# Absolutely continuous and singular spectral shift functions

- 2011

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topNurulla Azamov. Absolutely continuous and singular spectral shift functions. 2011. <http://eudml.org/doc/286027>.

@book{NurullaAzamov2011,

abstract = {Given a self-adjoint operator H₀, a self-adjoint trace-class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and cokernel, using the limiting absorption principle an explicit set Λ(H₀;F) ⊂ ℝ of full Lebesgue measure is defined, such that for all λ ∈ Λ(H₀+rV;F) ∩ Λ(H₀;F), where r ∈ ℝ, the wave $w_\{±\}(λ;H₀+rV,H₀)$ and the scattering matrices S(λ;H₀+rV,H₀) can be defined unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including the stationary formula for the scattering matrix. This version of abstract scattering theory allows us, in particular, to prove that
$det S(λ;H₀+V,H₀) = e^\{-2πiξ^\{(a)\}(λ)\}$, a.e. λ ∈ ℝ,
where $ξ^\{(a)\}(λ) = ξ^\{(a)\}_\{H₀+V,H₀\}(λ)$ is the so called absolutely continuous part of the spectral shift function defined by
$ξ^\{(a)\}_\{H₀+V,H₀\}(λ) := d/dλ ∫_\{0\}^\{1\} Tr(VE^\{(a)\}_\{H₀+rV\}(λ))dr$
and where $E_\{H\}^\{(a)\}(λ) = E^\{(a)\}_\{(-∞,λ)\}(H)$ denotes the absolutely continuous part of the spectral projection. Combined with the Birman-Kreĭn formula, this implies that the singular part of the spectral shift function,
$ξ^\{(s)\}_\{H₀+V,H₀\}(λ) := d/dλ ∫_\{0\}^\{1\} Tr(VE^\{(s)\}_\{H₀+rV\}(λ))dr$,
is an almost everywhere integer-valued function, where $E_\{H\}^\{(s)\}(λ) = E^\{(s)\}_\{(-∞,λ)\}(H)$ denotes the singular part of the spectral projection.
It is also shown that eigenvalues of the scattering matrix S(λ;H₀+V,H₀) can be connected to 1 in two natural ways, and that the singular spectral shift function measures the difference of the spectral flows of eigenvalues of the scattering matrix.},

author = {Nurulla Azamov},

keywords = {spectral shift function; scattering matrix; absolutely continuous spectral shift function; singular spectral shift function; Birman-Krein formula; Pushnitskij -invariant; infinitesimal spectral flow; infinitesimal scattering matrix},

language = {eng},

title = {Absolutely continuous and singular spectral shift functions},

url = {http://eudml.org/doc/286027},

year = {2011},

}

TY - BOOK

AU - Nurulla Azamov

TI - Absolutely continuous and singular spectral shift functions

PY - 2011

AB - Given a self-adjoint operator H₀, a self-adjoint trace-class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and cokernel, using the limiting absorption principle an explicit set Λ(H₀;F) ⊂ ℝ of full Lebesgue measure is defined, such that for all λ ∈ Λ(H₀+rV;F) ∩ Λ(H₀;F), where r ∈ ℝ, the wave $w_{±}(λ;H₀+rV,H₀)$ and the scattering matrices S(λ;H₀+rV,H₀) can be defined unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including the stationary formula for the scattering matrix. This version of abstract scattering theory allows us, in particular, to prove that
$det S(λ;H₀+V,H₀) = e^{-2πiξ^{(a)}(λ)}$, a.e. λ ∈ ℝ,
where $ξ^{(a)}(λ) = ξ^{(a)}_{H₀+V,H₀}(λ)$ is the so called absolutely continuous part of the spectral shift function defined by
$ξ^{(a)}_{H₀+V,H₀}(λ) := d/dλ ∫_{0}^{1} Tr(VE^{(a)}_{H₀+rV}(λ))dr$
and where $E_{H}^{(a)}(λ) = E^{(a)}_{(-∞,λ)}(H)$ denotes the absolutely continuous part of the spectral projection. Combined with the Birman-Kreĭn formula, this implies that the singular part of the spectral shift function,
$ξ^{(s)}_{H₀+V,H₀}(λ) := d/dλ ∫_{0}^{1} Tr(VE^{(s)}_{H₀+rV}(λ))dr$,
is an almost everywhere integer-valued function, where $E_{H}^{(s)}(λ) = E^{(s)}_{(-∞,λ)}(H)$ denotes the singular part of the spectral projection.
It is also shown that eigenvalues of the scattering matrix S(λ;H₀+V,H₀) can be connected to 1 in two natural ways, and that the singular spectral shift function measures the difference of the spectral flows of eigenvalues of the scattering matrix.

LA - eng

KW - spectral shift function; scattering matrix; absolutely continuous spectral shift function; singular spectral shift function; Birman-Krein formula; Pushnitskij -invariant; infinitesimal spectral flow; infinitesimal scattering matrix

UR - http://eudml.org/doc/286027

ER -

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