Abstract
AbstractThe article describes the construction of a parametrix, global in space, in the initial value problem for the Schrödinger equation i∂u/∂t = δu ‐ Q(x)u ‐ V(x)u, with Q a real quadratic form and V a real magnified image potential in magnified image. The hypothesis on V is that ∣V(x)∣ ≤ const. ∣x∣, and ∣V(α)(x)∣ ≤ const. ∣x∣1‐∣α∣(as ∣x∣ → + ∞). By conjugating the equation with the unitary group exp(itH)(H = ‐δ + Q(x)) the problem is reduced to a hyperbolic pseudodifferential operator to which a global version of the geometric optics method applies. The symbolic calculus used by the construction is based on weights that reflect the role of the propagation set.
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