Let A be a selfadjoint operator on a Hilbert space and \(u_ 1,u_ 2,...,u_ r\) an orthonormal set in H such that for some real \(\lambda_ 0\) we have \(\max \| (A-\lambda_ 0)u_ k\| =\epsilon\). Such \(u_ i\) and \(\lambda_ 0\) are called quasimodes and quasieigenvalues. Assuming that A has at most r eigenvalues in \([\lambda_ 0- 4\sqrt{3}r\epsilon, \lambda_ 0+4\sqrt{3}r\epsilon],\) then, as the author shows, one can get more precise information on the eigenvalues near \(\lambda_ 0\). This result is then applied to Schrödinger equations with potentials having special properties.