Quantum Spectrum of Some Anharmonic Central Potentials: Wave Functions Ansatz

Based on the ansatz to the wave functions, the quasi-exact solutions of the 2D Schrödinger equation with some anharmonic potentials are reviewed and analyzed if admitting restrictions on the parameters of the potential and the angular momentum m. These potentials are taken as the screened Coulomb potential V(r)=a/r+b/(r+), the singular one-fraction power one V(r)=ar ^–1/2+br ^–3/2 and the singular two-fraction one V(r)=ar ^2/3+br ^–2/3+cr ^–4/3. The latter one is found that the hidden symmetry exists if substituting r–ir. It will reverse the signs of E and c of quantum system, leaving the remaining parameters invariant.     

Determining the strengths of hydrogen bonds in solid-state ammonia and urea: insight from periodic DFT calculations

Plane-wave density functional theory has been applied to determine the strengths of hydrogen bonds in the phase I crystal structures of ammonia and urea. For ammonia, each component of the trifurcated hydrogen bond has been found to be almost as strong as a standard N-H.N interaction, and for urea the strengths of the two different N-H.O interactions have been determined by a quantum mechanical technique for the first time.     

Experiments on the interactions between impurities and solitary waves in lattice model

The interactions between solitary waves and impurities have been studied experimentally in a 1D nonlinear coupled pendulum chain under vertical excitation. The mass and the coupling are unique, except a single pendulum with length impurity in the chain. The experiment reveals: the long impurity repels breather and attracts kink while the short one attracts breather and repels kink under higher frequency driving, and the long impurity attracts breather and repels kink while the short one repels breather and attracts kink under the lower frequency driving. These results prove the current theoretical prediction based on continuum-limit approximation.     

Nonlinear Dynamics of a Two-Mode Wave Packet with Strong Linear Intermode

We investigate the dynamics of the wave packet formed by two codirected strongly interacting waves propagating in a medium with cubic nonlinearity. We obtain a soliton solution of the nonlinear Schrödinger equation in the degenerate case where the wave packet is described by a single partial momentum. In the nondegenerate case, we use the variational method to find the equation for the pulse duration, which turns out to be analogous to the equation for the coordinate in the Kepler problem. Solving it, we find the dependences of the pulse duration on the propagation distance in the cases of finite and infinite propagation regimes.     

Operator kernel estimates for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.

     

Retrieval Properties of Bidirectional Associative Memories

We study a two-layer neural network made of N and M(N) neurons, producing a two-way association search for a family of p(N) patterns, where each pattern is a pair of two independent sub-categories of information having respectively N and M(N) components. In terms of the ratio =lim _N M(N)/N, we study the retrieval capability of this network and show that there exists, at least, three regimes of association for which we determine the evolution of the threshold _c () of the storage capacity =lim _N p(N)/N.     

BLOWUP OF SOLUTIONS TO THE CAUCHY PROBLEM FOR NONLINEARWAVE EQUATIONS

gi. IntroductionThis paper deals with solutionS of certain nonlinear wave equationS Of the formcorresponding to Antial conditionSwuersis the wave OPerstor.we are interested in showing the ~ up" Of solutions to (1.1)--(1.2). For that, wereIf ac ~ 1)(n ~ 1) > 2, global solutions of ~ equation subject to very general perturbationsof order p exist Provided the initial data are swhciently small (see I6] and references therein).We are also interested in esthaattw the take when "blow up" occurs. …     

On the Determination of Free Evolution in the Lax--Phillips Scattering Scheme for Second-Order Operator-Differential Equations

The paper presents some necessary and sufficient conditions on abstract positive self-adjoint operators L under which the operator-differential equation determines a free evolution in the Lax--Phillips scattering scheme.     

Laser Stabilisation for the Measurement of Thermal Noise

In order to measure the thermal noise of a mirror suspended in a vacuum it is necessary for the length measurement error due to intensity and frequency noise of the probe laser to be reduced below the thermal noise level. Here we report on an experiment to reduce the frequency and intensity noise of a 40mW Nd:YAG laser for this purpose. The frequency is stabilised using the standard reflection locking technique. To stabilise the laser intensity a technique which uses the properties of an in loop light field has been developed. This technique is capable of suppressing the intensity noise below the shot noise limit without reducing the useful laser power. A servo based on this technique has been designed and tested. The experimental results indicate that the laser noise can be reduced to a level which will allow a displacement sensitivity of 1.5 × 10^-19m/ Hz for the detection of thermal noise in a frequency band of 10 to 500Hz.