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Control for Schrödinger equation on hyperbolic surfaces. (English) Zbl 1417.58013
Summary: We show that any nonempty open set on a hyperbolic surface provides observability and control for the time dependent Schrödinger equation. The only other manifolds for which this was previously known are flat tori [A. Haraux, J. Math. Pures Appl. (9) 68, No. 4, 457–465 (1989; Zbl 0685.93039); S. Jaffard, Port. Math. 47, No. 4, 423–429 (1990; Zbl 0718.49026); V. Komornik, J. Math. Pures Appl. (9) 71, No. 4, 331–342 (1992; Zbl 0889.34055)]. The proof is based on the main estimate in [S. Dyatlov and the author, Acta Math. 220, No. 2, 297–339 (2018; Zbl 1404.28010)] and standard arguments in control theory.

MSC:
58J05 Elliptic equations on manifolds, general theory
35B45 A priori estimates in context of PDEs
35Q41 Time-dependent Schrödinger equations and Dirac equations
35R01 PDEs on manifolds
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