On singularities of solutions of the Schrödinger equation for a free particle. (English. Russian original) Zbl 0831.35040

Shananin, N. A.

Math. Notes 55, No. 6, 626-631 (1994); translation from Mat. Zametki 55, No. 6, 116-123 (1994).

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.

Reviewer: H.Yamagata (Osaka)

MSC:

35J10	Schrödinger operator, Schrödinger equation
35B65	Smoothness and regularity of solutions to PDEs
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
42B10	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Keywords:

wave-front set; Cauchy problem

References:

[1]	N. A. Shananin, ”On a class of Fourier integrals,”Mat. Sb.,180, 750–761 (1989).
[2]	L. Hörmander,Analysis of Linear Partial Differential Operators [Russian translation], Mir, Moscow (1986).