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A New Proof of Okaji’s Theorem for a Class of Sum of Squares Operators

Paulo D. Cordaro, Nicholas Hanges (2009)

Annales de l’institut Fourier

Let P be a linear partial differential operator with analytic coefficients. We assume that P is of the form “sum of squares”, satisfying Hörmander’s bracket condition. Let q be a characteristic point for P . We assume that q lies on a symplectic Poisson stratum of codimension two. General results of Okaji show that P is analytic hypoelliptic at q . Hence Okaji has established the validity of Treves’ conjecture in the codimension two case. Our goal here is to give a simple, self-contained proof of...

Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients

Kenichi Ito, Shu Nakamura (2012)

Annales de l’institut Fourier

We consider Schrödinger operators H on n with variable coefficients. Let H 0 = - 1 2 be the free Schrödinger operator and we suppose H is a “short-range” perturbation of H 0 . Then, under the nontrapping condition, we show that the time evolution operator: e - i t H can be written as a product of the free evolution operator e - i t H 0 and a Fourier integral operator W ( t ) which is associated to the canonical relation given by the classical mechanical scattering. We also prove a similar result for the wave operators. These results...

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