Critical sets of eigenfunctions of the Laplacian

We give an upper bound for the (n−1)$(n-1)$-dimensional Hausdorff measure of the critical set of eigenfunctions of the Laplacian on compact analytic n$n$-dimensional Riemannian manifolds. This is the analog of a result on the nodal set of eigenfunctions by H. Donnelly and C. Fefferman.     

On the second eigenfunctions of the Laplacian in R2

A conjecture about the nodal line of a second eigenfunction states that the nodal line of a second eigenfunction divides the domain by intersecting with the boundary of transversely, where is a bounded convex domain ofR ². We prove this conjecture provided has a symmetry. Also, we prove the multiplicity of the second eigenvalue is two at most provided is a bounded convex domain ofR ².Supported in part by NSF DMS 84-09447Home Institution: Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA     

Exponential decay in the stark effect

LetH=–+V+Fx ₁ withV(x ₁,x ) analytic in the first variable andV(x ₁+ia, x ) bounded and decreasing to zero asx for eacha . Let be an eigenvector of –+V with negative eigenvalue. Among our results we show that forF0, (,e ^–H ) decays exponentially at a rate governed by the positions of the resonances ofH. This exponential decay is in marked contrast to conventional atomic resonances for which power law decay is the rule.Research supported by NSF Grant No. MCS 78-00101.     

Perturbation theory for the decay rate of eigenfunctions in the generalizedN-body problem

Simple examples are known where eigenfunctions decay faster than the usual upper bounds would lead one to believe. We develop aspects of the perturbation theory of the decay rate of eigenfunctions as measured by radial exponential weights. We show that generically (in a Baire category sense) eigenfunction decay rates are governed by the lowest threshold.Supported in part by NSF grant DMS-8807816     

Exponential decay of bound state wave functions

The spatial decay properties of the wave functions of multiparticle systems are investigated. The particles interact through pair potentials in the classR+L . The bound states lie below the bottom of the continuous spectrum of the system. Exponential decay, in anL ² sense, is proven for these wave functions. The result is the best possible one which will cover every potential in this class.Based on a thesis submitted to Princeton University in partial fulfillment of the degree of Doctor of Philosophy.     

Perturbed periodic Hamiltonians: Essential Spectrum and exponential decay of eigenfunctions

Spectral properties of – +V(x), whereV(x) lies in a neighbourhood of the periodic case and describes various models of disorder, are studied. We prove the exponential decay of generalized eigenfunctions corresponding to energies in the resolvent set of the unperturbed periodic Hamiltonian, as well as the stability of the essential spectrum for the dislocation disorder in two dimensions.     

On the localization of eigenfunctions of the Laplace operator in rectangular domains

The localization problem is considered for eigenfunctions of the Laplace operator in a domain that consists of two rectangles linked by a small hole. The localization of the eigenfunction is proven in a subdomain. The velocity is estimated for the convergence of an eigenvalue of the original problem to a subdomain eigenvalue.     

Exponential decay of connectivities in the two-dimensional ising model

We prove some results concerning the decay of connectivities in the low-temperature phase of the two-dimensional Ising model. These provide the bounds necessary to establish, nonperturbatively, large-deviation properties for block magnetizations in these systems. We also obtain estimates on the rate at which the finite-volume, plus-boundary-condition expectation of the spin at the origin converges to the spontaneous magnetization.On leave from São Paulo University, Brazil.     

Exponential decay of the power spectrum of turbulence

The analyticity on a strip of the solutions of Navier-Stokes equations in 2D is shown to explain the observed fast decay of the frequency power spectrum of the turbulent velocity field. Some subtleties in the application of the Wiener-Khinchine method to turbulence are resolved by showing that the frequency power spectrum of turbulent velocities is in fact a measure exponentially decaying for frequency ±. Our approach also shows that the conventional procedures used in analyzing data in turbulence experiments are valid even in the absence of the ergodic property in the flow.     

Exponential decay and scaling laws in noisy chaotic scattering

In this Letter we present a numerical study of the effect of noise on a chaotic scattering problem in open Hamiltonian systems. We use the second order Heun method for stochastic differential equations in order to integrate the equations of motion of a two-dimensional flow with additive white Gaussian noise. We use as a prototype model the paradigmatic Hénon-Heiles Hamiltonian with weak dissipation which is a well-known example of a system with escapes. We study the behavior of the scattering particles in the scattering region, finding an abrupt change of the decay law from algebraic to exponential due to the effects of noise. Moreover, we find a linear scaling law between the coefficient of the exponential law and the intensity of noise. These results are of a general nature in the sense that the same behavior appears when we choose as a model a two-dimensional discrete map with uniform noise (bounded in a particular interval and zero otherwise), showing the validity of the algorithm used. We believe the results of this work be useful for a better understanding of chaotic scattering in more realistic situations, where noise is presented.     

