Asymptotic behavior of solutions to certain nonlinear Schrödinger-Hartree equations

The asymptotic behavior of solutions to the Cauchy problem for the equation $$i\psi _\imath = \frac{1}{2}\Delta \psi - \upsilon (\psi )\psi , \upsilon = r^{ - 1} *|\psi |^2 ,$$ and for systems of similar form, is studied. It is shown that the norms $$\parallel \psi (t)\parallel _{L_2 (|x| \leqq R)}^2 + \parallel \nabla \psi (t)\parallel _{L_2 (|x| \leqq R)}^2 $$ are integrable in time for any fixedR>0, from which it follows that $$\mathop {\lim }\limits_{t \to \infty } \parallel \psi (t)\parallel _{L_2 (|x| \leqq R)} = 0.$$ \] Nevertheless, it is established that anL ₂-scattering theory is impossible.     

On Lévy (or stable) distributions and the Williams-Watts model of dielectric relaxation

This paper is concerned with the Lévy, or stable distribution function defined by the Fourier transform $$Q_\alpha \left( z \right) = \frac{1}{{2\pi }}\int {_{ - \infty }^\infty \exp \left( { - izu - \left| u \right|^\alpha } \right)du} with 0< \alpha \leqslant 2$$ Whenα=2 it becomes the Gauss distribution function and whenα=1, the Cauchy distribution. Whenα≠2 the distribution has a long inverse power tail $$Q_\alpha \left( z \right) \sim \frac{{\Gamma \left( {1 + \alpha } \right)\sin \tfrac{1}{2}\pi \alpha }}{{\pi \left| z \right|^{1 + \alpha } }}$$ In the regime of smallα, ifα¦logz¦?1, the distribution is mimicked by a log normal distribution. We have derived rapidly converging algorithms for the numerical calculation ofQ _α (z) for variousα in the range 0<α<1. The functionQ _α (z) appears naturally in the Williams-Watts model of dielectric relaxation. In that model one expresses the normalized dielectric parameter as $$ \in _n \left( \omega \right) \equiv \in '_n \left( \omega \right) - i \in ''_n \left( \omega \right) = - \int {_0^\infty e^{ - i\omega t} \left[ {{{d\phi \left( t \right)} \mathord{\left/ {\vphantom {{d\phi \left( t \right)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}}} \right]} dt$$ with $$\phi \left( t \right) = \exp - \left( {{t \mathord{\left/ {\vphantom {t \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)^\alpha $$ It has been found empirically by various authors that observed dielectric parameters of a wide variety of materials of a broad range of frequencies are fitted remarkably accurately by using this form ofφ(t).ε″ _n (ω) is shown to be directly related toQ _α (z). It is also shown that if the Williams-Watts exponential is expressed as a weighted average of exponential relaxation functions $$\exp - \left( {{t \mathord{\left/ {\vphantom {t \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)^\alpha = \int {_0^\infty } g\left( {\lambda , \alpha } \right)e^{ - \lambda t} dt$$ the weight functiong(λ, α) is expressible as a stable distribution. Some suggestions are made about physical models that might lead to the Williams-Watts form ofφ(t).     

Strongly positive semigroups and faithful invariant states

Let (?, τ, ω) denote aW*-algebra ?, a semigroupt>0?τ _t of linear maps of ? into ?, and a faithful τ-invariant normal state ω over ?. We assume that τ is strongly positive in the sense that $$\tau _t (A^ * A) \geqq \tau _t (A)^ * \tau _t (A)$$ for allA∈? andt>0. Therefore one can define a contraction semigroupT on ?= \(\overline {\mathcal{M}\Omega } \) by $$T_t A\Omega = \tau _t (A)\Omega ,{\rm A} \in \mathcal{M},$$ where Ω is the cyclic and separating vector associated with ω. We prove 1. the fixed points ?(τ) of τ are given by ?(τ)=?∩T′=?∩E′, whereE is the orthogonal projection onto the subspace ofT-invariant vectors, 2. the state ω has a unique decomposition into τ-ergodic states if, and only if, ?(τ) or {?υE}′ is abelian or, equivalently, if (?, τ, ω) is ?-abelian, 3. the state ω is τ-ergodic if, and only if, ?υE is irreducible or if $$\mathop {\inf }\limits_{\omega '' \in Co\omega 'o\tau } \left\| {\omega '' - \omega '} \right\| = 0$$ for all normal states ω′ where Coω′°τ denotes the convex hull of {ω′°τ _t } _t>0. Subsequently we assume that τ is 2-positive,T is normal, andT* _t ?₊Ω \( \subseteqq \overline {\mathcal{M}_ + \Omega } \) , and then prove 4. there exists a strongly positive semigroup |τ| which commutes with τ and is determined by $$\left| \tau \right|_t \left( A \right)\Omega = \left| {T_t } \right|A\Omega ,$$ 5. results similar to 1 and 2 apply to |τ| but the τ-invariant state ω is |τ|-ergodic if, and only if, $$\mathop {\lim }\limits_{t \to \infty } \left\| {\omega 'o\tau _t - \omega } \right\| = 0$$ for all normal states ω′.     

