Boundedness of total cross-sections in potential scattering

If a spherically symmetric potential belongs to the Rollnik class, i.e., if $$I = \frac{1}{{(4\pi )^2 }}\smallint \frac{{|V(x)||V(x')|}}{{|x - x'|^2 }}d^3 xd^3 x'$$ is finite, the total cross-section is finite, and an explicit bound on this quantity can be given in terms ofI. We also investigate the case of non-spherically symmetric potentials, and show that ifI is less than unity, the total cross-section averaged over the directions of the incident beam at a given energy is finite.     

Model of scalar mesons

A model containing a nonet of scalar mesonsS, a nonet of pseudoscalar mesonsP, and a nonet of baryons is constructed where the mesons enter in the form of the matrixM=e^i(S+iP), Several Lagrangians are introduced such that the mesons get their physical masses, and the decay widths of the scalar mesons are calculated. The model satisfies generalized PCAC. It is found that the coupling constants of the mesonsε(700) andε′(1060) to pions and nucleons satisfy the relations: $$\frac{{|G_{\varepsilon NN} |}}{{|G_{\varepsilon \pi \pi } |}} = 1 \cdot 4,\frac{{|G_{\varepsilon 'NN} |}}{{|G_{\varepsilon '\pi \pi } |}} = 0 \cdot 89,\frac{{|G_{\varepsilon \pi \pi } G_{\varepsilon NN} |}}{{4\pi }} = 4 \cdot 9,{\text{and}}\frac{{|G_{\varepsilon '\pi \pi } G_{\varepsilon 'NN} |}}{{4\pi }} = 0 \cdot 34.$$      

Time decay of solutions to the cauchy problem for time-dependent Schrödinger-Hartree equations

We consider the time-dependent Schrödinger-Hartree equation (1) $$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$ (2) $$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$ where λ≧0 and \(\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \} \) . We show that there exists a unique global solutionu of (1) and (2) such that $$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$ with $$u \in L^\infty (\mathbb{R};L^2 ).$$ Furthermore, we show thatu has the following estimates: $$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$ and $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$      

An analytical expression for the specific capacity of lithium ion cells

The concentration of lithium ions in the cathode of lithium ion cells has been obtained by solving the materials balance equation $$\frac{{\partial c}}{{\partial t}} = \varepsilon ^{1/2} D\frac{{\partial ^2 c}}{{\partial x^2 }} + \frac{{aj_n (1--t_ + )}}{\varepsilon }$$ by Laplace transform. On the assumption that the cell is fully discharged when there are zero lithium ions at the current collector of the cathode, the discharge timet _d is obtained as $$\tau = \frac{{r^2 }}{{\pi ^2 \varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{{r^2 }}\left( {\frac{{\varepsilon ^{1/2} }}{J} + \frac{{r^2 }}{6}} \right)} \right]$$ which, when substituted into the equationC=It _d /M, whereI is the discharge current andM is the mass of the separator and positive electrode, an analytical expression for the specific capacity of the lithium cell is given as $$C = \frac{{IL_c ^2 }}{{\pi {\rm M}D\varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{2}\left( {\frac{{FDc_0 \varepsilon ^{3/2} }}{{I(1 - t_ + )L_c }} + \frac{1}{6}} \right)} \right]$$      

One-dimensional random Ising systems with interaction decayr −(1+ɛ): A convergent cluster expansion

