engleski

What feeds oscillations in the Schroedinger type equations?

H-measures, as originally introduced by Luc Tartar and Patrick G\'erard, are well adapted to hyperbolic problems. In a recent study we gave an attempt to apply the H-measure theory to parabolic equations. In order to accomplish this task, we proposed a new variant of H-measures, called the parabolic H-measure, which is better suited to such kind of problems. The goal is to identify the microlocal energy, expressed by the parabolic H-measure, via oscillations in initial data and source terms. For the heat equation this can be achieved by means of the localisation property for the new variant. In fact, due to the hypoelliptic character of the equation, the microlocal energy contained in the initial data dies out. However, for the Schr\"odinger equation this is not the case. Similarly to the treatment of hyperbolic equations, the propagation property has to be developed. The preliminary results tell us that the microlocal energy is generated by the initial data, rather than by the source term function, as opposed to the situation involving the heat equation.

Parabolic H-measure; Propagation property; Schroedinger equation

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