Time boundedness of the energy for the charge transfer model

The energy behavior of the time-dependent Schrödinger equation $$i\frac{\partial }{{\partial t}}\psi = \frac{{ - 1}}{{2m}}\Delta \psi + \sum\limits_{j = 1}^N {V_j } (x - y_j (t))\psi $$ is discussed, where they _j (t) are trajectories of classical scattering. In particular, we prove that the energy cannot become arbitrarily large ast→∞.     

Inelastic Character of Solitons of Slowly Varying gKdV Equations

In this paper we study soliton-like solutions of the variable coefficients, the subcritical gKdV equation $$u_t + (u_{xx} -\lambda u + a(\varepsilon x) u^m )_x =0,\quad {\rm in} \quad \mathbb{R}_t\times\mathbb{R}_x, \quad m=2,3\,\, { \rm and }\,\, 4,$$ with ${\lambda\geq 0, a(\cdot ) \in (1,2)}$ a strictly increasing, positive and asymptotically flat potential, and ${\varepsilon}$ small enough. In previous works (Mu?oz in Anal PDE 4:573?C638, 2011; On the soliton dynamics under slowly varying medium for generalized KdV equations: refraction vs. reflection, SIAM J. Math. Anal. 44(1):1?C60, 2012) the existence of a pure, global in time, soliton u(t) of the above equation was proved, satisfying $$\lim_{t\to -\infty}\|u(t) - Q_1(\cdot -(1-\lambda)t) \|_{H^1(\mathbb{R})} =0,\quad 0\leq \lambda<1,$$ provided ${\varepsilon}$ is small enough. Here R(t, x) := Q _c (x ? (c ? ??)t) is the soliton of R _t +? (R _xx ??? R + R ^m ) _x =?0. In addition, there exists ${\tilde \lambda \in (0,1)}$ such that, for all 0?<??? <?1 with ${\lambda\neq \tilde \lambda}$ , the solution u(t) satisfies $$\sup_{t\gg \frac{1}{\varepsilon}}\|u(t) - \kappa(\lambda)Q_{c_\infty}(\cdot-\rho(t)) \|_{H^1(\mathbb{R})}\lesssim \varepsilon^{1/2}.$$ Here ${{\rho'(t) \sim (c_\infty(\lambda) -\lambda)}}$ , with ${{\kappa(\lambda)=2^{-1/(m-1)}}}$ and ${{c_\infty(\lambda)>\lambda}}$ in the case ${0<\lambda<\tilde\lambda}$ (refraction), and ${\kappa(\lambda) =1}$ and c _??(??)?<??? in the case ${\tilde \lambda<\lambda<1}$ (reflection). In this paper we improve our preceding results by proving that the soliton is far from being pure as t ?? +???. Indeed, we give a lower bound on the defect induced by the potential a(·), for all ${{0<\lambda<1, \lambda\neq \tilde \lambda}}$ . More precisely, one has $$\liminf_{t\to +\infty}\| u(t) - \kappa_m(\lambda)Q_{c_\infty}(\cdot-\rho(t)) \|_{H^1(\mathbb{R})}>rsim \varepsilon^{1 +\delta},$$ for any ${{\delta>0}}$ fixed. This bound clarifies the existence of a dispersive tail and the difference with the standard solitons of the constant coefficients, gKdV equation.     