Exponential decay lifetimes of excitons in individual single-walled carbon nanotubes

The dynamics of excitons in individual semiconducting single-walled carbon nanotubes was studied using time-resolved photoluminescence (PL) spectroscopy. The PL decay from tubes of the same (n,m) type was found to be monoexponential, however, with lifetimes varying between less than 20 and 200 ps from tube to tube. Competition of nonradiative decay of excitons is facilitated by a thermally activated process, most likely a transition to a low-lying optically inactive trap state that is promoted by a low-frequency phonon mode.     

Monopole decay in a variable external field

The rate of monopole decay into a dyon and an electron in an inhomogeneous external electric field is calculated by semiclassical methods. Comparison is made to an earlier result where this quantity was calculated for a constant field. Experimental and cosmological tests are suggested. The text was submitted by the authors in English.     

Exponential decay properties of Wannier functions and related quantities

The spatial decay properties of Wannier functions and related quantities have been investigated using analytical and numerical methods. We find that the form of the decay is a power law times an exponential, with a particular power-law exponent that is universal for each kind of quantity. In one dimension we find an exponent of -3/4 for Wannier functions, -1/2 for the density matrix and for energy matrix elements, and -1/2 or -3/2 for different constructions of nonorthonormal Wannier-like functions.     

Allowed electron-capture branches in the decay of 34mCl

The decay of ^34mCl has been studied with 36 and 100 cm³ Ge(Li) detectors and with a high-resolution large volume Ge(Li)-NaI(Tl) Compton-suppression spectrometer. The ^34mCl activity was produced with the reaction ²⁴Mg(¹²C, pn)³⁴Cl at $\text{E(}{}^{12}\text{C}\text{) = 35}\text{MeV}$ by bombarding thick natural Mg targets. The half-life was measured to be $\text{τ}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 32.06 ± 0.08}\text{min}$. Nine γ-ray transitions were observed including four γ-rays not seen previously. The measured γ-ray intensities determine new electron-capture branches of (0.030 ± 0.006) % and (0.032 ± 0.003) % to ³⁴S levels at 4.69 and 4.88 MeV with log ft values of 5.48 ± 0.08 and 5.26 ±0.04, respectively. These log ft values imply allowed transitions and are consistent with the known J_π values of J_π = 4⁺ and 3⁺ of the 4.69 and 4.88 MeV levels, respectively. A lower limit of log ft ? 6.9 is obtained for the allowed electron-capture branch to the J^π = 2⁺, 4.89 MeV level. Other previously observed decay branches have been confirmed. Since the decay of ^34mCl proceeds (46.9 ± 1.0)% to the ³⁴Cl ground state, the latter decay was studied concurrently; ³⁴Cl(0)with J^π = 0⁺, T = 1 decays exclusively to its analog ³⁴S(0) and the sum of the intensities of four other allowed branches is less than 1.2 × 10^?4 of the intensity of the ground-state branch. The experimental results are compared with recent shell-model calculations performed in a large configuration space.     

Exponential decay of derivatives of covariance operators on the lattice

We prove exponential decay for derivatives of covariance operators on the lattice. This result is obtained by using random walk methods on the v-dimensional lattice and a certain estimate on the generating function of the one-dimensional random walk. The result is useful in the frame of the cluster and mean field expansion of continuous spin models on the lattice.     

Exponential vs. slower decay of time dependent structure factors

This article deals with the possible forms of decay of time dependent structure factors in rather generic systems. It concentrates on the distinction between exponential and slower forms of decay. The common form of decay is assumed traditionally to be exponential or approximately so. Slow decay on the other hand, is characteristic of systems such as glasses that may be viewed as being very special. The purpose of the present article is to show, that slow decay is not related necessarily to the system under consideration but rather to the structure factor we prefer to measure. We may think of the decay of structure factors in certain systems as being exponential or slow but this may just reflect experimental practical preference to measure certain correlations and not others.     

Precision measurement of the half-life and the decay branches of 62Ga

In an experiment performed at the Accelerator Laboratory of the University of Jyväskylä, the -decay half-life of ⁶²Ga has been studied with high precision using the IGISOL technique. A half-life of T_1/2 = 116.09(17) ms was measured. Using - coincidences, the intensity of the 954 keV transition and an upper limit of the -decay feeding of the 0⁺₂ state have been extracted. The present experimental results are compared to previous measurements and their impact on our understanding of the weak interaction is discussed.     

Exponential speedup with a single bit of quantum information: measuring the average fidelity decay

We present an efficient quantum algorithm to measure the average fidelity decay of a quantum map under perturbation using a single bit of quantum information. Our algorithm scales only as the complexity of the map under investigation. Thus for those maps admitting an efficient gate decomposition, it provides an exponential speedup over known classical procedures. Fidelity decay is important in the study of complex dynamical systems, where it is conjectured to be a signature of eigenvector statistics. Our result also illustrates the role of chaos in the process of decoherence.     

Exponential decay of Green's functions for a class of long range Hamiltonians

We consider a class of long range Hamiltonians with diagonal disorder onl ² (Z). For anyergodic potentialV with non-empty essential range, we prove the exponential decay of the Green's functions for energies in the essential range. IfV is independent identically distributed, we obtain the exponential decay of the Green's functions for all coupling constant >0. Moreover the Hamiltonian has only pure point spectrum.