Asymptotic inverse spectral problem for anharmonic oscillators

We study perturbationsL=A+B of the harmonic oscillatorA=1/2(??²+x ²?1) on ?, when potentialB(x) has a prescribed asymptotics at ∞,B(x)～|x|^?α V(x) with a trigonometric even functionV(x)=Σa _mcosω _m x. The eigenvalues ofL are shown to be λ _k =k+μ _k with small μ _k =O(k ^?γ), γ=1/2+1/4. The main result of the paper is an asymptotic formula for spectral fluctuations {μ _k }, $$\mu _k \sim k^{ - \gamma } \tilde V(\sqrt {2k} ) + c/\sqrt {2k} ask \to \infty ,$$ whose leading term \(\tilde V\) represents the so-called “Radon transform” ofV, $$\tilde V(x) = const\sum {\frac{{a_m }}{{\sqrt {\omega _m } }}\cos (\omega _m x - \pi /4)} .$$ as a consequence we are able to solve explicitly the inverse spectral problem, i.e., recover asymptotic part |x ^?α|V(x) ofB from asymptotics of {µ _k }. ₁ ^∞      

On the rate of quantum ergodicity I: Upper bounds

One problem in quantum ergodicity is to estimate the rate of decay of the sums $$S_k (\lambda ;A) = \frac{1}{{N(\lambda )}}\sum\limits_{\sqrt {\lambda _j } \leqq \lambda } {\left| {(A\varphi _j ,\varphi _j ) - \bar \sigma _A } \right|^k } $$ on a compact Riemannian manifold (M, g) with ergodic geodesic flow. Here, {λ _j ,? _j } are the spectral data of the Δ of(M, g), A is a 0-th order ψDO, $\bar \sigma _A $ is the (Liouville) average of its principal symbol and $N(\lambda ) = \# \{ j:\sqrt {\lambda _j } \leqq \lambda \} $ . ThatS _k (λ;A)=o(1) is proved in [S, Z.1, CV.1]. Our purpose here is to show thatS _k (λ;A)=O((logλ) ^?k/2 ) on a manifold of (possibly variable) negative curvature. The main new ingredient is the central limit theorem for geodesic flows on such spaces ([R, Si]).     

Diffusion Effects on the Breakdown of a Linear Amplifier Model Driven by the Square of a Gaussian Field

We investigate solutions to the equation ? _t ?? $\mathcal{D}$ Δ?=λS ²?, where S(x, t) is a Gaussian stochastic field with covariance C(x?x′, t, t′), and x∈ $\mathbb{R}$ ^d . It is shown that the coupling λ _cN (t) at which the N-th moment <? ^N (x, t)> diverges at time t, is always less or equal for $\mathcal{D}$ >0 than for $\mathcal{D}$ =0. Equality holds under some reasonable assumptions on C and, in this case, λ _cN (t)=Nλ _c (t) where λ _c (t) is the value of λ at which <exp[λ ∫ ^t ₀ S ²(0, s) ds]> diverges. The $\mathcal{D}$ =0 case is solved for a class of S. The dependence of λ _cN (t) on d is analyzed. Similar behavior is conjectured when diffusion is replaced by diffraction, $\mathcal{D}$ →i $\mathcal{D}$ , the case of interest for backscattering instabilities in laser-plasma interaction.     

Search for Θ+(1540) in exclusive proton-induced reaction p + C(N) \to \Theta ^ + \bar \kappa ^0 + C(N) at the energy of 70 GeV

