TY - JOUR

T1 - Schrodinger's equation on a one-dimensional lattice with weak disorder

AU - Kuske, R.

AU - Schuss, Z.

AU - Goldhirsch, I.

AU - Noskowicz, S. H.

PY - 1993

Y1 - 1993

N2 - The stationary Schrodinger equation on a one-dimensional lattice endowed with a random potential is considered. Specifically, the equation studied is un+1 + un-1 = (E - εVn) Un, where εVn is the random potential at site n. When ε = 0, the band of allowed energies is given by E=2cosπr, |r| < 1, and only this band is considered. A singular perturbation expansion of the stationary probability density p(x,E,ε)of the random process Xn = Un/Un-1 is constructed in the limit of weak disorder (ε ≪ 1). The coefficients in the expansion are analytic functions of r for ε > 0. They contain internal layers at rational values of r, which were previously termed 'anomalies.' The expansion approximates p(x,E, ε) uniformly for all r inside the band, away from bandcenter (r=1/2) and bandedge (r=0). It is used to calculate the first term in the expansion of the Lyapunov exponent γ(E,ε), which determines the localization length of the wave function, thus confirming the Thouless formula for γ(E,ε) inside the band and the Kappus-Wegner formula in bandcenter. Band-center and band-edge expansions are constructed, which match the in-band limits, allowing a uniform approximation for the Lyapunov exponent in all regions of the energy band.

AB - The stationary Schrodinger equation on a one-dimensional lattice endowed with a random potential is considered. Specifically, the equation studied is un+1 + un-1 = (E - εVn) Un, where εVn is the random potential at site n. When ε = 0, the band of allowed energies is given by E=2cosπr, |r| < 1, and only this band is considered. A singular perturbation expansion of the stationary probability density p(x,E,ε)of the random process Xn = Un/Un-1 is constructed in the limit of weak disorder (ε ≪ 1). The coefficients in the expansion are analytic functions of r for ε > 0. They contain internal layers at rational values of r, which were previously termed 'anomalies.' The expansion approximates p(x,E, ε) uniformly for all r inside the band, away from bandcenter (r=1/2) and bandedge (r=0). It is used to calculate the first term in the expansion of the Lyapunov exponent γ(E,ε), which determines the localization length of the wave function, thus confirming the Thouless formula for γ(E,ε) inside the band and the Kappus-Wegner formula in bandcenter. Band-center and band-edge expansions are constructed, which match the in-band limits, allowing a uniform approximation for the Lyapunov exponent in all regions of the energy band.

UR - http://www.scopus.com/inward/record.url?scp=0027680745&partnerID=8YFLogxK

U2 - 10.1137/0153059

DO - 10.1137/0153059

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AN - SCOPUS:0027680745

SN - 0036-1399

VL - 53

SP - 1210

EP - 1252

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

IS - 5

ER -