Conformal properties of nonpeeling vacuum space-times

In a previous paper we investigated a class ofnonpeeling asymptotic vacuum solutions which were shown to admit finite expressions for the Winicour-Tamburino energy-momentum and angular momentum integrals. These solutions have the property that $$\psi _0 = O(r^{ - 3 - \in _0 } ), \in _0 \leqslant 2$$ and $$\psi _1 = O(r^{ - 3 - \in _1 } ), \in _1< \in _0 and \in _1< 1$$ withψ ₂,ψ ₃, andψ ₄ having the same asymptotic behavior as they do for peeling solutions. The above investigation was carried out in the physical space-time. In this paper we examine the conformal properties of these solutions, as well as the more general Couch-Torrence solutions, which include them as a subclass. For the Couch-Torrence solutions $$\begin{gathered} \psi _0 = O(r^{ - 2 - \in _0 } ) \hfill \\ \psi _1 = O(r^{ - 2 - \in _1 } ), \in _1< \in _0 {\text{ }}and \in _1 \leqslant 2 \hfill \\ \end{gathered} $$ and , $$\psi _2 = O(r^{ - 2 - \in _2 } ),{\text{ }} \in _2< \in _1 {\text{ }}and \in _2 \leqslant 1$$ withψ ₃ andψ ₄ behaving as they do for peeling solutions. It is our purpose to determine how much of the structure generally associated with peeling space-times is preserved by the nonpeeling solutions. We find that, in general, a three-dimensional null boundary (?⁺) can be defined and that the BMS group remains the asymptotic symmetry group. For the general Couch-Torrence solutions several physically and/or geometrically interesting quantities     

The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension: $$ i\frac{\rm d}{{\rm d}t}\psi^{\varepsilon}(t)=-\Delta\psi^{\varepsilon}(t) +\frac{1}{\varepsilon}V\left(\frac{x}{\varepsilon} \right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t)\quad \varepsilon > 0\,\quad V\in L^1(\mathbb{R},(1+|x|){\rm d}x) \cap L^\infty(\mathbb{R}).$$ This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit \({\varepsilon\to 0}\) the weak (integral) dynamics converges in \({H^1(\mathbb{R})}\) to the weak dynamics of the NLS with point-concentrated nonlinearity: $$ i\frac{{\rm d}}{{\rm d}t} \psi(t) =H_{\alpha} \psi(t) .$$ where H _α is the Laplacian with the nonlinear boundary condition at the origin \({\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)}\) and \({\alpha=\int_{\mathbb{R}}V{\rm d}x}\) . The convergence occurs for every \({\mu\in \mathbb{R}^+}\) if V ≥  0 and for every  \({\mu\in (0,1)}\) otherwise. The same result holds true for a nonlinearity with an arbitrary number N of concentration points.     

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

In this paper, we study the singularly perturbed Laguerre unitary ensemble $$\frac{1}{Z_n} ({\rm det}\,\, M)^\alpha e^{- {\rm tr}\, V_t(M)}dM, \qquad \alpha > 0,$$ with \({V_t(x) = x + t/x,\,\, x \in (0,+\infty)}\) and t >  0. Due to the effect of t/x for varying t, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves \({\psi}\) -functions associated with a special solution to a new third-order nonlinear differential equation, which is then shown to be equivalent to a particular Painlevé III equation. The transition of this limiting kernel to the Bessel and Airy kernels is also studied when the parameter t changes in a finite interval (0, d]. Our approach is based on Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.     

Flux Fluctuations in the One Dimensional Nearest Neighbors Symmetric Simple Exclusion Process

Let J(t) be the the integrated flux of particles in the symmetric simple exclusion process starting with the product invariant measure ν _ρ with density ρ. We compute its rescaled asymptotic variance: $$\mathop {\lim }\limits_{t \to \infty } t^{ - 1/2} \mathbb{V}J(t) = \sqrt {2/\pi } (1 - \rho )\rho$$ Furthermore we show that t ^?1/4 J(t) converges weakly to a centered normal random variable with this variance. From these results we compute the asymptotic variance of a tagged particle in the nearest neighbor case and show the corresponding central limit theorem.     

