A remark on tensor representations

The identity $$\sum\limits_{v = 0} {\left( {\begin{array}{*{20}c} {n + 1} \\ v \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} {n - v} \\ v \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {n - v} \\ {v - 1} \\ \end{array} } \right)} \right] = ( - 1)^n } $$ is proved and, by means of it, the coefficients of the decomposition ofD ₁ ⁿ into irreducible representations are found. It holds: ifD ₁ ⁿ \(\mathop {\sum ^n }\limits_{m = 0} A_{nm} D_m \) , then $$A_{nm} = \mathop \sum \limits_{\lambda = 0} \left( {\begin{array}{*{20}c} n \\ \lambda \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda - 1} \\ \end{array} } \right)} \right].$$      

Eine scheinbare Abschwächung der Lokalitätsbedingung. II

If for a relativistic field theory the expectation values of the commutator (Ω|[A (x),A(y)]|Ω) vanish in space-like direction like exp {? const|(x-y ²|^α/2#x007D; with α>1 for sufficiently many vectors Ω, it follows thatA(x) is a local field. Or more precisely: For a hermitean, scalar, tempered fieldA(x) the locality axiom can be replaced by the following conditions 1. For any natural numbern there exist a) a configurationX(n): $$X_1 ,...,X_{n - 1} X_1^i = \cdot \cdot \cdot = X_{n - 1}^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^1 - X_{i + 1}^1 )} \right]^2 + \left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^2 - X_{i + 1}^2 )} \right]^2 > 0\) for all λ _i ≧0i=1,...,n?2, \(\sum\limits_{i = 1}^{n - 2} {\lambda _i > 0} \) , b) neighbourhoods of theX _i 's:U _i (X _i )?R ⁴ i=1,...,n?1 (in the euclidean topology ofR ⁴) and c) a real number α>1 such that for all points (x):x ₁, ...,x _n?1:x _i ∈U _i (X _r ) there are positive constantsC ⁽ⁿ⁾{(x)},h ⁽ⁿ⁾{(x)} with: $$\left| {\left\langle {\left[ {A(x_1 )...A(x_{n - 1} ),A(x_n )} \right]} \right\rangle } \right|< C^{(n)} \left\{ {(x)} \right\}\exp \left\{ { - h^{(n)} \left\{ {(x)} \right\}r^\alpha } \right\}forx_n = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ r \\ \end{array} } \right),r > 1.$$ 2. For any natural numbern there exist a) a configurationY(n): $$Y_2 ,Y_3 ,...,Y_n Y_3^i = \cdot \cdot \cdot = Y_n^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^1 - Y_{i{\text{ + 1}}}^{\text{1}} } )} \right]^2 + \left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^2 - Y_{i{\text{ + 1}}}^{\text{2}} } )} \right]^2 > 0\) for all μ _i ≧0,i=3, ...,n?1, \(\sum\limits_{i = 3}^{n - 1} {\mu _i > 0} \) , b) neighbourhoods of theY _i 's:V _i(Y _i )?R ⁴ i=2, ...,n (in the euclidean topology ofR ⁴) and c) a real number β>1 such that for all points (y):y ₂, ...,y _n y _i ∈V _i (Y _i there are positive constantsC _(n){(y)},h _(n){(y)} and a real number γ_(n){(y)∈a closed subset ofR?{0}?{1} with: γ_(n){(y)}\y ₂,y ₃, ...,y _n totally space-like in the order 2, 3, ...,n and $$\left| {\left\langle {\left[ {A(x_1 ),A(x_2 )} \right]A(y_3 )...A(y_n )} \right\rangle } \right|< C_{(n)} \left\{ {(y)} \right\}\exp \left\{ { - h_{(n)} \left\{ {(y)} \right\}r^\beta } \right\}$$ for \(x_1 = \gamma _{(n)} \left\{ {(y)} \right\}r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right),x_2 = y_2 - [1 - \gamma _{(n)} \{ (y)\} ]r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right)\) and for sufficiently large values ofr.     