WE consider a one-dimensional random Ising model with Hamiltonian $$H = \sum\limits_{i\ddag j} {\frac{{J_{ij} }}{{\left| {i - j} \right|^{1 + \varepsilon } }}S_i S_j } + h\sum\limits_i {S_i } $$ , where ε>0 andJ _ij are independent, identically distributed random variables with distributiondF(x) such that i) $$\int {xdF\left( x \right) = 0} $$ , ii) $$\int {e^{tx} dF\left( x \right)< \infty \forall t \in \mathbb{R}} $$ . We construct a cluster expansion for the free energy and the Gibbs expectations of local observables. This expansion is convergent almost surely at every temperature. In this way we obtain that the free energy and the Gibbs expectations of local observables areC ^∞ functions of the temperature and of the magnetic fieldh. Moreover we can estimate the decay of truncated correlation functions. In particular for every ε′>0 there exists a random variablec(ω)m, finite almost everywhere, such that $$\left| {\left\langle {s_0 s_j } \right\rangle _H - \left\langle {s_0 } \right\rangle _H \left\langle {s_j } \right\rangle _H } \right| \leqq \frac{{c\left( \omega \right)}}{{\left| j \right|^{1 + \varepsilon - \varepsilon '} }}$$ , where 〈 〉 _H denotes the Gibbs average with respect to the HamiltonianH.     

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.     

General lower bounds for resonances in one dimension

Lower bounds are derived for the magnitude of the imaginary parts of the resonance eigenvalues of a Schrödinger operator $$ - \frac{{d^2 }}{{dx^2 }} + V(x)$$ on the line, depending only on the support and bounds ofV and on the real part of the resonance eigenvalue. For example, if the resonance eigenvalue is denotedE +i?, then there existC and ?₀ depending only on ‖E‖_∞ andE such that if the support ofV is contained in an interval of length ? > ?₀, then $$\left| \varepsilon \right| > \frac{{m^3 \sqrt E }}{{(m + \sqrt E )^2 }}\exp ( - m\ell )(1 - C\ell ^{ - 1} ),$$ wherem‖V(x)?E? _∞ ^1/2 .     

Study of the mechanisms of pre-equilibrium nuclear reactions in the microscopic approach

The mechanisms of pre-equilibrium nuclear reactions are investigated within the Statistical Multistep Direct Process (SMDP) + Statistical Multistep Compound Process (SMCP) formalism. It has been shown that from an analysis of linear part in such dependences as $$\ln \left[ {{{\frac{{d^2 \sigma }}{{d\varepsilon _b d\Omega _b }}} \mathord{\left/ {\vphantom {{\frac{{d^2 \sigma }}{{d\varepsilon _b d\Omega _b }}} {\varepsilon _b^{1/2} }}} \right. \kern-\nulldelimiterspace} {\varepsilon _b^{1/2} }}} \right]upon\varepsilon _b $$ and $$\ln \left[ {{{\frac{{d\sigma ^{SMDP \to SMCP} }}{{d\varepsilon _b }}} \mathord{\left/ {\vphantom {{\frac{{d\sigma ^{SMDP \to SMCP} }}{{d\varepsilon _b }}} {\frac{{d\sigma ^{SMDP} }}{{d\varepsilon _b }}}}} \right. \kern-\nulldelimiterspace} {\frac{{d\sigma ^{SMDP} }}{{d\varepsilon _b }}}}} \right]upon{{U_B } \mathord{\left/ {\vphantom {{U_B } {\left( {E_a - B_b } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {E_a - B_b } \right)}}$$ one can extract information about the type of mechanism (SMDP, SMCP, SMDP→SMCP) and the number of stages of the multistep emission of secondary particles. In the above approach, we have discussed the experimental data for a broad class of reactions in various entrance and exit channels.     

Remarks on the measurement of polarization of6Li-ions

The determination of the polarization of⁶Li-ions is discussed. It is shown, that independent of the reaction mechanism the following relations between the analysing powers for polarized deuterons and polarized⁶Li-ions hold for the⁶Li(d, α)⁴He-reaction: for all scattering angles \(\vartheta : A_{y y}^{(d)} (E, \vartheta ) = A_{y y}^{(Li)} (E, \vartheta )\) for the scattering angle \(\vartheta = \pi /2\) only: $$A_{z z}^{(d)} (E, \vartheta = \pi /2) = A_{z z}^{(Li)} (E, \vartheta = \pi /2)$$ and $$A_{x x - y y}^{(d)} (E, \vartheta = \pi /2) = A_{x x - y y}^{(Li)} (E, \vartheta = \pi /2)$$ . Using these identities the determination of the polarization of⁶Li-beams is reduced to the experimentally well established determination of the polarization of deuterons.     