Limit behavior of saturated approximations of nonlinear Schrödinger equation

We consider the solutionu _?(t) of the saturated nonlinear Schrödinger equation (1) $$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} u + \varepsilon \left| u \right|^{q - 1} uandu(0,.) = \varphi (.)$$ where \(N \geqslant 2,\varepsilon > 0,1 + 4/N< q< (N + 2)/(N - 2),u:\mathbb{R} \times \mathbb{R}^N \to \mathbb{C},\varphi \) , ? is a radially symmetric function inH ¹(R ^N ). We assume that the solution of the limit equation is not globally defined in time. There is aT>0 such that \(\mathop {\lim }\limits_{t \to T} \left\| {u(t)} \right\|_{H^1 } = + \infty \) , whereu(t) is solution of (1) $$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} uandu(0,.) = \varphi (.)$$ For ?>0 fixed,u _?(t) is defined for all time. We are interested in the limit behavior as ?→0 ofu _?(t) fort≥T. In the case where there is no loss of mass inu _? at infinity in a sense to be made precise, we describe the behavior ofu _? as ? goes to zero and we derive an existence result for a solution of (1) after the blow-up timeT in a certain sense. Nonlinear Schrödinger equation with supercritical exponents are also considered.     

Asymptotic observables on scattering states

In quantum mechanical potential scattering theory we use selected observables to describe the asymptotic properties of scattering states for long times. E.g., we show for the position and momentum operators that for Ωε?^cont(H), $$\left( {m\frac{x}{t} - p} \right)e^{ - iHt} \Psi \to 0,$$ and that the set of outgoing states is absorbing. This is obtained easily without any detailed analysis of the interacting time evolution. The class of forces includes highly singular and very long range potentials. The results may serve as an intermediate step in a proof of asymptotic completeness; as a particular application we present a simple proof of completeness for Coulomb systems.     

Probability Law for the Euclidean Distance Between Two Planar Random Flights

We consider two independent symmetric Markov random flights Z ₁(t) and Z ₂(t) performed by the particles that simultaneously start from the origin of the Euclidean plane $\mathbb{R}^{2}$ in random directions distributed uniformly on the unit circumference S ₁ and move with constant finite velocities c ₁>0, c ₂>0, respectively. The new random directions are taking uniformly on S ₁ at random time instants that form independent homogeneous Poisson flows of rates λ ₁>0, λ ₂>0. The probability distribution function $\varPhi(r,t)= \operatorname{Pr} \{ \rho(t)<r \}$ of the Euclidean distance $$\rho(t)=\big\Vert \mathbf{Z}_1(t) - \mathbf{Z}_2(t) \big\Vert , \quad t>0, $$ between Z ₁(t) and Z ₂(t) at arbitrary time instant t>0, is derived. Asymptotics of Φ(r,t), as r→0, and a numerical example are also given.     

Supremum of the Airy2 Process Minus a Parabola on a Half Line

Let $\mathcal {A}_{2}(t)$ be the Airy₂ process. We show that the random variable $$\sup_{t\leq\alpha} \bigl\{\mathcal {A}_2(t)-t^2 \bigr\}+\min\{0,\alpha \}^2 $$ has the same distribution as the one-point marginal of the Airy_2→1 process at time α. These marginals form a family of distributions crossing over from the GUE Tracy-Widom distribution F _GUE(x) for the Gaussian Unitary Ensemble of random matrices, to a rescaled version of the GOE Tracy-Widom distribution F _GOE(4^1/3 x) for the Gaussian Orthogonal Ensemble. Furthermore, we show that for every α the distribution has the same right tail decay $e^{-\frac{4}{3} x^{3/2} }$ .     

Dispersionskorrektur Δf′ für Nickel nahe derK-absorptionskante

Employing an X-ray interferometer the dispersion correction Δf′ of the forward atomic scattering amplitude \(f = z + \Delta f' + \Delta f''\) of Nickel is measured using CuK \(\bar \alpha \) -radiation. Since \(\lambda _{CuK\bar \alpha } /\lambda _{K - edge Ni} = 1.037 \approx 1\) a value of Δf′ close to theK-absorption edge is obtained. The result $$\Delta f\prime = - 1.45 \pm 7.6\% $$ is compared with experimental values by Doan, Kiessing and Lameris and Prins and also theoretical values by Hönl, Dauben and Templeton and Cromer. The present value is considerably smaller than all theoretical estimates. It is the most accurate value measured so far and-except for the value measured by Kiessing-aggrees with the other experimental values within their (comparatively large) error limits.     

