Generalized quadratic spectrum approximation in bounded and unbounded cases

The goal of this paper is to generalize concepts in spectral theory in order to define the quadratic spectrum associated to three bounded linear operators. This concept was initially defined for three matrices. Moreover, we construct a new method of spectral approximation to avoid the problem of spectral pollution. This problem is resolved with the obtention of property U under the norm convergence or the collectively compact convergence. Also, we make numerical tests on the quadratic pencil associated to Schrödinger's operator in order to validate our theoretical results and to show the efficiency of our method.