Upper bounds to the eigenvalues of the no-pair Hamiltonian

Abstract

A Dirac-like Hamiltonian H with two-body terms, and its no-pair Hamiltonian ${\mathrm{H}}^{+}$=${\mathrm{\ensuremath{\Lambda}}}^{++}$H${\mathrm{\ensuremath{\Lambda}}}^{++}$ where ${\mathrm{\ensuremath{\Lambda}}}^{++}$ is related to a one-particle Hamiltonian ${\mathrm{h}}_{0}$, are studied in finite-basis representations H and ${\mathbf{H}}^{+}$. Using finite-basis eigenfunctions of ${\mathrm{h}}_{0}$, it holds ${\mathrm{E}}_{\mathrm{i}}^{+}$\ensuremath{\leqslant}${\mathrm{E}}_{{\mathrm{N}}^{\mathrm{\ensuremath{-}}}+\mathrm{i}}$, i>0, where ${\mathrm{E}}_{{\mathrm{N}}^{\mathrm{\ensuremath{-}}}+\mathrm{i}}$ and ${\mathrm{E}}_{\mathrm{i}}^{+}$ are the ordered eigenvalues of H and ${\mathbf{H}}^{+}$, and ${\mathrm{N}}^{\mathrm{\ensuremath{-}}}$ is the difference between the dimensions of H and ${\mathbf{H}}^{+}$. The states of order i\ensuremath{\leqslant}${\mathrm{N}}^{\mathrm{\ensuremath{-}}}$ exhibit continuum dissolution. In contrast, those of order ${\mathrm{N}}^{\mathrm{\ensuremath{-}}}$+i, i>0, are bounded from below and after application of a variational principle they represent bound states.