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* The author was supported by NSF Grant No. DMS 9706883.
Let P be a bi-variate algebraic polynomial of degree n with the
real senior part, and Y = {yj }1,n an n-element collection of pairwise
noncolinear unit vectors on the real plane. It is proved that there exists a rigid
rotation Y^φ of Y by an angle φ = φ(P, Y ) ∈ [0, π/n] such that P equals the
sum of n plane wave polynomials, that propagate in the directions ∈ Y^φ .
We study the local properties of the time-dependent probability density function for the free quantum particle in a box, i.e. the squared magnitude of the solution of the Cauchy initial value problem for the Schrödinger equation with zero potential, and the periodic initial data. √δ-families of initial functions are considered whose squared magnitudes approximate the periodic Dirac δ-function. The focus is on the set of rectilinear domains where the density has a special character, in particular,...
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