Spectral resonances which become eigenvalues

The stationary Schrödinger equation is ? _x ² φ + λV(x)φ=zφ for φ∈?²(R ⁺,dx). If the potential is bounded below, singular only atx=0, negative on some compact interval and behaves likeV(x)～1/x ^μ asx→∞ with 2≧μ>0, then the system admits shape resonances which continuously become eigenvalues as λ increases. Here λ>0 and for μ=2 a sufficiently large λ is required. Exponential bounds are obtained on Im(z) as λ approaches a threshold. The group velocity near threshold is also estimated.     

Glueball mass spectrum and mass splitting in 2+1 strongly coupled lattice gauge theories

For a 2+1 strongly coupled (β=2/g ² small) Wilson action lattice gauge theory with complex character we analyze the mass spectrum of the associated quantum field theory restricted to the subspace generated by the plaquette function and its complex conjugate. It is shown that there is at least one but not more than two isolated masses and each mass admits a representation of the formm(β)=?4lnβ+r(β), wherer(β) is a gauge group representation dependent function analytic inβ ^1/2 orβ atβ=0. For the gauge group SU(3) there is mass splitting and the two massesm _± are given by $$m_ \pm (\beta ) = - 41n\beta + 16r^4 + \tfrac{1}{2}(2 \pm 1)\beta + \left( {d_ \pm (\beta )\sum\limits_{n = 2}^\infty {c_n^ \pm } \beta ^n } \right)$$ wherer=3 is the dimension of the representation andd _±(β) is analytic atβ=0.c _n ^± can be determined from a finite number of theβ=0 Taylor series coefficients of finite lattice truncated plaquette-plaquette correlation function at a finite number of points.     

Weitere Messungen zur Anisotropie der intermolekularen Wechselwirkung durch Streuung von Molekülen in definiertem Quantenzustand

The interaction between diatomic molecules and rare gas atoms can be described by the realistic, though simplified potentialV=?[(r_m/r)¹²(1 +q₁₂P₂(cosθ))?2(r_m/r)⁶(1+q₆P₂(cosθ))] The determination of the parameters?, r _m andq ₆has been treated in the two previous parts of this series. The present final paper describes the determination of the anisotropy parameter in the repulsive part of the potential, q₁₂, for the system CsF-He. Whileq ₆ could be derived using only the dependence of the total scattering cross section on the molecular rotational state, the determination ofq ₁₂ requires, in addition, knowledge of the velocity dependence. The comparison of the experimental data for CsF in the rotational states ¦J, M〉=¦1, 1〉 and ¦1, 0〉 with cross sections calculated by means of the “high energy” approximation yields the result:q ₁₂=0.9±0.2. The validity of the “high energy” approximation in the velocity range covered by the experiment is discussed.     

Analyticity properties and a convergent expansion for the inverse correlation length of the high-temperatured-dimensional Ising model

We show that the inverse correlation lengthm(β) (= mass of the fundamental particle of the associated lattice quantum field theory) of the spin-spin correlation function 〈s _x s _y 〉,x, y εZ ^d , of thed-dimensional Ising model admits the representation $$m(\beta ) = - ln\beta + r(\beta )$$ for small inverse temperaturesβ > 0.r(β) is ad-dependent function, analytic atβ = 0.c _n , the nth β = 0 Taylor series coefficient of r(β) can be computed explicitly from the Z^d limit of a finite number of finite lattice A spin-spin correlation functions 〈s₀s_x〉t>Afor a finite number ofx = (x ₁,x₂, ..., x_d), ¦x¦ = ∑ _i ^d 1¦x_i¦< R(n), where R(n) increases withn. Furthermore, there exists aβ' > 0, such that for eachβ ε (0,β')m(β) is analytic. Similar results are also obtained for the dispersion curve ω(p), ω(p)=ω(0)=m, pε(-π, π]^d?1, of the fundamental particle of the associated lattice quantum field theory.     

