engleski

Scale-free Uncertainty Principles And Wegner Estimates For Random Breather Potentials

We present new scale-free quantitative unique continuation principles for Schrödinger operators. They apply to linear combinations of eigenfunctions corresponding to eigenvalues below a predescribed energy, and can be formulated as an uncertainty principle for spectral projectors. This extends recent results of Rojas- Molina & Veselić, and Klein. We apply the scale- free unique continuation principle to obtain a Wegner estimate for a random Schrödinger operator of breather type. It holds for arbitrarily high energies. Schrödinger operators with random breather potentials have a non-linear dependence on random variables. We explain the challanges arising from this non-linear dependence. could be naturally implemented. Based on this criterion, a discretization procedure is constructed for the calculation of the optimal damping coefficient. If the internal damping is present, we show that this procedure can be used to obtain the optimal damping operator in the case of optimization over the set of all admissible damping operators.

scale-free unique continuation property ; equidistribution prop- erty ; observability estimate ; Carleman estimate ; Schrödinger operator ; quantitative uncer- tainty principle ; random breather potential ; Wegner estimate

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