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A fundamental result of Solomyak says that the number of negative eigenvaluesof a Schr\"odinger operator on a two-dimensional domain is bounded from aboveby a constant times a certain Orlicz norm of the potential. Here we show thatin the case of Dirichlet boundary conditions the constant in this bound can bechosen independently of the domain.

Bound on the number of negative eigenvalues of two-dimensional Schr\"odinger operators on domains