A search for narrow Θ⁺(1540), a candidate for a pentaquark baryon with positive strangeness, has been performed in an exclusive proton-induced reaction $p + C(N) \to \Theta ^ + \bar \kappa ^0 + C(N)$ on carbon nuclei or quasifree nucleons at $E_{beam} = 70GeV(\sqrt s = 11.5GeV)$ studying nK ⁺, pK _S ⁰ , and pK _L ⁰ decay channels of Θ⁺(1540) in four different final states of the $\Theta ^ + \bar K^0 $ system. In order to assess the quality of the identification of the final states with neutron or K _L ⁰ , we reconstructed Λ(1520) → nK _S ⁰ and ? → K _L ⁰ K _S ⁰ decays in the calibration reactions p + C(N) → Λ (1520)K ⁺+C(N) and p+C(N) → p?+C(N). We found no evidence for a narrow pentaquark peak in any of the studied final states and decay channels. Assuming that the production characteristics of the $\Theta ^ + \bar K^0 $ system are not drastically different from those of the Λ(1520)K ⁺ and p? systems, we established upper limits on the cross-section ratios $\sigma (\Theta ^ + \bar K^0 )/\sigma (\Lambda (1520)K^ + ) < 0.02$ and $\sigma (\Theta ^ + \bar K^0 )/\sigma (p\phi ) < 0.15$ at 90% C.L. and a preliminary upper limit for the forward-hemisphere cross section $\sigma (\Theta ^ + \bar K^0 )$ nb/nucleon.     

Radiative decays and quark content of f 0(980) and φ(1020)

The quark structure of φ(1020) and f ₀(980) is studied on the basis of data on the radiative decays φ(1020) → γπ ⁰, γη, γη′, γ a ₀(980), γ f ₀(980) and f ₀(980) → γγ. The partial widths are calculated under the assumption that all the mesons under consideration are $\bar qq$ states: φ(1020) is a dominantly $s\bar s$ state ( $\eta ' = n\bar nsin\theta + s\bar scos\theta $ component contributes not more than 1%); η, η′, and π ⁰ are standard $q\bar q$ states, $\eta = n\bar ncos\theta - s\bar ssin\theta $ and $\eta ' = n\bar nsin\theta + s\bar scos\theta $ with θ?37°; and f ₀(980) is a e5 meson with the flavor wave function $n\bar ncos\varphi + s\bar ssin\varphi $ . The transition φ → γπ ⁰ specifies the admixture of the $n\bar n$ component in the φ meson: it is on the order of 0.5%. We argue that this order of $n\bar n$ value does not contradict data on the decay φ(1020) → γ a ₀(980). The partial widths calculated for the decays φ → γη, γη? are in reasonable agreement with experimental data. The measured branching-ratio value Br(φ→γf ₀(980))=(3.4±0.4 _?0.5 ^+1.5 ×10^?4) requires 25°≤|?|≤90°. For the decay f ₀(980) → γγ, the agreement with data, Γ(f ₀(980) → γγ)=0.28 _?0.13 ^+0.09 keV, is attained at either ?=85°±8° or ?=?46°±8°. A simultaneous analysis of the decays φ(1020) → γ f ₀(980) and f ₀(980) → γγ favors the solution with the negative mixing angle of ?=?48°±6°, setting f ₀(980) very close to the flavor octet (? _octet=±54.7°).     

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.     

Angular asymmetries in the reactions \vec pp \to d\pi ^ + \eta and \vec pn \to d\pi ^0 \eta and a 0-f 0 mixing

The reactions pp → dπ ⁺ η and pn → dπ ⁰ η are of special interest for investigating the a ₀(980) (J ^P=0⁺) resonance in the process NN → da ₀ → dπη. We study some aspects of those reactions within a general formalism and also in a concrete phenomenological model. In particular, it is shown that the presence of nonresonant (i.e., without excitation of the a ₀ resonance) contributions to these reactions yields nonvanishing values for specific polarization observables, i.e., to effects like those generated by a ₀ ⁰ -f ₀ mixing. An experimental determination of these observables for the reaction $\vec pp \to d\pi ^ + \eta $ would provide concrete information on the magnitude of those nonresonant contributions to πη production. We also discuss the possibility of extracting information about a ₀ ⁰ -f ₀ mixing from the reaction $\vec pn \to d\pi ^0 \eta $ with a polarized proton beam.     

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 $e^{-\frac{4}{3} x^{3/2} }$ .     

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.     