the distribution of exit times for weakly colored noise

We analyze the exit time (first passage time) problem for the Ornstein-Uhlenbeck model of Brownian motion. Specifically, consider the positionX(t) of a particle whose velocity is an Ornstein-Uhlenbeck process with amplitudeσ/ρ and correlation time ε², $$dX/dt = \sigma Z/\varepsilon , dZ/dt = - Z/\varepsilon ^2 + 2^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \xi (t)/\varepsilon $$ whereξ(t) is Gaussian white noise. Let the exit timet _ex be the first time the particle escapes an interval ?A, given that it starts atX(0)=0 withZ(0)=z ₀. Here we determine the exit time probability distributionF(t)≡Prob {t _ex>t} by directly solving the Fokker-Planck equation. In brief, after taking a Laplace transform, we use singular perturbation methods to reduce the Fokker-Planck equation to a boundary layer problem. This boundary layer problem turns out to be a half-range expansion problem, which we solve via complex variable techniques. This yields the Laplace transform ofF(t) to within a transcendentally smallO(e ^?A/εσ +e ^?B/εσ error. We then obtainF(t) by inverting the transform order by order in ε. In particular, by lettingB→∞ we obtain the solution to Wang and Uhlenbeck's unsolved problem b; throughO(ε²σ²/A ¹) this solution is $$F(t) = Erf\left\{ {\frac{{A + \varepsilon \sigma \alpha + \varepsilon \sigma z_0 }}{{2\sigma (t - \varepsilon ^2 \kappa )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right\} + ... for \frac{t}{{\varepsilon ^2 }} > > 1$$ andF=1 otherwise. Here, α=∥ξ(1/2)∥=1.4603?, where ξ is the Riemann zeta function, and the constant κ is 0.22749?.     

Momentum distribution and one-body density matrix in liquid3He.

Summary The momentum distributionn(k) and the one-body density matrix ρ₁(r,r' have been calculated in normal liquid³He atT=0. A variational wave function containing two-, three-body and backflow correlations has been used. The Fermi hypernetted chain technique has been employed and the elementary diagrams have been evaluated by the scaling approximation. The present estimate ofn(k) is in good agreement with the Monte Carlo data obtained with similar wave functions.n(k) and the discontinuityZ ofn(k) at the Fermi surface have been computed at several values of the density. The density dependence of the effective massm ^* has been found to be mainly due to that ofZ.     

Large-scale structure of isotropic homogeneous turbulence

It is shown that the longitudinal correlation functionf is asymptotically proportional tor ^?3 asr→∞ and the energy spectrum function is asymptotically proportional toκ ² asκ→0 if and only if 0<〈(f u d ³ x)·u〉<∞. Moreover, the latter finiteness condition is shown to be essentially equivalent to 〈(fy·ud ³ x)²〉<∞ for nonstochasticyεL ²(R₃). Confirmed by recent experimental measurements, the larger dependencef ∝r ^?3 is concomitant with anO(r ^?6)=O(f ²) fall-off of the viscous force term in the Kármán-Howarth equation.     

Quark model relation for kaon charge radii

We derive the equation r²_E(K⁰)/r²_E(K⁺) = -(m²_s ? m²_n)/(2m²_s + m²_n) relating the kaon electric charge radii and the strange (m_s) and non-strange (m_n) quark masses in the nonrelativistic quark model, and suggest an inequality which we expect to hold in the presence of relativistic corrections. New data for these charge radii are presently being analysed by two experimental groups.     

Statistical analysis of24Mg+24Mg elastic and inelastic scattering

The data on the excitation functions of²⁴Mg+²⁴Mg elastic and inelastic (²⁴Mg +²⁴Mg^*(2⁺),²⁴Mg^*(2⁺)+²⁴Mg^*(2⁺),²⁴Mg+²⁴Mg^*(4⁺),²⁴Mg^*(4⁺)+²⁴Mg^*(2⁺),²⁴Mg+²⁴Mg^*(6⁺)) scattering fromE _c.m=42 to 56 MeV have been subjected to a statistical analysis consisting of calculations of deviation function, cross-correlation function, cross-channel correlation coefficients, coherence widths, and the distribution of cross sections. On the basis of the analysis resonant structures atE _c.m=45.70, 46.65, 47.35 and 47.75 MeV have been confirmed. Two new resonant structures atE _c.m=44.55 and 50.50 MeV have been identified.     

