The density of Andreev levels in a normal metal (N) in contact with two superconductors (S) is known to exhibit an induced minigap related to the inverse dwell time. We predict a small secondary gap just below the superconducting gap edge-a feature that has been overlooked so far in numerous microscopic studies of the density of states in S-N-S structures. In a generic structure with N being a chaotic cavity, the secondary gap is the widest at zero phase bias. It closes at some finite phase bias, forming the shape of a "smile". Asymmetric couplings give even richer gap structures near the phase difference π. All the features found should be amendable to experimental detection in high-resolution low-temperature tunneling spectroscopy. DOI: 10.1103/PhysRevLett.112.067001 PACS numbers: 74.45.+c, 74.78.Na The modification of the density of states (DOS) in a normal metal by a superconductor in its proximity was discovered almost 50 years ago [1]. Soon afterwards, it was predicted, theoretically, for diffusive structures that a socalled minigap of the order of the inverse dwell time in the normal metal (or the Thouless energy) appears in the spectrum [2]. The energy-dependent DOS reflects the energy scale of electron-hole decoherence, and is sensitive to the distance, the geometry, and the properties of the contact between the normal metal and the superconductor [3][4][5]. The details of the local density of states in proximity structures have been investigated experimentally many years later [6][7][8][9][10] and the theoretical predictions have been confirmed [11][12][13] Substantial interest has been paid to the density of states in a finite normal metal between two superconducting leads with different superconducting phases [14,15]. The difference between diffusive [5] and classical ballistic [16] dynamics has been investigated [17]. Many publications have addressed the dependence of the minigap on the competition between dwell and Ehrenfest time [18,19]. The most generic model in this context is that of a chaotic cavity, where a piece of normal metal is connected to the superconductors by means of ballistic point contacts that dominate the resistance of the structure in the normal state. The Thouless energy is given by E Th ¼ ðG Σ =G Q Þδ S , G Q ¼ e 2 =πℏ being the conductance quantum, G Σ ≫ G Q being the total conductance of the contacts, and δ S being the level spacing in the normal metal provided the contacts are closed. The DOS in chaotic cavities has been studied for years [19][20][21][22].The DOS depends on the ratio of E Th and the superconducting energy gap Δ, and on the superconducting phase difference. If the dwell time exceeds the Ehrenfest time, qualitative features of the DOS do not seem to depend much on the contact nature and are the same for ballistic, diffusive, and tunnel contacts. Mesoscopic fluctuations of the DOS [23,24] are small provided G ≫ G Q . It looks like everything is understood, perhaps except a small dip or peak in the DOS just at the gap edge for the diffusive case, which has ...