As a basis for electronic structure calculations, Gaussians are inconvenient because they show unsuitable behavior at larger distances, while Hankel functions are singular at the origin. This paper discusses a new set of special functions which combine many of the advantageous features of both families. At large distances from the origin, these “smoothed Hankel functions” resemble the standard Hankel functions and therefore show behavior similar to that of an electronic wave function. Near the origin, the functions are smooth and analytical. Analytical expressions are derived for two-center integrals for the overlap, the kinetic energy, and the electrostatic energy between two such functions. We also show how to expand such a function around some point in space and discuss how to evaluate the potential matrix elements efficiently by numerical integration. This supplies the elements needed for a practical application in an electronic structure calculation.

We consider a slowly rotating Timoshenko beam in a horizontal plane whose movement is controlled by the angular acceleration of the disk of a driving motor into which the beam is clamped. The problem to be solved is to transfer the beam from a position of rest into a position of rest under a given angle within a given time. We show that this problem is solvable, if the time of rotation prescribed is large enough.

A vibrating plate is here taken to satisfy the model equation: utt + A2u = 0 (where AZu := A(Au); A = Laplacian) with boundary conditions of the form: u~ = 0 and (Au), = cp = control. Thus, the state is the pair [u, ut] and controllability means existence of cp on Y..'= (0, T)× 0~2 transfering 'any' [u, ut] 0 to 'any' [u, ut] r. The formulation is given by eigenfunction expansion and duality. The substantive results apply to a rectangular plate. For large T one has such controllability with [[~[[ = O(T 1/2). More surprising is that (based on a harmonic analysis estimate [11]) one has controllability for arbitrarily short times (in contrast to the wave equation: utt = Au) with log llcpl] = O(T -1) as T~ 0. Some related results on minimum time control are also included. IntroductionThis paper may be viewed as a continuation of [10] in studying the boundary controllability properties of a vibrating plate. We note earlier results [2, 3] on the vibrating beam but it is only more recently [5,10,11] the results have become available other than in one dimension.A standard (linearized) model of a vibrating solid is given by PUtt q-A[aAut+aAu ] = f (0.1) *

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This work continues the authors' previous investigation on the controllability problem of a slowly rotating Timoshenko beam. We obtain conditions of exact controllability under the assumption that the parameter γ appearing in the model equation is rational. Our result rests on a generalization of the theorem by Ullrich on the Riesz basis property of exponential divided differences.