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Exponential decay of averaged Green functions for random Schrödinger operators. A direct approach. (English) Zbl 0934.35036
Summary: Under suitable analyticity conditions on the probability distribution, we study the expectation of the Green function. We give precise results about domains of holomorphic extensions in energy and exponential decay. The key ingredient is the construction of a probability measure in the complex domain after contour deformation. This permits us to avoid the use of perturbation series. Compared to the method in [the authors, Supersymmetric measures and maximum principles in complex space-Exponential decay of Green’s functions], the variant here seems limited to the random Schrödinger equation, in which case however it permits to treat more general probability distributions.

MSC:
35J10 Schrödinger operator, Schrödinger equation
60H25 Random operators and equations (aspects of stochastic analysis)
35A08 Fundamental solutions to PDEs
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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References:
[1] S. BELL , The Cauchy transform, potential theory, and conformal mapping , Studies in advanced mathematics, CRC Press, Boca Raton, Ann Arbor, London, Tokyo 1992 . MR 94k:30013
[2] F. CONSTANTINESCU , J. FRÖHLICH , T. SPENCER , Analyticity of the density of states and replica methods for random Schrödinger operators on a lattice , J. Stat. Phys. 34 ( 1984 ), 571-596. Zbl 0591.60060 · Zbl 0591.60060 · doi:10.1007/BF01018559
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[4] J. SJÖSTRAND , W.-M. WANG , Supersymmetric measures and maximum principles in complex space-Exponential decay of Green’s functions . Numdam | Zbl 0941.47033 · Zbl 0941.47033 · doi:10.1016/S0012-9593(99)80017-2 · numdam:ASENS_1999_4_32_3_347_0 · eudml:82491
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