## Smoothing effect for Schrödinger boundary value problems.(English)Zbl 1061.35024

The author proves the necessity of the nontrapping condition for the plain smoothing effect $$(H^{1/2})$$ for the Schrödinger equation with Dirichlet boundary conditions in exterior problems. The principal result reads as follows. Consider an arbitrary smooth domain with boundary $$\Omega\subset \mathbb{R}^d$$, with no infinite order contact with its boundary, and consider $$P$$ a second-order self-adjoint operator on $$L^2(\Omega)$$, with domain $$D\subset H^1_0(\Omega)$$ and such that the boundary is noncharacteristic. Denote by $$\varphi_s: {^bT^*}\Omega\setminus\{0\}\to {^bT^*}\Omega\setminus\{0\}$$ the bicharacteristic flow of the operator $$P$$ defined on the boundary cotangent bundle. Let $$A$$ be a classical tangential pseudodifferential operator of order $$1/2$$. Suppose that $$(z_0,\zeta_0)\in{^bT^*}\Omega\setminus\{0\}$$ satisfying the trapping assumption $\int^0_{-\infty} |\sigma_{1/2}(A)(\varphi_s(x_0, \zeta_0))|^2\,ds= +\infty,$ where $$\sigma_{1/2}(A)$$ is the principal symbol of $$A$$. Then for any $$t_0> 0$$ the map $u_0\in C^\infty_0\subset L^2(\Omega)\mapsto A^{itP}_e u_0\in L^2([0,t_0]; L^2(M)),\;M= \mathbb{R}\times\Omega,$ is not bounded (even for data with fixed compact support).
The author also gives a class of trapped obstacles (Ikawa’s example) for which he can prove a weak $$(H^{-{1\over 2}- \varepsilon})$$ smoothing effect.

### MSC:

 35B65 Smoothness and regularity of solutions to PDEs 35Q40 PDEs in connection with quantum mechanics 35P25 Scattering theory for PDEs
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### References:

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