Spectral inequalities for Schrödinger operators and parabolic observability

Abstract

This work examines the observability of abstract Cauchy problems corresponding to generators of a strongly continuous operator semigroup. A special focus is put on the investigation of sufficient conditions on the sensor sets that ensure observability. This is done by means of new spectral inequalities and dissipation estimates. First of all, the situation of selfadjoint operators is examined. Suitable spectral inequalities are proven for selfadjoint Schrödinger operators, where the potential has either mild local singularities or is (at least in certain coordinate directions) a power growth potential. In the second case, the prime examples are the partial harmonic oscillators. While the spectral inequality is proven in the case of mild local singularities for equidistributed sensor sets, in the case of potentials with power growth it is shown that these sets may become sparse in directions where the potential grows unboundedly. In particular, the presented results allow to consider sensor sets with finite measure in certain cases, e.g., for the harmonic oscillator. Subsequently, new dissipation estimates for semigroups generated by quadratic differential operators are shown. Here the associated so-called comparison operator is a partial harmonic oscillator and its specific form is characterized by the singular space of the generator. In this context, the aforementioned spectral inequalities allow to establish observability of these semigroups from sensor sets that become sparse. At last, the situation is investigated where the range of the semigroup is contained in a Gelfand-Shilov space. Establishing an uncertainty principle with error term for functions from these spaces, the observability is also proven in this situation. The work is concluded by an appendix discussing a unique continuation principle for the gradient and its applications in the theory of random divergence-type operators.