A limit law for the ground state of Hill's equation

It is proved that the ground state Λ(L) of (?1)x the Schrödinger operator with white noise potential, on an interval of lengthL, subject to Neumann, periodic, or Dirichlet conditions, satisfies the law $$\mathop {\lim }\limits_{L \uparrow \infty } P[(L/\pi )\Lambda ^{1/2} \exp ( - \tfrac{8}{3}\Lambda ^{3/2} ) > x] = \left\{ {\begin{array}{*{20}c} {1forx< 0} \\ {e^{ - x} forx \geqslant 0} \\ \end{array} } \right.$$      

An expansion of Hartree-Fock and correlation energies,relativistic corrections and the dipole matrix element in the powers of of 1/Z

Feynman diagrammatic technique was used for the calculation of Hartree-Fock and correlation energies, relativistic corrections, dipole matrix element. The whole energy of atomic system was defined as a polen-electron Green function. Breit operator was used for the calculation of relativistic corrections. The Feynman diagrammatic technique was developed for 〈H_B>. Analytical expressions for the contributions from diagrams were received. The calculations were carried out for the terms of such configurations as 1s² 2sⁿ¹ 2pⁿ² (2 ≧n₁≧ 0, 6≧ n₂ ≧ 0). Numerical results are presented for the energies of the terms in the form $$E = E_0 Z^2 + \Delta {\rm E}_2 + \frac{1}{Z}\Delta {\rm E}_3 + \frac{{\alpha ^2 }}{4}(E_0^r + \Delta {\rm E}_1^r Z^3 )$$ and for fine structure of the terms in the form $$\begin{gathered} \left\langle {1s^2 2s^{n_1 } 2p^{n_2 } LSJ|H_B |1s^2 2s^{n_1 \prime } 2p^{n_2 \prime } L\prime S\prime J} \right\rangle = \hfill \\ = ( - 1)^{\alpha + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 1} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 [E^{(0)} (Z - B) + \varepsilon _{co} ] + \hfill \\ + ( - 1)^{L + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 2} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 \varepsilon _{cc} . \hfill \\ \end{gathered} $$ Dipole matrix elements are necessary for calculations of oscillator strengths and transition probabilities. For dipole matrix elements two members of expansion by 1/Z have been obtained. Numerical results were presented in the form P(a,a′) = a/Z(1+τ/Z).     

Nonlinear Schrödinger equations and sharp interpolation estimates

A sharp sufficient condition for global existence is obtained for the nonlinear Schrödinger equation $$\begin{array}{*{20}c} {(NLS)} & {2i\phi _t + \Delta \phi + \left| \phi \right|^{2\sigma } \phi = 0,} & {x \in \mathbb{R}^N } & {t \in \mathbb{R}^ + } \\ \end{array} $$ in the case σ=2/N. This condition is in terms of an exact stationary solution (nonlinear ground state) of (NLS). It is derived by solving a variational problem to obtain the “best constant” for classical interpolation estimates of Nirenberg and Gagliardo.     

Exponential lower bounds to solutions of the Schrödinger equation: Lower bounds for the spherical average

For a large class of generalizedN-body-Schrödinger operators,H, we show that ifE<Σ=infσ_ess(H) and ψ is an eigenfunction ofH with eigenvalueE, then $$\begin{array}{*{20}c} {\lim } \\ {R \to \infty } \\ \end{array} R^{ - 1} \ln \left( {\int\limits_{S^{n - 1} } {|\psi (R\omega )|} ^2 d\omega } \right)^{1/2} = - \alpha _0 ,$$ with α ₀ ² +E a threshold. Similar results are given forE≧Σ.     

On resonant classical Hamiltonians with two equal frequencies

This paper contains a detailed study of the flow that the classical Hamiltonian $$H = \tfrac{1}{2}(x_1^2 + y_1^2 ) - \tfrac{1}{2}(x_2^2 + y_2^2 ) + \mathcal{O}_3 $$ induces near the origin of its phase spaceR ⁴. Here the perturbation term \(\mathcal{O}_3 \) represents a convergent power series. In particular, criteria for the existence and stability of periodic orbits are developed and expressed in terms of canonical invariants that are extracted from the perturbation term.     