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 ^∞      

Nuclear moments and optical isotope shifts of108m Ag and110m Ag

The hyperfine structure of the resonance lines of^108m Ag and^110m Ag has been investigated by means of optical interference spectroscopy. The metastable isotopes were produced by neutron irradiation in a reactor. From the measureda- andb-values we derive the value for the nuclear magnetic dipole moment of^108mAgμ _I (108m)=3.577 (20)μ _n and the values for the electric quadrupole momentsQ _I (108m)=1.52(8) bQ _I (110m)=1.65(10) b (without Sternheimer correction). The measured isotope shifts allow the determination of the changes in the mean square charge radii of the nuclei involved:δ 〈r ²〉(108m-107)=0.022(3)fm² δ 〈r ²〉(110m-109)=0.029(2)fm². The isotope positions show odd-even staggering comparable with those of neighbouring isotonic nuclides.     

Der Zerfall des Mn52m

Mn^52m(T _1/2=21 m) was produced by irradiating iron-foiles with deuterons ofE _d =9 MeV. Coincidences ofγ-rays with the main 1434 keV-transition had been sought for with the aid of a scintillation fast-slow-coincidence circuit. There was no evidence ofγ-lines which had been found earlier byKatoh et al. atγ-energies 700, 940, 1020, 1150, 1370, 1520 keV. If these transitions exist, their relative intensities are less than 0.7% (700 keV) and 0.3% (all others) per decay of Mn^52m.     

An inequality onS wave bound states,with correct coupling constant dependence

We prove that the number ofS wave bound states in a spherically symmetric potentialgV(r) is less than 1 $$g^{1/2} \left[ {\int\limits_0^\infty {r^2 V^ - (r)dr} \int\limits_0^\infty {V^ - (r)dr} } \right]^{1/4}$$ whereV ^? is the attractive part of the potential, in units where ?²/2M=1.     

Aussagen der Theorie von Kraichnan für die isotrope hydromagnetische Turbulenz

The theory ofKraichnan is applied to quasi-stationary isotropic hydromagnetic turbulence. The average infinitesimal-impulse-response functionsg(k, τ), g _m (k, τ) and the time-correlationsr(k, τ), r _m (k, τ) are evaluated by the non-local direct-interaction approximation within the inertial range. For the range of ohmic but no viscous dissipation it is found that the magnetic energy spectrumE _m (k) obeys aE(k)k ^?2-law in accordance with results ofGolitsyn andMoffatt.     

An application of hypervirial perturbation theory to calculate energy eigenvalues of the Morse potential for various diatomic molecules

The Schrödinger equation with the potentialV(r)=D[exp(?2β(r?r _e))?2exp(?β(r?r _e))] is treated in the framework of the hypervirial-renormalization parameter scheme. The energy eigenvalues of various eigenstates for different molecules are calculated.     

A family of angle-moments proportional to r-n,n = 1,2,…, in free space

The moments M_n(r) ≡ 1/2 ∝^2π₀ dθ sinⁿ θ I(r,θ) of the intensity I(r, θ) in free space surrounding a spherical object emitting radiation with an arbitrary directional dependence are shown to be exactly proportional to r^-(n+1), n = 0, 1,….     

On the ground state energy of a two-dimensional electron fluid

A new theory of the ground state energy of a two-dimensional electron fluid is presented. It is shown that the ring diagram contribution changes its analytical behavior atr _s =2^1/2, wherer _s is the usual density parameter defined by r_S = 1/a ₀(π n)^1/2,a ₀ being the Bohr radius andn is the electron density. For smallr _s , a high density series is obtained in agreement with the previous calculation. For larger _s , a hitherto unknown low density series is obtained. In the low density region, the first order exchange energy is completely cancelled out by a term from the ring contribution so that the ground state energy decreases in proportion tor _s ^?2/3 , followed byr _s ^/?4/3 and higher order terms. The energy is found to be minimum atr _s=1.4757, the minimum value being ?0.481915 Rydbergs.