Optimal bounds for ratios of eigenvalues of one-dimensional Schrödinger operators with Dirichlet boundary conditions and positive potentials

Consider the Schrödinger equation –u+V(x)u=u on the intervalI, whereV(x)0 forxI and where Dirichlet boundary conditions are imposed at the endpoints ofI. We prove the optimal bound

Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzschild Space

We provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzschild background. Our estimate implies that such solutions have asymptotic behavior as long as the source term is bounded in the norm . In particular this gives scattering at small amplitudes for non-linear scalar fields of the form for all 2 < p. This paper is dedicated to the memory of Hope Machedon The second author would like thank MSRI and Princeton University, where a portion of this research was conducted during the Fall of 2005. The second author was also supported by a NSF postdoctoral fellowship.     

Determination of the low ω cusp in the velocity autocorrelation spectrum of liquid sodium

Recently performed high resolution incoherent neutron scattering experiments on liquid sodium at smallA and low make the precise evaluation of the frequency distribution of the liquid feasible. An appropriate extension of the well known Egelstaff formula for the evaluation of the frequency distribution in a liquid is deduced. The new analysis is shown to be exact in the smallQ low region, removing the singularity of the above formula at 0 andQ0. The easy applicability of the evaluation is demonstrated on neutron scattering data of liquid sodium, where the long discussed low frequency cusp ofz() could be revealed.     

Photophysical Behavior of β-Carboline-3-Carboxylic Acid N-Methylamide

The solvent and temperature effect of fluorescence and phosphorescence and its polarization characteristics for -carboline-3-carboxylic acid N-methylamide were studied. From the results obtained it is concluded that the first excited singlet state changes from the * type in a polar solvent (EPA) to the n* type in a nonpolar solvent (MC) and that the lowest singlet excited states (* type and n* type) interact by vibronic coupling. In the nonpolar solvent (MC) the lowest singlet excited states are very close in energy and consequently the vibronic coupling is stronger. It has been found that the 0–0 band of the phosphorescence emission in EPA is polarized out of plane, using the excitations corresponding to absorptions into the first and second excited * singlet states. This fact and the order of lifetime measured indicate an emitting triplet that is *. The origin of this polarization is very likely mixing as a result of spin–orbit coupling. However, the vibrational structures and changing polarization outside the 0–0 band indicate that second-order effects involving spin–orbit coupling and vibronic interactions are important and the mixing scheme appears to be and . The proximity effect can qualitatively explain the absence of phosphorescence and the stronger vibronic interaction between the n* and the * states of -carboline-3-carboxylic acid .N-methylamide in a hydrocarbon solvent (MC).     

A Uniform Quantum Version of the Cherry Theorem

Consider in the operator family . P ₀ is the quantum harmonic oscillator with diophantine frequency vector ω, F ₀ a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and . Then there exist independent of and an open set such that if and , the quantum normal form near P ₀ converges uniformly with respect to . This yields an exact quantization formula for the eigenvalues, and for the classical Cherry theorem on convergence of Birkhoff’s normal form for complex frequencies is recovered. Partially supported by PAPIIT-UNAM IN106106-2.     

CP asymmetry in $B \rightarrow \phi K_{S}$ in a general two-Higgs-doublet model with fourth-generation quarks

We discuss the time-dependent CP asymmetry of the decay in an extension of the standard model with both a two Higgs doublet and additional fourth-generation quarks. We show that, although the standard model with a two Higgs doublet and the standard model with fourth-generation quarks alone are not likely to largely change the effective from the decay , the model with both an additional Higgs doublet and fourth-generation quarks can easily account for the possible large negative value of without conflicting with other experimental constraints. In this model, additional large CP violating effects may arise from the flavor-changing Yukawa interactions between neutral Higgs bosons and the heavy fourth-generation down type quark, which can modify the QCD penguin contributions. With the constraints obtained from processes such as and , this model can lead to an effective as large as - 0.4 in the CP asymmetry of .Received: 25 March 2004, Revised: 20 April 2004, Published online: 18 June 2004     