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.     

Three-loop relation of quark\overline {MS} and pole masses

We calculate, exactly, the next-to-leading correction to the relation between the \(\overline {MS} \) quark mass, \(\bar m\) , and the scheme-independent pole mass,M, and obtain $$\begin{gathered} \frac{M}{{\bar m(M)}} \approx 1 + \frac{4}{3}\frac{{\bar \alpha _s (M)}}{\pi } + \left[ {16.11 - 1.04\sum\limits_{i = 1}^{N_F - 1} {(1 - M_i /M)} } \right] \hfill \\ \cdot \left( {\frac{{\bar \alpha _s (M)}}{\pi }} \right)^2 + 0(\bar \alpha _s^3 (M)), \hfill \\ \end{gathered} $$ as an accurate approximation forN _F?1 light quarks of massesM _i <M. Combining this new result with known three-loop results for \(\overline {MS} \) coupling constant and mass renormalization, we relate the pole mass to the \(\overline {MS} \) mass, \(\bar m\) (μ), renormalized at arbitrary μ. The dominant next-to-leading correction comes from the finite part of on-shell two-loop mass renormalization, evaluated using integration by parts and checked by gauge invariance and infrared finiteness. Numerical results are given for charm and bottom \(\overline {MS} \) masses at μ=1 GeV. The next-to-leading corrections are comparable to the leading corrections.     

Equations of electrodynamics with noninvariant velocity of light

A five-dimensional spaceV ^{5(t, x, c)}, in which the velocity of light c is the fifth coordinate of the event point together with the radius vectorx and time t is is introduced. The equations rot $$rot E = - \frac{1}{c}\frac{{\partial H}}{{\partial t}}; div E = 4\pi \rho ;$$ $$rot H = - \frac{1}{c}\frac{{\partial E}}{{\partial t}} + 4\pi \rho \frac{{o - Nx\dot c/c}}{{c + (1 - N)\dot ct}}; div H = 0,$$ invariant relative to the symmetry group c ₁₆₀₀ ^tf of D'Alembert's equation are constructed in the space V⁵ (E andH are the electric and magnetic fields;ν= dx/dt, c=dc/dt,N and N is a numerical parameter). The transformation properties of the electromagnetic field and the electric charge density ρ are examined for the finite group of transformationsP ₁₀ ^N ×Δ ₁ ^N ·T ₁ ^N . A theorem for adding velocities is formulated. The limiting transition to a Minkowski space M⁴(t,x)?V⁵(t, x,c) is examined.     

Uniqueness and the global Markov property for Euclidean fields: The case of general exponential interaction

The uniqueness and the global Markov property for the regular Gibbs measure corresponding to the interaction $$U_\Lambda (\varphi ): = \lambda \int\limits_\Lambda {d_2 x\int {d\varrho (\alpha ):e^{\alpha \varphi } :_0 (x)} } $$ [forλ>0,d?(α) a probability measure with support in \(( - 2\sqrt {\pi ,} 2\sqrt \pi )\) ] is proved.     

Measurement of the Λ→p μ− ¯v branching ratio

A sample of 1.2× 10⁶ Λ-hyperons was analyzed in order to detect the rare decay mode: Λ →pμ^?¯v. The Λ-hyperons were produced by stoppingK ^?-mesons in the 81 cm Saclay hydrogen bubble chamber at the CERN PS. We obtained for the branching ratio: $$\frac{{\Gamma {\text{(}}\Lambda \to p\mu ^ - \bar \nu )}}{{\Gamma {\text{(}}\Lambda \to {\text{all)}}}} = (1.4 \pm 0.5) \times 10^{ - 4} ,$$ based on 20 events, of which 6 have to be attributed to the background from the reaction $$\Lambda \to p\pi ^ - , \pi ^ - \to \mu ^ - \bar v.$$ The background was determined by a Monte Carlo calculation.     

