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.     

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→∞.     

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.     

Existence of excited states for a nonlinear Dirac field

We prove the existence of infinitely many stationary states for the following nonlinear Dirac equation $$i\gamma ^\mu \partial _\mu \psi - m\psi + (\bar \psi \psi )\psi = 0.$$ Seeking for eigenfunctions splitted in spherical coordinates leads us to analyze a nonautonomous dynamical system inR ². The number of eigenfunctions is given by the number of intersections of the stable manifold of the origin with the curve of admissible datum. This proves the existence of infinitely many stationary states, ordered by the number of nodes of each component.     

A global attracting set for the Kuramoto-Sivashinsky equation

New bounds are given for the L²-norm of the solution of the Kuramoto-Sivashinsky equation $$\partial _t U(x,t) = - (\partial _x^2 + \partial _x^4 )U(x,t) - U(x,t)\partial _x U(x,t)$$ , for initial data which are periodic with periodL. There is no requirement on the antisymmetry of the initial data. The result is $$\mathop {\lim \sup }\limits_{t \to \infty } \left\| {U( \cdot ,t)} \right\|_2 \leqslant const. L^{8/5} $$ .     

On concentration of positive bound states of nonlinear Schrödinger equations

We study the concentration behavior of positive bound states of the nonlinear Schrödinger equation $$ih\frac{{\partial \psi }}{{\partial t}} = \frac{{ - h^2 }}{{2m}}\Delta \psi + V\left( x \right)\psi - \gamma \left| \psi \right|^{p - 1} \psi .$$ Under certain condition ofV, we show that positive ground state solutions must concentrate at global minimum points ofV ash→0⁺; moreover, a point at which a sequence of positive bound states concentrates must be a critical point ofV. In cases thatV is radial, we prove that the positive radial solutions with least energy among all nontrivial radial solutions must concentrate at the origin ash→0⁺.     

Existence of localized solutions for a classical nonlinear Dirac field

We prove the existence of stationary states for nonlinear Dirac equations of the form: $$i\gamma ^\mu \partial _\mu \psi - m\psi + F(\bar \psi \psi )\psi = 0.$$ We seek solutions which are separable in spherical coordinates and we use a shooting method for solving the associated problem of ordinary differential equations.     

One-range Addition Theorems for Noninteger n Slater Functions using Complete Orthonormal Sets of Exponential Type Orbitals in Standard Convention

By the use of complete orthonormal sets of ${\psi ^{(\alpha^{\ast})}}$ -exponential type orbitals ( ${\psi ^{(\alpha^{\ast})}}$ -ETOs) with integer (for α ^* = α) and noninteger self-frictional quantum number α ^*(for α ^* ≠ α) in standard convention introduced by the author, the one-range addition theorems for ${\chi }$ -noninteger n Slater type orbitals ${(\chi}$ -NISTOs) are established. These orbitals are defined as follows $$\begin{array}{ll}\psi _{nlm}^{(\alpha^*)} (\zeta ,\vec {r}) = \frac{(2\zeta )^{3/2}}{\Gamma (p_l ^* + 1)} \left[{\frac{\Gamma (q_l ^* + )}{(2n)^{\alpha ^*}(n - l - 1)!}} \right]^{1/2}e^{-\frac{x}{2}}x^{l}_1 F_1 ({-[ {n - l - 1}]; p_l ^* + 1; x})S_{lm} (\theta ,\varphi )\\ \chi _{n^*lm} (\zeta ,\vec {r}) = (2\zeta )^{3/2}\left[ {\Gamma(2n^* + 1)}\right]^{{-1}/2}x^{n^*-1}e^{-\frac{x}{2}}S_{lm}(\theta ,\varphi ),\end{array}$$ where ${x=2\zeta r, 0<\zeta <\infty , p_l ^{\ast}=2l+2-\alpha ^{\ast}, q_l ^{\ast}=n+l+1-\alpha ^{\ast}, -\infty <\alpha ^{\ast} <3 , -\infty <\alpha \leq 2,_1 F_1 }$ is the confluent hypergeometric function and ${S_{lm} (\theta ,\varphi )}$ are the complex or real spherical harmonics. The origin of the ${\psi ^{(\alpha ^{\ast})} }$ -ETOs, therefore, of the one-range addition theorems obtained in this work for ${\chi}$ -NISTOs is the self-frictional potential of the field produced by the particle itself. The obtained formulas can be useful especially in the electronic structure calculations of atoms, molecules and solids when Hartree–Fock–Roothan approximation is employed.     