On the Schrödinger equation and the eigenvalue problem

If _k is thek ^th eigenvalue for the Dirichlet boundary problem on a bounded domain in ⁿ , H. Weyl's asymptotic formula asserts that , hence . We prove that for any domain and for all . A simple proof for the upper bound of the number of eigenvalues less than or equal to - for the operator –V(x) defined on ⁿ (n3) in terms of is also provided.Research partially supported by a Sloan Fellowship and NSF Grant No. 81-07911-A1     

Limiting behavior of asymptotically flat gravitational fields

Couch and Torrence suggest that the vacuum Einstein equations admit a larger class of asymptotically flat solutions than those exhibiting the peeling property. Starting with the assumption that , (d/dr) and (/x ^A ) , wherex ^A (A = 2, 3) are angular coordinates, they show that , where ₁ 2 and ₁<₀; , where ₂ 1 and ₁< ₁; and ₄ and ₃ peel as they would under the stronger peeling conditions. The Winicour-Tamburino energy-momentun and angular momentum integrals for these solutions, in general, diverge. In fact, since Couch and Torrence determine only the radial dependence of the solution, it is not clear that the solutions are well defined. We find that the stronger assumption , (d/dr) , and (/x ^A ) does result in well-defined solutions for which both the energy-momentum and angular momentum intergrals are not only finite but result in the same expressions as are obtained for peeling space-times. This assumption appears to be the minimal assumption that is necessary for investigating outgoing radiation at null infinity.In part based on a dissertation by Stephanie Novak and submitted to Syracuse University in partial fulfillment of the requirement for the Ph.D. degree.     

Poincaré-Invariant Markov Processes and Gaussian Random Fields on Relativistic Phase Space

We give a complete characterization, including a Lévy–Itô decomposition, of Poincaré-invariant Markov processes on , the relativistic phase space in 1+1 spacetime dimensions. Then, by means of such processes, we construct Poincaré-invariant Gaussian random fields, and we prove a no-go theorem for the random fields corresponding to Brownian motions on .     

Conformally Invariant Quantization at Order Three

Let (M, g) be a pseudo-Riemannian manifold and the space of densities of degree on M. Denote the space of differential operators from to of order k and S _k with = – the corresponding space of symbols. We construct (the unique) conformally invariant quantization map . This result generalizes that of Duval and Ovsienko.     

New bounds on the number of bound states for Schrödinger operators

We consider the Schrödinger operatorH = – +V(|x|) onR ³. Letn denote the number of bound states with angular momentum (not counting the 2 + 1 degeneracy). We prove the following bounds onn . LetV 0 and d/dr r ^1-2p (-V)^{1 –p} 0 for somep [1/2, 1) then

Scaling characteristics of ac conductivity and dielectric function in fractal regime of the metal-insulator transition

An analysis of the ac conductivity _ac(), and the ac dielectric constant, (), of the metal-insulator percolation systems is presented in the critical regime near the transition threshold. It is argued that the polarization and relaxation of the finite fractal metallic clusters play dominant roles in controlling the dynamic response of the system on both sides of the threshold. The relaxation time constant of a fractal cluster is shown to scale with its size as withd _t = 4 – 2d +d _c + /, whered is tge Euclidean dimension, andd _c , , and are the scaling indices for the charging, the dc conductivity, and the correlation length respectively. The average time dependent response of the system is shown to scale with a new time scale , where is the correlation length and ₀ is a microscopic time constant. It is shown that at frequencies and with /d_t 1, in close agreement with experiments. The effects of the anomalous transport along the infinite cluster and the medium polarizability are also discussed.     

N\bar N annihilation at rest into five pions

annihilation at rest into five pions was studied in the ASTERIX spectrometer by stopping antiprotrons from LEAR in a H₂ or D₂ gas target. In annihilations in H₂ leading to the ^+–+–0 final state, the invariant mass spectra of two, three, and four pions show no evidence for any new narrow states. In the difference of the four pion mass spectra recoiling against a ^– and those recoiling against a ⁺ shows a resonance-like structure ( (1480)), confirming a previous analysis with a smaller event sample. The (1480) mass is shown to depend strongly on the momentum of the spectator proton. In H₂, we searched for the (1480) in the reaction (1480), (1480) ^+–+– to clarify whether the (1480) is identical to the 4-mode of thef ₂ (1520). No evidence for this decay mode was found. A prominent resonance in innihilation into five pions is the meson. We find two quasi-two-body intermediate states: and (1235). The mass and width of theb ₁ (1235) are determined to be:

