Classical Rayleigh-Ritz min-max characterizes eigenvalues below the essential spectrum of a semibounded operator, but a self-adjoint operator with a gap in its essential spectrum has eigenvalues that are neither minima nor maxima, so no counting from the bottom is available. This concept covers the variational principle that repairs that: fix an orthogonal splitting of the Hilbert space into two subspaces and a core F, define levels by maximizing the Rayleigh quotient over one summand and minimizing over subspaces of the other, and state the hypotheses on the splitting under which those levels equal the eigenvalues in the gap, together with the continuation argument that propagates the identity along a continuous family of operators. The framework is instantiated on Dirac operators with Coulomb-like potentials, where the upper/lower-spinor (Talman) splitting and the free-energy spectral-projector splitting both satisfy the hypotheses up to the optimal coupling threshold, and where the same inequalities yield Hardy-type bounds as by-products. Also covered is the form-theoretic care the statement requires: the quadratic form on the chosen summand must be closable for the levels to be well defined.
A self-adjoint operator on a Hilbert space can have a gap in its essential spectrum -- the Dirac operator of relativistic quantum mechanics is the model case, with essential spectrum (-infinity,-1] u…