The localization properties of an electron in the tight-binding potential A, cos(man ), where 0 & v & 1, are investigated. Wave functions approaching the metal-insulator transition (MIT} A. , =2 -~E ( (or the mobility edges E, =+~2 -iL~) from the metallic region are found to localize with multiple well-separated power-law decaying peaks, contradicting previous conclusions obtained with the help of the Lyapunov exponent. The general expectation of exponential localization with divergent localization length at the MIT cannot be detected convincingly.The electron localization in a three-dimensional medium of a random potential is well established. ' The conductivity in such systems vanishes if the measure of disorder exceeds a certain critical value. This phenomenon is a manifestation of the metal-insulator transition (MIT). In the spectrum of three-dimensional (3D) random systems there exist mobility edges E, that separate extended states from localized ones. The MIT takes place when E, crosses the Fermi energy as the disorder is increased.On the metallic side of the MIT, the states are extended functions, while in an insulator they are assumed to have an exponential decay with a localization length which is divergent at the MIT. Therefore at the MIT one expects an abrupt change in the spatial characteristics of the states. Thus it is especially of great importance to find the properties of the wave function at the mobility edge suggesting the way how the conductivity tends to zero at the MIT.A random potential-induced MIT appears only in 3D systems, while in 1D an infinitesimal amount of disorder localizes all the states, and it is probably the same in two dimensions, as well. In 1D systems, ho~ever, one may obtain a MIT induced by deterministic quasiperiodic potentials. ' These potentials are of special experimental importance, since they appear naturally in crystals containing charge-density or spin-density waves, as well as in superlattices grown with molecular-beam epitaxy. In this paper we will study the localization properties of the wave functions of 1D quasiperiodic potentials at and in the vicinity of the mobility edge in the spectrum, i.e. , at the MIT.The discretized one-dimensional tight-binding Schrodinger equation within the nearest-neighbor approximation is written as t (u"~i+u",) -( V"E)u"=O, where u" is the amplitude of the eigenstate on the nth lattice site, V" is the on-site (diagonal}, quasiperiodic potential, and E is the eigenenergy. For sake of simplicity t = l is chosen as the unit of the energy scale without any loss of generality. V"=Xcos(~ctn'),(2) which produces a gapless spectrum containing mobility edges' ' in contrast to the more complicated spectrum of other quasiperiodic models. ' A perturbation theory calculation' yields localized states for v~2 and extended states for 0& v 1 near the band center. %KB theory, ' however, indicates a clear mobility edge at the critical energy E, =+ 2 -k~if 0& v & 1, which means that there is a MIT at A, , =2 -~E~f or a fixed energy.(assuming expo...

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