Equivalence of Deviau's, Hestenes', and Parra's Formulations of Dirac Theory

Daviau showed the equivalence of matrix Dirac theory, formulated within a spinor bundle , to a Clifford algebraic formulation within space Clifford algebra Pauli algebra (matrices) biquaternions. We will show, that Daviau's map : is an isomorphism. It is shown that Hestenes' and Parra's formulations are equivalent to Daviau's Clifford algebra formulation, which uses outer automorphisms. The connection between different formulations is quite remarkable, since it connects the left and right action on the Pauli algebra itself viewed as a bi-module with the left (resp. right) action of the enveloping algebra . The isomorphism established in this article and given by Daviau's map does clearly show that right and left actions are of similar type. This should be compared with attempts of Hestenes, Daviau, and others to interprete the right action as the iso-spin freedom.     

Glueball Spectroscopy in Regge Phenomenology

We show that linear Regge trajectories for mesons and glueballs, and the cubic mass spectrum associated with them, determine a relation between the masses of the meson and the scalar glueball, , which implies MeV. We also discuss relations between the masses of the scalar, tensor and 3^-- glueballs, , which imply MeV.     

Neutrino dipole moments and charge radii in non-commutative space-time

In this paper we obtain a bound TeV on the scale of space-time non-commutativity considering photon-neutrino interactions. We compute *-dipole moments and *-charge radii originating from space-time non-commutativity and compare them with the dipole moments calculated in the neutrino-mass extended standard model (SM). The computation depends on the nature of the neutrinos, Dirac versus Majorana, their mass and the energy scale. We focus on Majorana neutrinos. The *-charge radius is found to be at TeV.Received: 17 June 2004, Published online: 18 August 2004     

The sphaleron at non-zero Weinberg angle

The sphaleron solution is calculated as a perturbation expansion in (g/g), to . For the physical value (g/g)=0.55 the original spherically symmetric sphaleron is a good approximation. However we expect the expansion to break down at (g/g)2 and the true solution to deviate strongly from the original solution above this value.     

Cantor Spectrum for the Almost Mathieu Operator

In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, on l ²() and its associated eigenvalue equation to deduce that for b0, ±2 and Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called Ten Martini Problem for these values of b and . Moreover, we prove that for |b|0 small or large enough all spectral gaps predicted by the Gap Labelling theorem are open.     

The decays of theρ′1 andω′1 mesons

New measurements of the reactione ⁺ e ^– confirm previous evidence for the decay₁ and provide a strong constraint on the mass of the₁. The results from a detailed analysis of these data are compared with those from a corresponding analysis of new data on the reactione ⁺ e ^–. It is shown that in accord with expectation. This identification of a strong₁ decay supports a previous claim that the₁ and theC(1480), observed in the channel, cannot be different manifestations of the same particle.     

A study of the decays of tau leptons produced on theZ resonance at LEP

From the analysis of a data sample corresponding to an integrated luminosity of 4.63 pb^–1 taken during the 1990 run of LEP at centre of mass energies between 88.2 GeV an 94.2 GeV, the tau decays and their charge conjugates have been studied. The following branching ratios have been measured; , , Br(^{– –} (K^–)v)=11.9±0.7±0.7%, BR (^{– –} v)= 22.4±0.8±1.3%, in good agreement with world averages. The measured electronic and muonic branching ratios lead to a measurement of the strong coupling constant, _s (m) = 0.26 _–0.12 ^+0.09 . Extrapolating the _s value fromm tom _Z yields _s (m_Z) = 0.109 _–0.028 ^+0.012 .The average polarizationP of taus produced in Z ^{s s} decays has also been measured using the above decay modes. The weighted mean of the polarizations obtained from the four decay modes isP =–0.24±0.07. This value ofP gives, in the improved Born approximation, a ratio between the axial and vector coupling constants of the tau of /a = 0.12 ± 0.04, and hence a value of the effective electroweak mixing parameter sin² _W(m _Z ² ).