Exact saturation and degeneracy of isospin bounds and zeros trajectories in pion-nucleon scattering: D. B. Ion. Joint Institute for Nuclear Research,Dubna, Moscow,P. O. Box 79, U.S.S.R.

By exploiting the analogy between the classical equation of heat conduction and the Bloch equation, the canonical density matrix is calculated exactly for noninteracting electrons in a finite step model of a metal surface. This result is then used to calculate the spatial variation of the electron density ?(z) as a function of distance z from the planar metal surface. In particular an expansion in inverse powers of $\text{V}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and V, V being the step height, is derived. The low-order terms are used to compute the electron density at the surface of Al metal: There is good agreement with a full numerical evaluation. Finally, in the limit $\text{V}\text{E}{}_{\mathrm{f}}\text{→ ∞}$, E_f being the Fermi energy, the change in the shape and size of the exchange (Fermi) hole surrounding an electron is studied as the electron moves into the vicinity of a metal surface. The anisotropy of the Fermi hole is large and the implications for one-body potential theory are briefly discussed.