Surface density of states in superconductors with inhomogeneous pairing constant: Analytical results

We consider a superconductor with surface suppression of the BCS pairing constant $\ensuremath{\lambda}(x)$. We analytically find the gap in the surface density of states (DOS), behavior of the DOS $\ensuremath{\nu}(E)$ above the gap, a ``vertical'' peculiarity of the DOS around an energy equal to the bulk order parameter ${\mathrm{\ensuremath{\Delta}}}_{0}$, and a perturbative correction to the DOS at higher energies. The surface gap in the DOS is parametrically different from the surface value of the order parameter due to a difference between the spatial scale ${r}_{c}$ at which $\ensuremath{\lambda}(x)$ is suppressed and the coherence length. The vertical peculiarity implies an infinite-derivative inflection point of the DOS curve at $E={\mathrm{\ensuremath{\Delta}}}_{0}$ with square-root behavior as $E$ deviates from ${\mathrm{\ensuremath{\Delta}}}_{0}$. The coefficients of this dependence are different at $El{\mathrm{\ensuremath{\Delta}}}_{0}$ and $Eg{\mathrm{\ensuremath{\Delta}}}_{0}$, so the peculiarity is asymmetric.