Effect of collision and magnetic field on quantum beat in biacetyl

Molecular quantum beats are shown for the case of biacetyl, to vary strongly under the influence of weak magnetic fields. Values for dephasing constants for a gas-phase molecular system were determined. Information on sub-Doppler level splittings in the intersystem-crossing process is obtained.     

Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

We investigate boundedness of the evolutione ^itH in the sense ofL ²(ℝ³→L ²(ℝ³) as well asL ¹(ℝ³→L ^∞(ℝ³) for the non-selfadjoint operator where μ>0 andV ₁, V₂ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave, and the aforementioned bounds are needed in the study of nonlinear asymptotic stability of such standing waves. We derive our results under some natural spectral assumptions (corresponding to a ground state soliton of NLS), see A1)–A4) below, but without imposing any restrictions on the edges±μ of the essential spectrum. Our goal is to develop an “axiomatic approach,” which frees the linear theory from any nonlinear context in which it may have arisen. This work was initiated in June of 2004, while the first author visited Caltech, and he wishes to thank that institution for its hospitality and support. The first author was partially supported by the NSF grant DMS-0303413. The second author was partially supported by a Sloan fellowship and the NSF grant DMS-0300081. The authors thank Avy Soffer for his interest in this work.     

Fine Properties of the Integrated Density of States and a Quantitative Separation Property of the Dirichlet Eigenvalues

We consider one-dimensional difference Schr?dinger equations with real analytic function V(x). Suppose V(x) is a small perturbation of a trigonometric polynomial V ₀(x) of degree k ₀, and assume positive Lyapunov exponents and Diophantine ω. We prove that the integrated density of states is H?lder continuous for any k > 0. Moreover, we show that is absolutely continuous for a.e. ω. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval , does not exceed , k > 0. Moreover, for , this averaged number does not exceed exp , for any . For the integrated density of states of the problem this implies that for any . To investigate the distribution of the Dirichlet eigenvalues of on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants with complexified phase x, and frozen ω, E. We prove equidistribution of these zeros in some annulus and show also that no more than 2k ₀ of them fall into any disk of radius exp. In addition, we obtain the lower bound (with δ > 0 arbitrary) for the separation of the eigenvalues of the Dirichlet eigenvalues over the interval [0, N]. This necessarily requires the removal of a small set of energies. Received: February 2006, Accepted: December 2007     

Evolutionarily stable sets

This paper provides definitions for the evolutionary stability of sets of strategies based on simple fitness comparisons in the spirit of the definition of an evolutionarily stable strategy (ESS) by Taylor and Jonker (1978). It compares these with the set-valued notions of Thomas (1985d) and Swinkels (1992). Provided only that the fitness function is analytic, our approach yields an alternative characterization of Thomas' evolutionarily stable sets (ES Sets) which does not rely on the structure or topology of the underlying strategy space. Moreover, these sets are shown to have a very special geometric structure and to be finite in number. For bimatrix games ES Sets are shown to be more uniformly robust against mutations than apparent from the definition and hence to be equilibrium evolutionarily stable sets in the sense of Swinkels (1992). Received November 1995/Final version December 2000     

A Geometric Inequality with Applications to the Kakeya Problem in Three Dimensions

((Without abstract)) Submitted: May 1997, Revised version: August 1997     

Femtosecond dynamics after ionization: 2-phenylethyl-N,N-dimethylamine as a model system for nonresonant downhill charge transfer in peptides

The cation of 2-phenylethyl-N,N-dimethylamine (PENNA) offers two local sites for the charge: the amine group and 0.7 eV higher in energy the phenyl chromophore. In this paper, we investigate the dynamics of the charge transfer (CT) from the phenyl to the amine site. We present a femtosecond resonant two-color photoionization spectrum which shows that the femtosecond pump laser pulse is resonant in the phenyl chromophore. As shown previously with resonant wavelengths the aromatic phenyl chromophore can be then selectively ionized. Because the state "charge in the phenyl chromophore" is the first excited state in the PENNA cation, it can relax to the lower-energetic state "charge in the amine site". To follow this CT dynamics, femtosecond probe photoabsorption of green light (vis) is used. The vis light is absorbed by the charged phenyl chromophore, but not by the neutral phenyl and the neutral or cationic amine group. Thus, the absorption of vis photons of the probe laser pulse is switched off by the CT process. For detection of the resonant absorption of two or more vis photons in the cation the intensity of a fragmentation channel is monitored which opens only at high internal energy. The CT dynamics in PENNA cations has a time constant of 80 +/- 28 fs and is therefore not a purely electronic process. Because of its structural similarity to phenylalanine, PENNA is a model system for a downhill charge transfer in peptide cations.     

A novel mechanism of proton transfer in protonated peptides

The study presents quantum-chemical calculations on proton transfer in protonated N-acetylglycyl-N1-methylglycinamide (AGA) as a short oligopeptide model. All calculations employ the B3LYP functional and the 6-31++G** basis set. Two different mechanisms of proton transfer are discussed. The rate-determining step of the first mechanism exhibits an energy barrier of about 17.7 kcal mol-1, and it is represented by an isomerization of the proton around the double bond of the carbonyl group. The second mechanism is based on the large conformational flexibility of AGA, where all carbonyl oxygens cooperate. The rate-determining step of this mechanism exhibits an energy barrier of only 8.3 kcal mol-1.     

Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension

Standing wave solutions of the one-dimensional nonlinear Schrödinger equation

with are well known to be unstable. In this paper we show that asymptotic stability can be achieved provided the perturbations of these standing waves are small and chosen to belong to a codimension one Lipschitz surface. Thus, we construct codimension one asymptotically stable manifolds for all supercritical NLS in one dimension. The considerably more difficult -critical case, for which one wishes to understand the conditional stability of the pseudo-conformal blow-up solutions, is studied in the authors' companion paper Non-generic blow-up solutions for the critical focusing NLS in 1-d, preprint, 2005.

     

On the Integrated Density of States for Schrödinger Operators on ℤ2 with Quasi Periodic Potential

In this paper we consider discrete Schr?dinger operators on the lattice ℤ² with quasi periodic potential. We establish new regularity results for the integrated density of states, as well as a quantitative version of a “Thouless formula”, as previously considered by Craig and Simon, for real energies and with rates of convergence. The main ingredient is a large deviation theorem for the Green's function that was recently established by Bourgain, Goldstein, and the author. For the integrated density of states an argument of Bourgain is used. Finally, we establish certain fine properties of separately subharmonic functions of two variables that might be of independent interest. Received: 13 February 2001 / Accepted: 21 May 2001