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

We consider a superconductor with surface suppression of the BCS pairing constant $lambda(x)$. We analytically find the gap in the surface density of states (DOS), behavior of the DOS $ u(E)$ above the gap, a vertical peculiarity of the DOS around an energy equal to the bulk order parameter $Delta_0$, 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 $lambda(x)$ is suppressed, and the coherence length. The vertical peculiarity implies an infinite-derivative inflection point of the DOS curve at $E=Delta_0$ with square-root behavior as $E$ deviates from $Delta_0$. The coefficients of this dependence are different at $E<Delta_0$ and $E>Delta_0$, so the peculiarity is asymmetric.