Positivity and monotonicity properties ofC 0-semigroups. II

IfS _t =exp{?tH},T _t =exp{?tK}, are self-adjoint positivity preserving semigroups on a Hilbert space ?=L ²(X; dμ) we write (*) $$T_t \succ 0$$ ifT _t is positivity improving and (**) $$S_t \succ T_t $$ if the differenceS _t ?T _t is positivity improving. We derive a variety of characterizations of (*) and (**). In particular (*) is valid for allt>0 if, and only if,T _t ∪L ^∞ (X; dμ) is irreducible for somet>0. Similarly if the semigroups are ordered the strict order (**) is valid if, and only if, {S _t ?T _t }∪L ^∞(X; dμ) is irreducible for somet>0. These criteria are used to prove that if (*) is valid for allt>0 then $$e^{ - tf(K)} \succ 0,t > 0,$$ and if (**) is valid for allt>0 then $$e^{ - tf(H)} \succ e^{ - tf(K)} ,t > 0$$ for each non-constantf in the class characterized in the preceding paper. We discuss the decomposition of positivity preserving semigroups in terms of positivity improving semigroups on subspaces. Various applications to monotonicity properties of Green's functions are given.     

The isosinglet and isovector ππ scattering lengths

The ππ scattering lengthsa ₀ ⁰ ,a ₂ ⁰ anda ₁ ¹ are determined from πN elastic scattering data using interior dispersion relations. The importance of the Born-Term contribution, via unitarity, to the imaginary part of all amplitudes is discussed. Proper consideration of these contributions and the analytic properties of the amplitudes near threshold allows us to obtain from the recent πN partial wave analysis of Pietarinen the following scattering lengths $$\begin{gathered} \mu a_0^0 = 0.27 \pm 0.03,\mu ^3 a_1^1 = 0.032 \pm 0.005, \hfill \\ \mu ^5 a_2^0 = 0.002 \pm 0.001. \hfill \\ \end{gathered} $$      

On the freund-witten adelic formula for Veneziano amplitudes

On the basis of the analysis of the adele group (Tate's formula), a regularization for the divergent infinite product ofp-adic Г-functions $$\Gamma _p (\alpha ) = \frac{{1 - p^{\alpha - 1} }}{{[ - p^{ - \alpha } }}$$ is proposed, and the adelic formula is proved $$reg\coprod\limits_{p = 2}^\infty {\Gamma _p (\alpha )} = \frac{{\zeta (\alpha )}}{{\zeta (1 - \alpha )}}$$ whereζ(α) is the Riemannζ-function.     

Metastable states for the Becker-Döring cluster equations

The Becker-Döring equations, in whichc _l (t) can represent the concentration ofl-particle clusters or droplets in (say) a condensing vapour at timet, are $$\begin{array}{*{20}c} {{{dc_l (t)} \mathord{\left/ {\vphantom {{dc_l (t)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = J_{l - 1} (t) - J_l (t)} & {(l = 2,3,...)} \\ \end{array} $$ with $$J_l (t): = a_l c_1 (t)c_l (t) - b_{l + 1} c_{l + 1} (t)$$ and eitherc ₁=const. (‘case A’) or \(\rho : = \sum\limits_1^\infty {lc_l } \) =const. (‘case B’). The equilibrium solutions arec _l =Q _l z ^l , where \(Q_l : = \prod\limits_2^l {({{a_{r - 1} } \mathord{\left/ {\vphantom {{a_{r - 1} } {b_r }}} \right. \kern-0em} {b_r }})} \) . The density of the saturated vapour, defined as \(\rho _s : = \sum\limits_1^\infty {lQ_l z_s ^l } \) , wherez _s is the radius of convergence of the series, is assumed finite. It is proved here that, subject to some further plausible conditions on the kinetic coefficientsa _l andb _l , there is a class of “metastable” solutions of the equations, withc ₁?z _s small and positive, which take an exponentially long time to decay to their asymptotic steady states. (An “exponentially long time” means one that increases more rapidly than any negative power of the given value ofc ₁?z _s (or, in caseB,ρ?ρ _s ) as the latter tends to zero). The main ingredients in the proof are (i) a time-independent upper bound on the solution of the kinetic equations (this upper bound is a steady-state solution of case A of the equations, of the type used in the Becker-Döring theory of nucleation), and (ii) an upper bound on the total concentration of particles in clusters greater than a certain critical size, which (with suitable initial conditions) remains exponentially small until the time becomes exponentially large.     

A local time approach to the self-intersections of Brownian paths in space

We study the Brownian functional $$\alpha (x,B) = \int\limits_B {\int {\delta _x (W_t - W_s )dsdt} } $$ whereW _t is a Brownian path in two or three dimensions. ForB off the diagonal we identify α(x, B) with a local time, and establish the Hölder continuity of α(x, B) in bothx andB.