Tunable picosecond pulses from a short-cavity dye laser under ultra-high pressure using diamond-anvil cell

It is demonstrated that broadly tunable picosecond pulses are generated from a dye laser of very short cavity utilizing a diamond-anvil cell, which operates at pressures up to 10 GPa. The pulses as short as 5 ps are obtained from the rhodamine 6G dye laser pumped by a frequency-doubled Q-switched Nd: YAG laser. The way of tuning is based on the pressure induced shift of the emission peak of the dye. The laser is tunable over 20 nm by changing the pressure of the cell within 4 GPa.     

Parametric simplex algorithms for a class of NP-Complete problems whose average number of steps is polynomial

We will show that the average number of steps of parametric simplex algorithms for obtaining global minima of rank-one and rank-two bilinear-programming problems are lower-order polynomial functions of the problem size under the standard assumptions on the distribution of the data imposed in the probabilistic analysis of the simplex method. This means that there exist algorithms for some special class of NP-complete problems, whose average number of arithmetics are polynomial order of the problem size.     

Estimation of the degree of differencing of an ARIMA process

Summary A method of estimating the degree of differencing of an ARIMA process is proposed. This is based on fitting an AR model to the original and to each differenced series and calculating the residual sum of squares. As an application, we suggest an identification method of an ARI (p, d) process combining our method of estimating the degree of differencing with Akaike's Information Criterion.     

A Remark on Lp-Boundedness of Wave Operators¶for Two Dimensional Schrödinger Operators

Let H=−Δ+V be a two dimensional Schr?dinger operator with a real potential V(x) satisfying the decay condition , δ > 6. Let H ₀=−Δ. We show that the wave operators are bounded in L ^p (R ²) under the condition that H has no zero resonances or bound states. In this paper the condition , imposed in a previous paper (K. Yajima, Commun. Math. Phys. 208, 125–152 (1999)), is removed. Received: 13 September 2001 / Accepted: 15 October 2001     

Schrödinger evolution equations with magnetic fields

Dedicated to Professor H. Fujita on his 60th birthday.     

Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations

The fundamental solutionE(t,s,x,y) of time dependent Schrödinger equationsi?u/?t=?(1/2)Δu+V(t,x)u is studied. It is shown that

?E(t,s,x,y) is smooth and bounded fort≠s if the potential is sub-quadratic in the sense thatV(t,x)=o(|x|²) at infinity;

? in one dimension, ifV(t,x)=V(x) is time independent and super-quadratic in the sense thatV(x)≧C(1+|x|)^2+ε at infinity,C>0 and ε>0, thenE(t,s,x,y) is nowhereC ¹.

The result is explained in terms of the limiting behavior as the energy tends to infinity of the corresponding classical particle.