Abstract
The exactly solvable Keldysh model of disordered electron system in a random scattering field with extremely long correlation length is converted to the time-dependent model with extremely long relaxation. The dynamical problem is solved for the ensemble of two-level systems (TLS) with fluctuating well depths having the discrete Z 2 symmetry. It is shown also that the symmetric TLS with fluctuating barrier transparency may be described in terms of the vector Keldysh model with dime-dependent random planar rotations in xy plane having continuous SO(2) symmetry. Application of this model to description of dynamic fluctuations in quantum dots and optical lattices is discussed.
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\( (2n - 1)!! = 2^n \frac{1} {{\sqrt \pi }}\Gamma (n + 1/2) = \frac{1} {{\sqrt {2\pi } }}\smallint _{ - \infty }^\infty dtt^{2n} e^{^{ - \frac{{t^2 }} {2}} } \).
The correlators \( \overline {\Delta (t)\Delta (t + \tau )} \) and \( \overline {\Delta *(t)\Delta *(t + \tau )} \) are transformed under “local” time-independent gauge transformation and therefore average to zero according to Elitzur theorem [S. Elitzur, Phys. Rev. D 12, 3978 (1975)].
\( n! = G(n + 1) = \smallint _0^\infty dzz^n e^{ - z} \).
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