Resonance studies at STAR

Preliminary results from measurements of resonances (K ^*0(892), $\overline {K*^0 } (892)$ , Φ(1020), and ρ(770)) and weakly decaying particles (Λ(1116), $\bar \Lambda (1116)$ , and K _S ⁰ (498)) are presented. The measurements are performed at mid-rapidity by the STAR detector in $\sqrt {s_{NN} } = 130$ GeV Au?Au collisions at RHIC. The ratios K ^*0/h?, $\overline {K*^0 } /K$ , and $\bar \Lambda /\Lambda $ are compared to measurements at different energies and colliding systems. Estimates of thermal parameters, such as temperature and baryon chemical potential, are also presented.     

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.     

Conserved-vector-current hypothesis and the\bar v_e e^ - \to \pi ^ - \pi ^0 process

Based on the conserved-vector-current (CVC) hypothesis and a four-ρ-resonance unitary and analytic VMD model of the pion electromagnetic form factor, theσ _tot(E _v ^lab ) and dσdE _π ^lab of the weak \(\bar v_e e^ - \to \pi ^ - \pi ^0\) process are predicted theoretically for the first time. Their experimental approval could verify the CVC hypothesis for all energies above the two-pion threshold. Since, unlike the electromagnetic e⁺e^?→π⁺π^? process, there is no isoscalar vector-meson contribution to the weak \(\bar v_e e^ - \to \pi ^ - \pi ^0\) reaction, accurate measurements of theσ _tot(E _v ^lab ) that moreover is strengthened with energyE _v ^lab linearly could solve now a widely discussed problem of the mass specification of the first excited state of theρ(770) meson. As a by-product, an equality \(\sigma _{tot} (\bar v_e e^ - \to \pi ^ - \pi ^0 ) = \sigma _{tot} (e^ + e^ - \to \pi ^ - \pi ^0 )\) is predicted for \(\sqrt s \approx 70 GeV\) .     

The local structure of the spectrum of the one-dimensional Schrödinger operator

Let \(H_V = - \frac{{d^{\text{2}} }}{{dt^{\text{2}} }} + q(t,\omega )\) be an one-dimensional random Schrödinger operator in ?²(?V,V) with the classical boundary conditions. The random potentialq(t, ω) has a formq(t, ω)=F(x _t ), wherex _t is a Brownian motion on the compact Riemannian manifoldK andF:K→R ¹ is a smooth Morse function, \(\mathop {\min }\limits_K F = 0\) . Let \(N_V (\Delta ) = \sum\limits_{Ei(V) \in \Delta } 1 \) , where Δ∈(0, ∞),E _i (V) are the eigenvalues ofH _V . The main result (Theorem 1) of this paper is the following. IfV→∞,E ₀>0,k∈Z ₊ anda>0 (a is a fixed constant) then $$P\left\{ {N_V \left( {E_0 - \frac{a}{{2V}},E_0 + \frac{a}{{2V}}} \right) = k} \right\}\xrightarrow[{V \to \infty }]{}e^{ - an(E_0 )} (an(E_0 ))^k |k!,$$ wheren(E ₀) is a limit state density ofH _V ,V→∞. This theorem mean that there is no repulsion between energy levels of the operatorH _V ,V→∞. The second result (Theorem 2) describes the phenomen of the repulsion of the corresponding wave functions.     

The decay of theT=1 isospin triplet inA=12 systems: IV. The energy dependence of the asymmetry coefficients\alpha _{\beta ^ \mp } of the beta-ray angular distribution in aligned12B and12N and the induced pseudotensor interaction

The asymmetry parameters \(\alpha _{\beta ^ \mp } \) of the beta-ray emitted from aligned¹²B and¹²N are evaluated as a function of the energy. The agreement with experimental differential data is excellent for both \(\alpha _{\beta ^ - } \) (W) and \(\alpha _{\beta ^ + } \) (W). This work confirms, using available nuclear model information, that no induced pseudotensor (IPT) interaction is required for a correct theoretical interpretation of the data. An upper limit for the IPT coupling constantf _T is determined from a simultaneous fit of \(\alpha _{\beta ^ - } \) (W) and \(\alpha _{\beta ^ + } \) (W).     

