8> =4tg, with the only singularity at t = 0. One would expect that dispersive smoothing should survive "small" perturbations of the free Hamiltonian H 0 = \Gamma\Delta. The problem is to determine what perturbations are "small". The case when the perturbed Hamiltonian has the form H = H 0 +V with a potential V = V (x), has been examined in [Ze], [OF], [Ki], [CFKS]. The dispersive smoothing takes place, for example, if the potential is infinitely differentiable, and it and all its derivatives are bounded, [Ze], [OF]. On the other hand, if V (x) grows quadratically or faster at infinity, then the singularities may resurrect, as the example of the quantum harmonic oscillator and Mehler's formula show ([Ze], [We], [CFKS], [MF]). On leave from St.Petersburg Branch of Steklov
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