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Asymptotics for optimal design problems for the Schrödinger equation with a potential. (English) Zbl 1460.93020

Summary: We study the problem of optimal observability and prove time asymptotic observability estimates for the Schrödinger equation with a potential in \(L^{\infty} (\Omega)\), with \(\Omega \subset \mathbb{R}^d\), using spectral theory. An elegant way to model the problem using a time asymptotic observability constant is presented. For certain small potentials, we demonstrate the existence of a nonzero asymptotic observability constant under given conditions and describe its explicit properties and optimal values. Moreover, we give a precise description of numerical models to analyze the properties of important examples of potentials wells, including that of the modified harmonic oscillator.

MSC:

93B07 Observability
93B60 Eigenvalue problems
93C20 Control/observation systems governed by partial differential equations
35Q41 Time-dependent Schrödinger equations and Dirac equations
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