Constraints on the Quasiparticle Density of States in High-$T_c$ Superconductors

In this Letter we present new tunneling data on YBa${}_2$Cu${}_3$O${}_7$ thin films by low temperature scanning tunneling spectroscopy. Unusual peak-dip-hump features, previously reported in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$, are also found in YBa${}_2$Cu${}_3$O${}_7$. To analyse these common signatures we propose a new heuristic model in which, in addition to the d-wave symmetry, the gap function is energy dependent. A simple expression for the quasiparticle density of states is derived, giving an excellent agreement with the experiment. The dynamics of the quasiparticle states and the energy scales involved in the superconducting transition are discussed.