On non-round points of the boundary of the numerical range and an application to non-selfadjoint Schrödinger operators

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We show that non-round boundary points of the numerical range of an unbounded operator (i.e. points where the boundary has infinite curvature) are contained in the spectrum of the operator. Moreover, we show that non-round boundary points, which are not corner points, lie in the essential spectrum. This generalizes results of Hübner, Farid, Spitkovsky and Salinas and Velasco for the case of bounded operators. We apply our results to non-selfadjoint Schrödinger operators, showing that in this case the boundary of the numerical range can be non-round only at points where it hits the essential spectrum.