Thin films of Ag(111) with two-dimensional crystallinity of large lateral coherence grow on Ge(111), free of in-plane registry with the underlying substrate. Ag s-p electrons forming two-dimensional quantum well states scatter coherently at the buried interface potential, resulting in an unexpected set of new quasiparticle states, as observed by angle-resolved photoemission. These new features originate from interactions among Ag quantum well bands, gaining a momentum equivalent to a reciprocal vector of the substrate lattice. DOI: 10.1103/PhysRevLett.97.206802 PACS numbers: 73.21.Fg, 73.22.Gk, 73.40.Ns Translational symmetry is an essential property in crystals, which gave birth to fundamental concepts such as periodic potentials, crystal momentum, Brillouin zones, and electronic bands. Breaking this symmetry challenges our understanding of the electronic structure and opens new perspectives in modifying the electronic, magnetic, growth, and transport properties as well as low-temperature phenomena such as the superconductivity and the Kondo physics. In a self-standing layer with a thickness of the order of the de Broglie wavelength, the breaking of the translational symmetry along the direction normal to the surface planes leads to a quantization of the energy levels. Similarly, in a thin metal film on a substrate the interface potential can act as a reflecting wall on the electron wave functions, giving rise to two-dimensional (2D) quantum well (QW) states, partially or completely confined within the film [1]. The degree of localization of QW states inside the film depends on their hybridization with the substrate bands, which is defined by the overlap of the states in the two materials with corresponding energy and symmetry. Because of the boundary conditions, the properties of QW states reflect thus in detail the electronic interactions at the interface with the supporting substrate [2 -6].A much more complex scenario arises when the translational symmetry is broken such that electrons reside in potentials with disparate translational periodicities. For instance, competing electron-electron or electron-phonon interactions drive particularly low-dimensional systems into charge-or spin-density wave states [7][8][9][10][11][12], which are incommensurable with the underlying crystal lattice. Alternatively, in artificially ordered 2D heterojunctions [13] or tunnel junctions [14], relevant for nanoelectronics and spintronics, electrons are coherently transported between two structures of different periodicities. Besides being of practical interest, these systems raise attractive and unresolved fundamental questions on how their electronic structure depends on the translational mismatch, the difference in the potential strength between the crystalline structure of the two constituents, and the dissimilarity of the electronic structure.We notice that recently, several photoemission experiments focused on QW states of noble and simple metal layers on semiconductors, aiming at the understanding of coupling effect...