  Optical properties of materials related to light absorption and scattering
are explained by the excitation of electrons. The Bethe-Salpeter equation is
the state-of-the-art approach to describe these processes from first principles
(ab initio), i.e. without the need for empirical data in the model. To harness
the predictive power of the equation, it is mapped to an eigenvalue problem via
an appropriate discretization scheme. The eigenpairs of the resulting large,
dense, structured matrix can be used to compute dielectric properties of the
considered crystalline or molecular system. The matrix always shows a $2\times
2$ block structure. Additionally, certain definiteness properties typically
hold. One form can be acquired for crystalline systems, another one is more
general and can for example be used to study molecules. In this work, we
present new theoretical results characterizing the structure of the two forms
in the language of non-standard scalar products. These results enable us to
develop a new perspective on the state-of-the-art solution approach for
crystalline systems. This new viewpoint is used to develop two new methods for
solving the eigenvalue problem. One requires less computational effort while
providing the same degree of accuracy. The other one improves the expected
accuracy, compared to methods currently in use, with a comparable performance.
Both methods are well suited for high performance environments and only rely on
basic numerical linear algebra building blocks.
