The notion of capacity is a useful tool in measuring the magnitude of the shift of the spectral bound for the Laplace operator in \(L^{2}(\Omega)\) \((\Omega\subset R^{d}\) or a Riemannian manifold) at the domain perturbation [\textit{M. Flucher}, J. Math. Anal. Appl. 193, No. 1, 169-199 (1995; Zbl 0836.35105)]. In the author's previous work [Commun. Partial Differ. Equations 24, No. 3-4, 759-775 (1999; Zbl 0929.47023)] this notion has been extended for selfadjoint operators acting in arbitrary real or complex Hilbert spaces, and some upper and lower bounds for the shift of the spectral bound have been obtained. In the reviewed article this upper bound is improved and another lower bound is proved which leads to a generalization of Thirring's inequality for \(L^{2}\)-space. Capacitary upper bound for the second eigenvalue is obtained. Such results are applied to elliptic constant coefficient differential operators of arbitrary order. It is given a capacity characterization for the existence of a shift of the spectral bound of positive length which works for operators with spectral bound of arbitrary type.