Layered structure of superfluid4He at supercritical motion

Landau’s criterion plays an important role in the theory of superfluidity. According to this criterion, superfluid motion is possible if \(\tilde \varepsilon \left( p \right) \equiv \varepsilon \left( p \right) + pV > 0\) along the curve of the spectrum?(p) of excitations. For⁴He it means thatv<v _c,v _c≈60 m/sec.v _s is equal to the tangent of the slope to the roton part of the spectrum. The question of what happens to the liquid when this velocity is exceeded, as far as we know, remains unclear. We shall show that for small excesses abovev _c a one-dimensional periodic structure appears in the helium. A wave vector of this structure oriented opposite to the flow and equal toρ _c/h whereρ _c is the momentum at the tangent point. The quantity \(\tilde \varepsilon \left( p \right)\) is the energy of excitation in the liquid moving with velocity v. Inequality of Landau ensures that \(\tilde \varepsilon \) is positive. If \(\tilde \varepsilon \) becomes negative, then the boson distribution function \(n\left( {\tilde \varepsilon } \right)\) becomes negative, indicating the impossibility of thermodynamic equilibrium of the ideal gas of rotons; therefore the interaction between them must be taken into account. The final form of the energy operator is $$\hat H = \int {\left\{ {\hat \psi + \tilde \varepsilon \left( p \right)\hat \psi + \tfrac{g}{2}\hat \psi + \hat \psi + \hat \psi \hat \psi } \right\}} d^3 x, g \sim 2 \cdot 10^{ - 38} erg.cm.$$ Then we can seek the rotonψ-operator in the formψ=ηexp(i p _c r/h), determiningη from the condition that the energy is minimized. The result is (η)²=(v?v _c)ρ _c/g, forv>v _c. The plane waveψ corresponds to a uniform distribution of rotons. It leads, however, to a spatial modulation of the density of the helium, since the density operator \(\hat n\) contains a term which is linear in the operator \(\psi :\hat n = n_0 + \left( {n_0 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} {A \mathord{\left/ {\vphantom {A {\hat \psi \to \hat \psi ^ + }}} \right. \kern-0em} {\hat \psi \to \hat \psi ^ + }}\) ), where |A|²～ρ _c ² /2m?(ρ _c). Finally we find that the density of helium is modulated according to the law $$\frac{{n - n_0 }}{{n_0 }} = \left[ {\frac{{\left| A \right|^2 \left( {\nu - \nu _c } \right)\rho _c }}{{n_0 g}}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \sin \rho _c x \approx 2,6\left[ {\frac{{\nu - \nu _c }}{{\nu _c }}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \sin \rho _c x$$ . This phenomenon can be observed, in principle, in the experiments on scattering ofx-rays in moving helium.     

Monopoles,baryon decay and charge conservation

The mechanism of baryon decay via monopoles is analyzed. For this purpose we quantize isodoublet fermion fields in the presence of a 't Hooft-Polyakov monopole. When the electromagnetic interactions are switched off, we find a condensation of a fermion pair \(\bar \psi _ - ^{(i)} \gamma ^0 \gamma ^5 \psi _ + ^{(i)} \) as well as that of \(\bar \psi _ - ^{(i)} \gamma ^0 \gamma ^5 \psi _ + ^{(i)} \bar \psi _ + ^{(j)} \gamma ^0 \gamma ^5 \psi _ - ^{(j)} \) . Here, the indices ± stand for the electric charge and (i,j) for the flavour. Hence, the charge symmetry is spontaneously broken. However, when the Coulomb interactions are switched on, it is proved that all fermion condensates carrying non-zero electric charges are removed; the condensates carrying zero electric charge, which induce baryon decay in the standardSU (5) model, are not removed by switching on the Coulomb interactions. In these analyses, the key element is the charge mixing boundary condition imposed on the fermion wave-function at the monopole center; the chiral anomaly does not play any role.     

