On singularities of solutions of the Schrödinger equation for a free particle
The author studies the wave-front set \(WFu (\cdot, t)\) of the distribution solution \(u(x,t) \in H^{- \infty} (\mathbb{R}^n)\) for the Cauchy problem: \(i \partial_t u = \Delta u,\;u(x,0) = u_0 (x)\); \(x \in \mathbb{R}^n\). Here \(WFu (\cdot,t)\), \(t \neq 0\), is the complement of the set of \((x^0, \xi^0)\) such that the estimate \(|\int u (x,t) \varphi (x) e^{- \lambda \langle x, \xi \rangle} dx |< C_N \lambda^{- N}\), holds for any \(\lambda > \lambda^0\), any \(\xi \in U (\xi^0)\), any natural number \(N\) and any \(\varphi (x) \in {\mathcal C}_0^\infty\) with support concentrated in a neighbourhood of \(x^0\). He gives the set \(Sp_\infty u_0 (x)\) including \(\{(x, \xi, t); (x, \xi) \in WFu (x,t)\}\) in Theorem 1. The condition \((U_\varepsilon (x^0, \xi, t_0)) \cap Sp_\infty u_0 (x) = \emptyset\) derives a neighbourhood \(V\) of \(x^0\) such that \(u (x,t_0)\) is infinitely differentiable in \(V\). Next he treats the Cauchy problem with initial data of the special form \(u_0(x) = a(x) \exp (i \varphi (x))\). In Theorem 2 he gives the set including \(WFu (\cdot, t)\) by using the characteristic set \(\text{Char} (\varphi : x^0, \eta^0)\) etc. Finally he gives some examples.
scientific article; zbMATH DE number 797879