Domain perturbations, capacity and shift of eigenvalues

Noll, André

Journées équations aux dérivées partielles (1999), article no. 8, 10 p.

After introducing the notion of capacity in a general Hilbert space setting we look at the spectral bound of an arbitrary self-adjoint and semi-bounded operator $H$. If $H$ is subjected to a domain perturbation the spectrum is shifted to the right. We show that the magnitude of this shift can be estimated in terms of the capacity. We improve the upper bound on the shift which was given in Capacity in abstract Hilbert spaces and applications to higher order differential operators (Comm. P. D. E., 24:759–775, 1999) and obtain a lower bound which leads to a generalization of Thirring’s inequality if the underlying Hilbert space is an $L^2$-space. Moreover, a similar capacitary upper bound for the second eigenvalue is established. The results are finally applied to higher-order partial differential operators.

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