For a semibounded, self-adjoint operator the eigenvalues below the essential spectrum can be computed in terms of the well-known min-max principle. This variational principle takes a particularly simple form for the Friedrichs extension of a semibounded, symmetric operator. The prime application concerns the Schrödinger operator in non-relativistic quantum mechanics where the principle does not only provide a way to numerically compute the bound states but is also a key ingredient in the proof of many important spectral bounds relating to the stability of matter. In this talk I will start with an introduction to this min-max principle. I will subsequently present a generalisation of the principle to the setting of self-adjoint operators satisfying only a gap condition. Again, the principle takes a particularly simple form for a certain distinguished self-adjoint extension. The results apply to the Dirac operator in relativistic quantum mechanics.

This talk is partly based on joint work with Jan Philip Solovej and Sabiha Tokus.