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We give a new representation of solutions to a class of time-dependent Schrödinger type equations via the short-time Fourier transform and the method of characteristics. Moreover, we also establish some novel estimates for oscillatory integrals which are associated with the fractional power of negative Laplacian $${( - \Delta )^{\kappa /2}}$$ with 1 ⩽ κ ⩽ 2. Consequently the classical Hamiltonian...
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