Radiative Σ± decays and search for neutral currents

The decay modesΣ ^±→nπ ^± γ, Σ ⁺→pγ,Σ ⁺ →pe ⁺ e ^}- were studied in the 81 cm Saclay hydrogen bubble chamber. In the radiative decayΣ ^±→nπ ^± γ only low momentum pions which stop in the chamber were accepted. We obtain the following branching ratios: (1) $$\frac{{\Gamma {\text{(}}\sum ^{\text{ + }} \to n\pi ^ + \gamma , p_{\pi + }^*< 110{\text{ MeV/c)}}}}{{\Gamma {\text{(}}\sum ^{\text{ + }} \to n\pi ^ + )}} = (2.7 \pm 0.5) \times 10^{ - 4} ,$$ (2) $$\frac{{\Gamma {\text{(}}\sum ^ - \to n\pi ^ - \gamma , p_{\pi - }^*< 110{\text{ MeV/c)}}}}{{\Gamma {\text{(}}\sum ^ - \to n\pi ^ - )}} = (1.0 \pm 0.2) \times 10^{ - 4} ,$$ (3) $$\frac{{\Gamma {\text{(}}\sum ^ + \to p\gamma {\text{)}}}}{{\Gamma {\text{(}}\sum ^ + \to p\pi ^0 )}} = (2.1 \pm 0.3) \times 10^{ - 3} ,$$ (4) $$\frac{{\Gamma {\text{(}}\sum ^ + \to pe^ + e^ - {\text{)}}}}{{\Gamma {\text{(}}\sum ^ + \to p\pi ^0 )}} = (1.5 \pm 0.9) \times 10^{ - 5} .$$ The radiative branching ratios (1) and (2) agree well with theoretical calculations and confirm very strongly the assignmentS wave toΣ ^? →nπ ^? andP wave toΣ ⁺→nπ ⁺ decay. The branching ratio (4) is based on 3 events with very low invariant masses of the electron-positron pair, being most probably radiative decays with internal conversion of theγ-ray. Combining (3) and (4) we obtain for the conversion coefficientρ: in agreement with predictions from electrodynamics.     

Perturbation theory for shape resonances and large barrier potentials

We develop a systematic perturbation and resonance theory for the one-dimensional Schrödinger equation of the form $$( - d^2 /dx^2 + U(x) + \lambda V(x) - E)\psi (x) = 0,0 \leqq x< \infty ,$$ where the barrier potentialV(x) is supported only wherex≧1 and is non-negative there, and λ is a real parameter tending to infinity. We prove that every λ=∞ eigenvalue turns into a resonance or an eigenvalue for finite λ.     

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.     

The incompressible limit in nonlinear elasticity

The incompressible limit in nonlinear elasticity is shown to fall under the theory of singular limits of quasilinear symmetric hyperbolic systems developed by Klainerman and Majda. Specifically, initial-value problems for a family of hyperelastic materials with stored energy functions $$W\left( {\frac{{\partial x}}{{\partial X}}} \right) = W_\infty \left( {\frac{{\partial x}}{{\partial X}}} \right) + \lambda ^2 w\left( {\det \frac{{\partial x}}{{\partial X}}} \right)$$ are considered, whereX andx are reference and deformed coordinates respectively. Under the assumption that the elasticity tensor $$A_{kl}^{ij} \equiv \frac{{\partial ^2 W_\infty }}{{\partial \left( {\frac{{\partial x^i }}{{\partial X^k }}} \right)\partial \left( {\frac{{\partial x^j }}{{\partial X^l }}} \right)}}$$ is positive definite near the identity matrix and thatw″(1)>0, the following results are proven for appropriate initial data: i) existence of solutions of the corresponding evolution equations on a time interval independent of λ as λ→∞, and ii) convergence as λ → ∞ of the solutions to a solution of the incompressible elastodynamics equations.     

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 $$\mathop {\lim \sup }\limits_{t \to \infty } \left\| {U( \cdot ,t)} \right\|_2 \leqslant const. L^{8/5} $$ .