Logarithmic two-dimensional saddle joint density of states ($J(E)\sim g_s{m}^{\ast }/(2{\pi }^2{\hslash }^2)\ln (E_{\Lambda }/|E-E_s|)$)

A single two-dimensional quadratic saddle with $A={\hslash }^2/(2m^{\ast })$ contributes this leading joint density of states when both contour branches are included and $g_s$ counts independent spin channels. The contour integral is proportional to ${\int }_{-\Lambda }^{\Lambda }dy/\sqrt{y^2+\delta /A}=2\mathrm{arsinh}(\Lambda \sqrt{A/\delta })$ above the saddle. Its asymptote gives the logarithm with $E_{\Lambda }$ of order $A{\Lambda }^2$. The cutoff-dependent constant is not universal. This is a density per area per energy; changing to a density per frequency requires a Jacobian. ^[ref]

A spin factor and disconnected contour branches count different independent contributions. For the transition energy $E_s+{\hslash }^2(k_x^2-k_y^2)/(2m^{\ast })$, one branch contributes $g_s{m}^{\ast }I/(4{\pi }^2{\hslash }^2)$, with $I=\int d{k}_y/\sqrt{k_y^2+\delta /A}$. The full two-branch saddle doubles this. A formula for one branch with spin degeneracy two can therefore equal a full-contour formula with spin degeneracy one, but those are distinct counting conventions. An extra valley or inequivalent saddle requires its own multiplicity.