Electronic density of levels in a disordered system

Three-, two-, and one-dimensional disordered systems with randomly distributed, purely repulsive scattering centers, known as Lorentz models, are studied in the low energy limit [1]. Using a functional integral representation and a version of the “replica trick”, we have found in the D-dimensional system the density of electronic levels of the form

$\text{n(E)=b}{}_0\text{Eγ}\text{exp}\text{(?b}{}_1\text{E}{}^{\mathrm{<mtext>?(</mtext><mtext>D</mtext><mtext>2</mtext><mtext>)</mtext>}}\text{+b}{}_2\text{E}{}^{\mathrm{<mtext>?(</mtext><mtext>D</mtext><mtext>2</mtext><mtext>)+1</mtext>}}\text{+·+b}{}_{\mathrm{D}}\text{E}{}^{\mathrm{<mtext>?(</mtext><mtext>1</mtext><mtext>2</mtext><mtext>)</mtext>}}\text{)(1+O(}\text{E}\text{))}$

and the constants b₀, b₁,…, b_D, and γ have been determined.