Burq, N. Smoothing effect for Schrödinger boundary value problems. (English) Zbl 1061.35024 Duke Math. J. 123, No. 2, 403-427 (2004). 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. Reviewer: Viorel Iftimie (Bucureşti) Cited in 2 ReviewsCited in 46 Documents MSC: 35B65 Smoothness and regularity of solutions to PDEs 35Q40 PDEs in connection with quantum mechanics 35P25 Scattering theory for PDEs Keywords:nontrapping condition; Dirichlet boundary conditions; exterior problems; tangential pseudodifferential operator; trapped obstacles × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] M. Ben-Artzi and A. Devinatz, Regularity and decay of solutions to the Stark evolution equation , J. Funct. Anal. 154 (1998), 501-512. · Zbl 0914.35110 · doi:10.1006/jfan.1997.3211 [2] M. Ben-Artzi and S. Klainerman, Decay and regularity for the Schrödinger equation , J. Anal. Math. 58 (1992), 25-37. · Zbl 0802.35057 · doi:10.1007/BF02790356 [3] N. 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