New exact travelling wave solutions of bidirectional wave equations

The surface water waves in a water tunnel can be described by systems of the form [Bona and Chen, Physica D116, 191 (1998)] 1 $$ \label{BWE} \left\{ \begin{array}{l} v_t+u_x+(uv)_x+au_{xxx}-bv_{xxt}=0, \\ u_t+v_x+uu_x+cv_{xxx}-du_{xxt}=0, \end{array} \right. $$ where a, b, c and d are real constants. In general, the exact travelling wave solutions will be helpful in the theoretical and numerical study of the nonlinear evolution systems. In this paper, we obtain exact travelling wave solutions of system (1) using the modified $\tanh$ – $\coth$ function method with computerized symbolic computation.     

Nuclear orientation and NMR/ON of205,207Po

^205,207Po have keen implanted with an isotope separator on-line into cold host matrices of Fe, Ni, Zn and Be. Nuclear magnetic resonance of oriented²⁰⁷Po has been observed in Fe and Ni, of²⁰⁵Po in Fe. The resonance frequencies for zero external field are $$\begin{gathered} v_L (^{207} Po\underline {Fe} ) = 575.08(20)MHz \hfill \\ v_L (^{207} Po\underline {Ni} ) = 160.1(8)MHz \hfill \\ v_L (^{205} Po\underline {Fe} ) = 551.7(8)MHz. \hfill \\ \end{gathered} $$ From the dependence of the resonance frequency on external magnetic field theg-factor of²⁰⁷Po was derived as $$g(^{207} Po) = + 0.31(22).$$ Using this value the magnetic hyperfine fields of Po in Fe and Ni were obtained as $$\begin{gathered} B_{hf} (Po\underline {Fe} ) = + 238(16)T \hfill \\ B_{hf} (Po\underline {Ni} ) = 66.3(4.6)T. \hfill \\ \end{gathered}$$ Theg-factor of²⁰⁵Po follows as $$g(^{205} Po) = + 0.304(22).$$ From the temperature dependence of the anisotropies ofγ-lines in the decay of^205,207Po the multipole mixing of several transitions was derived. The electric interaction frequenciesv _Q=eQV_zz/h in the hosts Zn and Be were measured as $$\begin{gathered} v_Q (^{207} Po\underline {Zn} ) = + 42(3)MHz \hfill \\ v_Q (^{207} Po\underline {Be} ) = - 70(20)MHz \hfill \\ v_Q (^{205} Po\underline {Be} ) = - 42(17)MHz. \hfill \\ \end{gathered}$$      

A rigorous upper bound in electrostatics on a random lattice ensemble

We consider a Kirchhoff network on a random two-dimensional lattice with links and weights as previously specified, and a circular boundary of radiusR. We show rigorously that the resistance between the central point and the boundary, averaged over all placements of the remaining sites with site density ?, is bounded above by $$\begin{array}{*{20}c} {(4\pi )^{ - 1} [\ln (4\pi \rho R^2 ) + 1] + 16[\tan ^{ - 1} 5^{ - {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} + 5^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} /(\sqrt 5 + 1)^2 ]} \\ { \simeq (4\pi )^{ - 1} \ln (4\pi \rho R^2 ) + 12.0.} \\ \end{array} $$      

Rigorous bounds on quark masses within a nonrelativistic approach

By applying the Feynman-Hellmann theorem to \(q\bar q\) systems we find the following bounds on quark mass differences from the spectrum ofall quarkonium states $$\begin{gathered} 0.27 \leqq m_s - m_u \leqq 0.45GeV \hfill \\ 1.23 \leqq m_c - m_s \leqq 1.46GeV \hfill \\ 3.30 \leqq m_b - m_c \leqq 3.55GeV. \hfill \\ \end{gathered}$$ As best values we derive $$\begin{gathered} m_u = m_d = 0.31GeV,m_s = 0.62GeV, \hfill \\ m_c = 1.91GeV,m_b = 5.27GeV. \hfill \\ \end{gathered}$$      

The final statesl ± K ± K ± X in jets as signatures ofB_s^0 - \bar B_s^0 mixings

Significant mixing is expected between the neutral bottom mesons \(B_s^0 - \bar B_s^0 \) in the standard model of weak interactions. We propose measurements of the processes \(\left\{ {\begin{array}{*{20}c} {e^ + e^ - } \\ {p\bar p} \\ \end{array} } \right\} \to \begin{array}{*{20}c} {b\bar b} \\ {} \\ \end{array} \to l^ + K^ - K^ - X\) as a measure of such mixing. Rates are presented for energetic bottom quark jets, produced ine ⁺ e ^? annihilation.     

