. Hence the standard theory of AREs does not apply. The analogous finite horizon problem has been studied in the companion paper (Acquistapace and Gozzi, 2017). Here, similarly to such paper, we prove that the linear selfadjoint operator associated with the value function is a solution of the above mentioned ARE. Moreover, differently to Acquistapace and Gozzi (2017), we prove that such solution is the maximal one. The first main result (Theorem 5.8) is proved by approximating the problem with suitable auxiliary finite horizon problems (which are different from the one studied in Acquistapace and Gozzi (2017)). Finally in the special case where the involved operators commute we characterize all solutions of the ARE (Theorem 6.5) and we apply this to the Landau–Ginzburg model.     

Transition states near rank-two saddles: Correlated electron dynamics of helium

The phase space structure in the vicinity of a rank-two saddle is explored both analytically and numerically. In particular, the geometry of electron dynamics in the neighborhood of the rank-two saddle associated with the nonsequentual double ionization of helium is analyzed. As in the rank-one saddle case, codimension-one normally hyperbolic invariant manifolds turn out to control the rates of the correlated electron dynamics within each energy surface. The construction of these manifolds, however, is more involved here and requires the use of pseudo-hyperbolic invariant manifold theory. Two distinct correlated motions occur, the nonsequential double ionization and the nonsequential exchange of electrons. The relative rates of crossing the barrier are related to the Lyapunov exponents. The dynamics associated with the larger of the two Lyapunov exponents predominates.     

On asymptotic stability in energy space of ground states of NLS in 2D

We transpose work by K. Yajima and by T. Mizumachi to prove dispersive and smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 2D. As an application we extend to dimension 2D a result on asymptotic stability of ground states of NLS proved in the literature for all dimensions different from 2.     

Local Stochastic Channeling Theory: Kinetic Functions in the Case of Interaction Between Fast Particles and Lattice Atoms

We investigate the motion of high-energy particles in a crystal with regard to their interaction with the thermal vibrations of the lattice atoms using analytic methods in the theory of Markov processes including the local Fokker–Planck equation. We construct a local matrix of random actions, which is used to introduce the main kinetic functions in the traverse-energy space, namely, the function a() of energy losses due to the dynamic friction and the diffusion function b(). We show that the singularities of the functions a() and b() are related to the distinction between the contributions to the kinetics from particles moving in three different regimes, namely, in the channeling, quasichanneling, and chaotic motion modes.     

Fuzzy expectation and energy states in fuzzy media

This paper is concerned with energy levels and the density of states in fuzzy crystals. The determination of eigenvalue spectrum, or the density of states, for a particle in a fuzzy crystal, is obtained by using the concepts developed in fuzzy statistics. Various analytical results are found, using the fuzzy expectation; these results can be applied to a variety of fields such as decision making under uncertainty, pattern analysis, and quantum mechanics.     

The solution of Lyumkis energy transport model in semiconductor science

The existence and uniqueness of solution to Lyumkis energy transport model is discussed for N+2<p≤q<∞ and 1≤N≤3. Copyright © 2003 John Wiley & Sons, Ltd.     

On the competition of elastic energy and surface energy in discrete numerical schemes

The Γ-limit of certain discrete free energy functionals related to the numerical approximation of Ginzburg–Landau models is analysed when the distance h between neighbouring points tends to zero. The main focus lies on cases where there is competition between surface energy and elastic energy. Two discrete approximation schemes are compared, one of them shows a surface energy in the Γ-limit. Finally, numerical solutions for the sharp interface Cahn–Hilliard model with linear elasticity are investigated. It is demonstrated how the viscosity of the numerical scheme introduces an artificial surface energy that leads to unphysical solutions.      

Ground states of two-component attractive Bose–Einstein condensates I: Existence and uniqueness

We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in ${\mathbb{R}}^2$, where the intraspecies interaction $(?a_1,?a_2)$ and the interspecies interaction ?β are both attractive, $i.e$, $a_1$, $a_2$ and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated $L^2$-critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as $\beta \nearrow {\beta }^{?}=a^{?}+\sqrt{(a^{?}?a_1)(a^{?}?a_2)}$, where $0<a_i<a^{?}:={6w6}_2^2$ ($i=1,2$) is fixed and w is the unique positive solution of $\mathrm{\Delta }w?w+w^3=0$ in ${\mathbb{R}}^2$. The semi-trivial limit behavior of ground states is tackled in the companion paper [12].     

Stable and unstable equilibrium states in a fishery–aquaculture model

We study interactions between fishery and aquaculture using a 3D generalized Lotka–Volterra model, where we assume that the aquaculture production may affect the growth rate in the fish stock and the productivity in harvesting. In addition, input demands from both marine industries may result in effort competition. We identify conditions for the coexistence of a unique equilibrium state inside the first octant of the phase space and equilibrium states on its boundary. Conditions for stability and instability of these states are also given, thus showing the possibility of having bistability. The equilibrium point inside the first octant is stable if the growth impact on fishery from sea farming is below the potential productivity in harvesting. In the complementary case, we have an unstable interior equilibrium, and we may then end up in stable equilibrium states on the boundary, where either the fishery or the aquaculture is wiped out. Recommendations for Resource Managers

• More empirical and theoretical research is needed to reveal types of interrelations between fisheries and aquaculture, and their importance for long run stability between the sectors.

• When designing policies for the aquaculture industries, managers should in particular be aware of possible long‐term harmful effects from aquaculture to fisheries.

• Increased areas for sea farming reduce the relative profitability of the fishery, and if the area increases above a certain level, this could wipe out the fishery.

     

Statistics and probability theory in renewable energy: Teaching and research

In this paper, the key-role and utility of statistics and probability theory in the field of renewable energy are emphasized and illustrated via specific examples. It is demonstrated that renewable energy is a very suitable field to effectively teach and implement many statistical and probabilistic concepts and techniques. From a research point of view, statistical and probabilistic methods have been successfully employed in evaluating renewable energy systems. These methods will continue to be of core interest for the renewable energy sector in the future, as new and more complex renewable energy systems are developed and installed. In this context, some future research directions in relation to the evaluation of renewable energy systems are also presented.     

Productive efficiency and energy use: An historical perspective

The paper discusses long-term trends in relationships between energy use and the overall productive efficiency of the American economy. While total energy consumption grew strongly during the twentieth century, the intensity of energy use (i.e. the energy/GNP ratio) fell persistently much of the time. Thus, there were simultaneous long-term improvements in labor productivity, total factor productivity,and energy productivity. The historical record appears to be at odds with conventional beliefs that gains in productive efficiency depend upon the rising intensity of energy use in production processes. A key role in bringing about these counter-intuitive results is assigned to what is referred to as the energy-technology-productivity nexus, in which the quality of particular energy forms such as electricity and liquid fuels (along with closely linked changes in energy-using technologies) played a critical part in leveraging the overall efficiency of production. As a result of these energy form-dependent improvements in productive efficiency, outputs grew more rapidly than all inputs, including the inputs of energy. The more recent past stands in sharp contrast to the long-term record. While energy efficiency (as measured by energy/GNP) showed strong gains during the late 1970's and early 1980's, the growth in overall productive efficiency was severely retarded. Implications for the future of suggested linkages between the quality of particular energy forms and technological progress are considered.     

Uniqueness and stability of steady states for a predator-prey model in heterogeneous environment

In this paper, we deal with a predator-prey model with diffusion in a heterogeneous environment, and we study the uniqueness and stability of positive steady states as the diffusion coefficient of the predator is small enough.