The theoretical part of this monograph considers the distribution of the spectrum of operator polynomials, focussing on quadratic operator polynomials with discrete spectrum. Standard spectral problems in Hilbert spaces are of the form A- I for an operator A, and self-adjoint operators are of particular interest and importance, both theoretically and in applications. A characteristic feature of self-adjoint operators is that their spectra are real. Many spectral problems in theoretical physics and engineering can be described by such operators. However, a large class of problems, in...