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.     

A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description

We consider the canonical Gibbs measure associated to aN-vortex system in a bounded domain Λ, at inverse temperature \(\widetilde\beta \) and prove that, in the limitN→∞, \(\widetilde\beta \) /N→β, αN→1, where β∈(?8π, + ∞) (here α denotes the vorticity intensity of each vortex), the one particle distribution function ?^N = ? ^N x,x∈Λ converges to a superposition of solutions ? _α of the following Mean Field Equation: $$\left\{ {\begin{array}{*{20}c} {\varrho _{\beta (x) = } \frac{{e^{ - \beta \psi } }}{{\mathop \smallint \limits_\Lambda e^{ - \beta \psi } }}; - \Delta \psi = \varrho _\beta in\Lambda } \\ {\psi |_{\partial \Lambda } = 0.} \\ \end{array} } \right.$$ Moreover, we study the variational principles associated to Eq. (A.1) and prove thai, when β→?8π⁺, either ?_β → δ _{x 0} (weakly in the sense of measures) wherex ₀ denotes and equilibrium point of a single point vortex in Λ, or ?_β converges to a smooth solution of (A.1) for β=?8π. Examples of both possibilities are given, although we are not able to solve the alternative for a given Λ. Finally, we discuss a possible connection of the present analysis with the 2-D turbulence.     

Charged Higgs boson discovery potential at a 500 GeV e+e− linear collider

The discovery potential for charged Higgs bosons has been studied with full-statistics background simulations for $\sqrt s$ =500 GeV and ?=10fb^?1. For the hadronic decay channels $H^ + H^ - \to \operatorname{c} \bar s\bar cs$ , a microvertex detector is crucial for establishing a signal over the $e^ + e^ - \to t\bar t$ background. A combination with a search in the channels $H^ + H^ - \to c\bar s\tau ^ - \nu ,\tau ^ + \nu \tau ^ - \bar \nu$ allows detection sensitivity for charged Higgs bosons up to a mass of about 210GeV, independent of the charged Higgs decay modes. Sensitivity regions in them _A-tanβ parameter space of the Minimal Supersymmetric extention of the Standard Model (MSSM) are given.     

Front solutions for the Ginzburg-Landau equation

We prove the existence of front solutions for the Ginzburg-Landau equation $$\partial _t u(x,t) = \partial _x^2 u(x,t) + (1 - |u(x,t)|^2 )u(x,t)$$ , interpolating between two stationary solutions of the form \(u(x) = \sqrt {1 - q^2 } e^{iqx}\) with different values ofq atx=±∞. Such fronts are shown to exist when at least one of theq is in the Eckhaus-unstable domain.     

Chiral symmetry breaking in the colour dielectric model

Approximating the long-distance gluon dynamics ofSU(3)_colour by colour-dielectric block-spin variables, we obtain an effective QCD theory of a scalar colour-dielectric field and a massive colour-bleached gluon field coupled to light quarks. The massive vector field produces a strong attraction betweenq \(\bar q\) pairs, which leads toq \(\bar q\) condensation when the colour-dielectric field becomes small. We calculate \(\left\langle {\bar \psi \psi } \right\rangle\) and the pion decay constantf _n as a function of the dielectric field expectation value, by evaluating the fermion determinant in a derivative expansion, and integrating out the bosonic variables. We find that the effective quark-gluon coupling,α _s ^eff , including quark effects, is large on the surface of bags, where \(\left\langle {\bar \psi \psi } \right\rangle\) ±0, but decreases inside hadronic bags, where | \(\left\langle {\bar \psi \psi } \right\rangle\) | is decreasing.     