Boundary values of analytic functions

It is known that a complex — valued continuous functionS(x) as well as a Schwartz distribution on the real axis can be extended in the complex plane minus the support ofS to an analytic function?(z). In the case of a continuous function the jump of?(z) on the real axis represents exactlyS(x): $$\mathop {\lim }\limits_{\varepsilon \to 0 + } [\hat S(x + i\varepsilon ) - \hat S(x - i\varepsilon )] = S(x)$$ . We call regular a pointx on the support ofS such that \(\mathop {\lim }\limits_{\varepsilon \to 0 + } [\hat S(x + i\varepsilon ) - \hat S(x - i\varepsilon )]\) exists. Conditions are found for the existence of regular points on the support of a distribution. It is possible to call this limit (if this exists) the valueS(x) of the distributionS in the pointx. Properties of this type occur in the theory of dispersion relations.     

Exact solution of the Boltzmann equation in the relaxation approximation,existence of negative differential conductivity for certain types of energy bands and its implication for the fluctuation spectrum and the noise temperature

The Boltzmann equation for the distributionf _k of a system of charged particles obeying classical statistics in a uniform fieldF, $$\frac{{\partial f_k }}{{\partial t}} + F\frac{{\partial f_k }}{{\partial k}} = \smallint d^3 k'(W_{kk'} f_{k'} - W_{k'k} f_k ),$$ will be solved analytically for a special class of transition ratesW _kk′=const·h _k ·ν _k ·ν _k′ for any initial distribution.h _k is the Maxwell distribution andν _k >0 can be interpreted as ak-dependent relaxation frequency. The constant relaxation approximation (ν _k =ν) will be used to discuss the drift velocitiesu for all the fields and temperaturesT for certain types of band structuresE(k). Bands with lineark-dependence for largek give rise to drift velocities saturating for large fields. For bands with the periodicity of the reciprocal lattice, the zero drift-theorem has been proved. It states that $$\mathop {\lim }\limits_{F \to \infty } u (F,T) = \mathop {\lim }\limits_{T \to \infty } u (F,T) = 0$$ for all the periodic band structures. This theorem is even correct for a generalW _kk′ if certain restrictions are made. Finally, making use of the Markov character of the conditional probability (Green's function) solution of the Boltzmann equation, the velocity fluctuation spectrumS is calculated forE(k)=A(1?cosa k). It will be shown thatS(F, T, 0) remains positive for the critical field and all temperatures, and therefore the noise temperature diverges on approaching the critical field.     

Study of the iterations of a mapping associated to a spin glass model

We study the iterations of the mapping $$\mathcal{N}[F(s)] = \frac{{(F(s))^2 - (F(0))^2 }}{s} + (F(0))^2 ,$$ with the constraintsF(1)=1,F(s)=∑a _nsⁿ,a _n≧0, and find that, except ifF(s)≡s,N[F(s)] approaches either 0 or 1 for |s|<1 ask→∞.     

How exotic is thea 0 (980)?

The mass and hadronic width of the scalar isovector mesona ₀(980) are estimated in QCD for two possible quark assignments: (a \(\bar qq\) and (b) \(\bar qq\bar qq\) . The two-photon width of thea ₀(980) is also discussed.     

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.     

Spectral signs of the internal heavy-atom effect in aliphatic carbonyl compounds

The CNDO/S method has been applied to the internal effect of Si on the electronic spectrum of the acetone molecule; there is a considerable bathochromic shift and an increase in the \(S_0 \to S_{n\pi ^ * } \) intensity for theα-silyl ketones, while theβ-silyl ketons give only an increase in the intensity of \(S_0 \to S_{n\pi ^ * } \) absorption relative to acetone. The heavy atom substantially alters \(f_{S_0 \to T_{n\sigma ^* } } \) and \(\tau _{T_{n\sigma ^* } }^0 \) but has little effect on \(f_{S_0 \to T_{n\pi ^* } } \) and \(\tau _{T_{n\pi ^* } }^0 \) .