Limiting set of second order spectra

M.Levitin and E.Shargorodsky purposed in a recent article, [math.SP/0212087], the use of the so called ``second order relative spectrum'', to find eigenvalues of self-adjoint operators in gaps of the essential spectrum. Let $M$ be a self-adjoint operator acting on the Hilbert space $H$. A complex number $z$ is in the second order spectrum of $M$ relative to a finite dimensional subspace $\Lambda\subset dom M^2$ if and only if the truncation to $\Lambda$ of $(M-z)^2$ is not invertible. It is remarkable that these sets seem to provide a general method to estimating eigenvalues free from the problems of spectral pollution present in most linear methods. In this notes we investigate rigorously various aspects of the use of second order spectrum to finding eigenvalues. Our main result shows that, under certain fairly mild hypothesis on $M$, the uniform limit of the second order spectra, as $\Lambda$ increases toward $H$, contains the isolated eigenvalues of $M$ of finite multiplicity. In applications the essential spectrum can be computed analytically, while precisely these eigenvalues are the ones that should be approximated numerically. Hence this method seems to combine non-pollution and approximation at a very high level of generality.