On linear and nonlinear elliptic eigenvalue problems

The numerical computation of the eigenvalues and eigenvectors of a self-adjoint operator on an infinite dimensional separable Hilbert space, is a standard problem of numerical analysis and scientific computing, with a wide range of applications in science and engineering. Such problems are encountered in particular in mechanics (vibrations of elastic structures), electromagnetism and acoustics (resonant modes of cavities), and quantum mechanics (bound states of quantum systems).

Galerkin methods provide an efficient way to compute the discrete eigenvalues of a bounded-from-below self-adjoint operator A laying below the bottom of the essential spectrum of A. On the other hand, Galerkin methods may fail to approximate discrete eigenvalues located in spectal gaps, that is between two points of the essential spectrum. In some cases, the Galerkin method cannot find some of the eigenvalues of A located in spectral gaps (lack of approximation); in other cases, the limit set of the spectrum of the Galerkin approximations of A contains points which do not belong to the spectrum of A (spectral pollution). Such problems arise in various applications, such as the numerical simulation of photonic crystals, of doped semiconductors, or of heavy atoms with relativistic models.

Quantum physics and chemistry also give rise to a variety of nonlinear elliptic eigenvalue problems, such as the stationary Gross-Pitaevskii equation modeling Bose-Einstein condensates, and the Hartree-Fock and Kohn-Sham models commonly used in electronic structure calculation.

In this talk, I will review various results obtained in collaboration with Rachida Chakir, Virginie Ehrlacher and Yvon Maday, on the numerical analysis of linear and nonlinear elliptic eigenvalue problems encountered in quantum physics and chemistry.