The Isospin Mixing in the X(3872) Decay Spectrum

The large isospin symmetry breaking found in the X(3872) decay is investigated by looking into the transfer strength from the \({{c}\bar{c}}\) quarkonium to the two-meson states: \({c\bar{c} \rightarrow D^{0}\overline{D}^{*0}, D^{+} D^{*-} , J /\psi\omega, {\rm and} \, J /\psi\rho}\) . The widths of the \({\rho}\) and \({\omega}\) mesons are taken into account in the calculation. It is found that very narrow \({J /\psi\omega}\) and \({J /\psi\rho}\) peaks appear at the \({D^{0}\overline{D}^{*0}}\) threshold. These narrow peaks appear provided that the strength of the \({D^{0}\overline{D}^{*0}}\) component is large around the threshold. The large width of the \({\rho}\) meson enhances the isospin-one component in the transfer strength considerably, which reduces the ratio \({{\rm Br}(X \rightarrow J /\psi\omega)/{\rm Br}(X \rightarrow J /\psi\rho)}\) down to 2.5.     

Charmonium and Meson–Molecule Hybrid Tetraquarks

In the X (3872) decay, both of the ${{J/{\psi\pi\pi}}}$ and ${{J/{\psi\pi\pi\pi}}}$ branching fractions are observed experimentally, and their sizes are comparable to each other. In order to clarify the mechanism to cause such a large isospin violation, we investigate X(3872) employing a model of coupled-channel two-meson scattering with a ${{\rm c}\bar{c}}$ core. The two-meson states consist of ${{D^0\overline{D}^{*0}}}$ , D ⁺ D ^*?, ${{J/{\psi\rho}}}$ , and ${{J/{\psi\omega}}}$ . The effects of the ρ and ω meson width are also taken into account. We calculate the transfer strength from the ${{{\rm c}\bar{c}}}$ core to the final two-meson states. It is found that very narrow ${{J/{\psi\rho}}}$ and ${{J/{\psi\omega}}}$ peaks appear very close to the ${{D^0\overline{D}^{*0}}}$ threshold for a wide range of variation in the parameter sets. The size of the ${{J/{\psi\rho}}}$ peak is almost the same as that of ${{J/{\psi\omega}}}$ , which is consistent with the experiments. The large width of the ρ meson makes the originally small isospin violation by about five times larger.     

On boundary integral operators for diffraction problems on graphs with finitely many exits at infinity

We consider a diffraction problem in a multi-connected domain ?² \ Γ, where Γ is an oriented graph with finitely many edges some of which are infinite. The problem is described by the Helmholtz equation (1) $\mathcal{H}u(x) = \rho (x)\nabla \cdot \rho ^{ - 1} (x)\nabla u(x) + k^2 (x)u(x) = 0,x \in \mathbb{R}^2 \backslash \Gamma ,$ where ρ and k are functions bounded together with all derivatives, and by the transmission conditions (2) $u_ + (t) - u_ - (t) = 0,t \in \Gamma \backslash \mathcal{V},$ (3) $a_ + (t)(\partial u/\partial n_t )_ + (t) - a_ - (t)(\partial u/\partial n_t )_ - (t) + a_0 (t)u(t) = f(t),t \in \Gamma \backslash \mathcal{V},$ where V is the set of vertices, a _± and a ₀ are functions bounded on Γ, slowly oscillating discontinuous at the vertices in V, and slowly oscillating at infinity, and f ∈ L ²(Γ). Using Green’s function for the Helmholtz operator H, we introduce simple- and double-layer potentials and reduce the diffraction problem (1)–(3) to a boundary integral equation. The main objective of the paper is to study the essential spectrum, the Fredholm property, and the index of boundary operators on Γ associated with the problem (1)–(3).     

Blow Up Dynamics for Equivariant Critical Schrödinger Maps

For the Schrödinger map equation \({u_t = u \times \triangle u \, {\rm in} \, \mathbb{R}^{2+1}}\) , with values in S ², we prove for any \({\nu > 1}\) the existence of equivariant finite time blow up solutions of the form \({u(x, t) = \phi(\lambda(t) x) + \zeta(x, t)}\) , where \({\phi}\) is a lowest energy steady state, \({\lambda(t) = t^{-1/2-\nu}}\) and \({\zeta(t)}\) is arbitrary small in \({\dot H^1 \cap \dot H^2}\) .     

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