Analyticity and global existence of small solutions to some nonlinear Schrödinger equations

In this paper we will study the nonlinear Schrödinger equations: $$\begin{gathered} i\partial _t u + \tfrac{1}{2}\Delta u = \left| u \right|^2 u, (t,x) \in \mathbb{R} \times \mathbb{R}_x^n , \hfill \\ u(0,x) = \phi (x), x \in \mathbb{R}_x^n \hfill \\ \end{gathered} $$ . It is shown that the solutions of (*) exist and are analytic in space variables fort∈??{0} if φ(x) (∈H ^2n+1,2(? _x ⁿ )) decay exponentially as |x|→∞ andn≧2.     

Search for rare radiative decays ofΦ-meson at VEPP-2M

Results of the search for rare radiative decay modes of the ?-meson performed with the Neutral Detector at the VEPP-2M collider are presented. For the first time upper limits for the branching ratios of the following decay modes have been placed at 90% confidence level: $$\begin{gathered} B(\phi \to \eta '\gamma )< 4 \cdot 10^{ - 4} , \hfill \\ B(\phi \to \pi ^0 \pi ^0 \gamma )< 10^{ - 3} , \hfill \\ B(\phi \to f_0 (975)\gamma )< 2 \cdot 10^{ - 3} , \hfill \\ B(\phi \to H\gamma )< 3 \cdot 10^{ - 4} , \hfill \\ \end{gathered} $$ whereH is a scalar (Higgs) boson with a mass 600 MeV<m _H <1000 MeV, the real measurement isB(φ→H γ)·B(H→2π⁰)<0.8·10^-4, the quoted result is model dependent, as explained in the text, $$\begin{gathered} B(\phi \to a\gamma ) \cdot B(a \to e^ + e^ - )< 5 \cdot 10^{ - 5} , \hfill \\ B(\phi \to a\gamma ) \cdot B(a \to \gamma \gamma )< 2 \cdot 10^{ - 3} , \hfill \\ \end{gathered} $$ wherea is a particle with a low mass and a short lifetime, $$B(\phi \to a\gamma )< 0.7 \cdot 10^{ - 5} ,$$ wherea is a particle with a low mass not observed in the detector.     

Lifetimes andg J factors of 6s7p and 5d6p levels of Ba I

Excited states of Ba have been investigated with optical double resonance and Hanle effect. The followingg _J factors and natural lifetimes (in 10^?9 sec) have been measured $$\begin{gathered} 6s7p\left\{ {\begin{array}{*{20}c} {^1 P_1 :g_J = 1.003(2)\tau = 13.5(6)} \\ {^3 P_1 :g_J = 1.4971(8)\tau = 85.0(8.0)} \\ \end{array} } \right. \hfill \\ 5d6p\left\{ {\begin{array}{*{20}c} {^1 P_1 :g_J = 1.004(2)\tau = 12.4(9)} \\ {^3 P_1 :g_J = 1.4847(15)\tau = 11.7(9)} \\ {^3 D_1 :g_J = 0.5064(3)\tau = 17.0(5).} \\ \end{array} } \right. \hfill \\ \end{gathered}$$ g _J is utilized to test the mixing coefficients of the wave functions in the intermediate coupling model. The lifetimes are converted into absolute transition probabilities for all the decays originating from the states investigated under the assumption that their branching ratios obtained elsewhere are correct. This assumption is not unquestionable, however.     

