Transmission of waves and particles through long random barriers

Марченко, А. В.; Pastur, L. А.

doi:10.1007/bf01019395

Cited by 5 publications

(5 citation statements)

References 2 publications

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“…A very similar statement was proven earlier by Marchenko and Pastur [42]. Our proof is a slight modification of that given in [42].…”

Section: Density Of the Transmission Coefficient And The Lyapunov Exp...supporting

confidence: 84%

“…A very similar statement was proven earlier by Marchenko and Pastur [42]. Our proof is a slight modification of that given in [42]. We start with Lemma 5.2 For all E ≥ 0 the transmission amplitude for the Hamiltonian (3.1) satisfies the inequality…”

Section: Density Of the Transmission Coefficient And The Lyapunov Exp...supporting

confidence: 64%

“…The connection between the solutions of (2.26) and scattering characteristics was noted earlier in [42,47].…”

Section: Theorem 24 For Arbitrary E > 0 Consider the Matrix U(x Xmentioning

confidence: 97%

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Scattering Theory Approach to Random Schrödinger Operators in One Dimension

Kostrykin

Schrader

1999

Rev. Math. Phys.

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Methods from scattering theory are introduced to analyze random Schrödinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz-Krein spectral shift function, which is related to the scattering phase by the theorem of Birman and Krein. The spectral shift density is defined as the "thermodynamic limit" of the spectral shift function per unit length of the interaction region. This density is shown to be equal to the difference of the densities of states for the free and the interacting Hamiltonians. Based on this construction, we give a new proof of the Thouless formula. We provide a prescription how to obtain the Lyapunov exponent from the scattering matrix, which suggest a way how to extend this notion to the higher dimensional case. This prescription also allows a characterization of those energies which have vanishing Lyapunov exponent. *

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“…A very similar statement was proven earlier by Marchenko and Pastur [42]. Our proof is a slight modification of that given in [42].…”

Section: Density Of the Transmission Coefficient And The Lyapunov Exp...supporting

confidence: 84%

Section: Density Of the Transmission Coefficient And The Lyapunov Exp...supporting

confidence: 64%

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Scattering Theory Approach to Random Schrödinger Operators in One Dimension

Kostrykin

Schrader

1999

Rev. Math. Phys.

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“…where φ m,n (E; ω) is the scattering phase for the pair of operators (−∆ m,n (E; ω), −∆). In the next section we use equation (21) to prove the Thouless formula in the present context. Now we are in position to prove Proposition 2.1.…”

Section: The Integrated Density Of States and Scattering Amplitudesmentioning

confidence: 99%

A random necklace model

Kostrykin¹,

Schrader²

2004

Waves in Random Media

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We consider a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length. The arc lengths of the loops are considered to be independent, identically distributed random variables. The integrated density of states of this Laplace operator is shown to have discontinuities provided that the distribution of arc lengths of the loops has a nontrivial pure point part. Some numerical illustrations are also presented.

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“…In [41,43,44,46] we proved that in arbitrary dimensions the scattering phase (or more generally of the spectral shift function) per interaction volume equals (up to a factor π) the difference of the integrated densities of states for the free and interaction theories respectively. In the strictly one-dimensional situation (Schrödinger operators on the line) the Lyapunov exponent is known to be related to the logarithmic density of transmission probability [52,53,54,41]. Due to the Ishii-Pastur-Kotani theorem (see e.g.…”

Section: Introductionmentioning

confidence: 99%

The generalized star product and the factorization of scattering matrices on graphs

Kostrykin

Schrader

2001

Journal of Mathematical Physics

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In this article we continue our analysis of Schrodinger operators on arbitrary graphs given as certain Laplace operators. In the present article we give the proof of the composition rule for the scattering matrices. This composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to its subgraphs. We perform a detailed analysis of the generalized star product for arbitrary unitary matrices. The relation to the theory of transfer matrices is also discusse

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Transmission of waves and particles through long random barriers

Cited by 5 publications

References 2 publications

Scattering Theory Approach to Random Schrödinger Operators in One Dimension

Scattering Theory Approach to Random Schrödinger Operators in One Dimension

A random necklace model

The generalized star product and the factorization of scattering matrices on graphs

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