TAILIEUCHUNG - Đề tài " On Mott’s formula for the acconductivity in the Anderson model "

We study the ac-conductivity in linear response theory in the general framework of ergodic magnetic Schr¨dinger operators. For the Anderson model, o if the Fermi energy lies in the localization regime, we prove that the ac1 conductivity is bounded from above by Cν 2 (log ν )d+2 at small frequencies ν. This is to be compared to Mott’s formula, which predicts the leading term to 1 be Cν 2 (log ν )d+1 . | Annals of Mathematics On Mott s formula for the ac-conductivity in the Anderson model By Abel Klein Olivier Lenoble and Peter M uller Annals of Mathematics 166 2007 549 577 On Mott s formula for the ac-conductivity in the Anderson model By Abel Klein Olivier Lenoble and Peter Muller Abstract We study the ac-conductivity in linear response theory in the general framework of ergodic magnetic Schrodinger operators. For the Anderson model if the Fermi energy lies in the localization regime we prove that the ac-conductivity is bounded from above by Cv2 log V d 2 at small frequencies V. This is to be compared to Mott s formula which predicts the leading term to be Cv2 log V d 1. 1. Introduction The occurrence of localized electronic states in disordered systems was first noted by Anderson in 1958 An who argued that for a simple Schrodinger operator in a disordered medium at sufficiently low densities transport does not take place the exact wave functions are localized in a small region of space. This phenomenon was then studied by Mott who wrote in 1968 Mol The idea that one can have a continuous range of energy values in which all the wave functions are localized is surprising and does not seem to have gained universal acceptance. This led Mott to examine Anderson s result in terms of the Kubo-Greenwood formula for ơEp v the electrical alternating current ac conductivity at Fermi energy Ef and zero temperature with V being the frequency. Mott used its value at V 0 to reformulate localization If a range of values of the Fermi energy Ef exists in which ơEp 0 0 the states with these energies are said to be localized if ơEf 0 0 the states are nonlocalized. Mott then argued that the direct current dc conductivity ơEf 0 indeed vanishes in the localized regime. In the context of Anderson s model he studied the behavior of Re ƠEF v as V 0 at Fermi energies Ef in the localization region note Im ơEf 0 0 . The result was the well-known Mott s formula for the ac-conductivity at .

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