The ground state energy of a Bose gas with Coulomb interaction

LetH _N be the 2N particle Hamiltonian $$\begin{array}{*{20}c} {H_N = \sum\limits_{i = 1}^{2N} {( - \Delta _\iota ) + \sum\limits_{i< j = 1}^N {\left| {x_i - x_j } \right|^{ - 1} + } \sum\limits_{i< j = 1}^N {\left| {x_{i + N} - x_{j + N} } \right|^{ - 1} } } } \\ { - \sum\limits_{i,j< j = 1}^N {\left| {x_i - x_{j + N} } \right|^{ - 1} ,} } \\ \end{array} $$ whereΔ _i is the Laplacian in the variablex _i ∈?³, 1≦i≦2N. The operatorH _N is assumed to act on wave functionsΨ(x ₁, ...,x _N ;x _N+1, ...,x _2N ) which are symmetric in the variables (x ₁, ...,x _N ) and (x _N+1, ...,x _2N ). SupposeΨ is supported in a setΛ ^2N , whereΛ is a cube in ?³. It is shown that if a normalized wave functionΨ can be written as a product of two wave functions $$\psi (x_1 ,...,x_N ;x_{N + 1} ,...,x_{2N} ) = \psi _1 (x_2 ,...,x_N )\psi _2 (x_{N + 1} ,...,x_{2N} ),$$ and the density of particles inΛ is constant, then 〈Ψ|H _N |Ψ〉≧?CN ^7/5 for some universal constantC.     

Energy spectrum of photons produced by neutrino (antineutrino) scattering on electrons. Effect of possible neutrino oscillations

The energy spectrum of photons emitted in neutrino (antineutrino) scattering on electrons at E_v ? m_v is calculated with the assumption that the neutral electron flow has an arbitrary (V, A) structure. The result obtained is generalized to the case of possible neutrino oscillations, \(v_e \begin{array}{*{20}c} { - \to } \\ { \leftarrow - } \\ \end{array} v_\mu , \overline v _e \begin{array}{*{20}c} { - \to } \\ { \leftarrow - } \\ \end{array} \overline v _\mu \) , at an arbitrary neutrino mixing angle. Using the Weinberg-Salam model (sin²θ_W = 0.23) estimates of the sections dσ_γ/dω and σ_γ are obtained with consideration of the reactor antineutrino flux \(\bar v_e \) . The contributions from charged and neutral lepton fluxes and their interference to dσ_γ/dω are compared.     

Centrifugal distortion constants of pentafluorobenzene from its pure rotation spectrum

The centrifugal distortion analysis of the pure rotational spectrum of pentafluorobenzene in the frequency region of 8 to 18 GHz involvingJ upto 54 has yielded the following rotational and quartic centrifugal distortion constants: $$\begin{gathered} A'' = 1480 \cdot 8665 \pm 0 \cdot 0026 MHz, \tau = - 1 \cdot 751 \pm 0 \cdot 20 kHz, \hfill \\ B'' = 1030 \cdot 0782 \pm 0 \cdot 0025 MHz, \tau _2 = - 0 \cdot 567 \pm 0 \cdot 066 kHz, \hfill \\ C'' = 607 \cdot 5152 \pm 0 \cdot 0026 MHz, \tau _{aaaa} = - 0 \cdot 765 \pm 0 \cdot 068 kHz, \hfill \\ \tau _{bbbb} = - 0 \cdot 612 \pm 0 \cdot 065 kHz, \hfill \\ \tau _{cccc} = - 0 \cdot 547 \pm 0 \cdot 068 kHz. \hfill \\ \end{gathered} $$      

BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one

We show that for most non-scalar systems of conservation laws in dimension greater than one, one does not have BV estimates of the form $$\begin{gathered} \parallel \overline V u(\overline t )\parallel _{T.V.} \leqq F(\parallel \overline V u(0)\parallel _{T.V.} ), \hfill \\ F \in C(\mathbb{R}),F(0) = 0,F Lipshitzean at 0, \hfill \\ \end{gathered} $$ even for smooth solutions close to constants. Analogous estimates forL ^p norms $$\parallel u(\overline t ) - \overline u \parallel _{L^p } \leqq F(\parallel u(0) - \overline u \parallel _{L^p } ),p \ne 2$$ withF as above are also false. In one dimension such estimates are the backbone of the existing theory.