Semiclassical expansion of the Slater sum for position dependent mass distributions in d dimensions

We consider hamiltonian systems with spatially varying effective mass and slowly varying local potential in d dimensions. The Slater sum is defined as the diagonal element of the Bloch propagator. We derive a gradient expansion of the Slater sum up to the second order. We will show that the derived analytical expression is valid for d=1,2,3 and 4. A numerical example is shown to highlight the effect of the spatially varying effective mass.