Two discrete-time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time-dependent Schrr odinger equation in two space variables. These are Crank-Nicolson and alternating direction implicit (ADI) schemes employing C 1 piecewise polynomial spaces of arbitrary degree 3 in each space variable. The stability of the schemes is examined and optimal order a prior...

The Motzkin numberMn is the cardinality of the set of sequencesMn = f(xi)ni=1 : xi 2 f 1; 0; 1g; all partial sums P k i=1 xi are nonnegative and P n i=1 xi = 0g. These sequences have a pictorial representation as Motzkin paths: lattice paths of upsteps (corresponding to +1), downsteps ( 1) and atsteps (0) that begin and end at, but never dip below, \ground level". For example, the M3 = 4 Motzki...

Wavelet applications to date have been dominated by signal and image processing. While perhaps not immediately obvious as a means of solving diierential equations, the growing body of literature in this area indicates that wavelets have a role to play here, too. Particularly attractive is their multilevel structure which leads to the possibility of data compression. In this paper we give some o...

We study the application of squeezed states in a quantum optical scheme for direct sampling of the phase space by photon counting. We prove that the detection setup with a squeezed coherent probe field is equivalent to the probing of the squeezed signal field with a coherent state. An example of the Schr odinger cat state measurement shows that the use of squeezed states allows one to detect cl...