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Uncertainty principle, minimal escape velocities, and observability inequalities for Schrödinger equations. (English) Zbl 1472.35324

Summary: We develop a new abstract derivation of the observability inequalities at two points in time for Schrödinger type equations. Our approach consists of two steps. In the first step we prove a Nazarov type uncertainty principle associated with a non-negative self-adjoint operator \(H\) on \(L^2(\mathbb{R}^n)\). In the second step we use results on asymptotic behavior of \(e^{-itH} \), in particular, minimal velocity estimates introduced by I. M. Sigal and the second author [Invent. Math. 99, No. 1, 115–143 (1990; Zbl 0702.35197)]. Such observability inequalities are closely related to unique continuation problems as well as controllability for the Schrödinger equation.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
35B40 Asymptotic behavior of solutions to PDEs
93B05 Controllability

Citations:

Zbl 0702.35197
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