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Erik Skibsted

Quantum scattering at low energies

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Quantum scattering at low energies. / Derezinski, Jan; Skibsted, Erik.

In: Journal of Functional Analysis, Vol. 257, No. 6, 2009, p. 1828-1920.

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

Harvard

Derezinski, J & Skibsted, E 2009, 'Quantum scattering at low energies', Journal of Functional Analysis, vol. 257, no. 6, pp. 1828-1920. https://doi.org/10.1016/j.jfa.2009.05.026

APA

Derezinski, J., & Skibsted, E. (2009). Quantum scattering at low energies. Journal of Functional Analysis, 257(6), 1828-1920. https://doi.org/10.1016/j.jfa.2009.05.026

CBE

Derezinski J, Skibsted E. 2009. Quantum scattering at low energies. Journal of Functional Analysis. 257(6):1828-1920. https://doi.org/10.1016/j.jfa.2009.05.026

MLA

Derezinski, Jan and Erik Skibsted. "Quantum scattering at low energies". Journal of Functional Analysis. 2009, 257(6). 1828-1920. https://doi.org/10.1016/j.jfa.2009.05.026

Vancouver

Derezinski J, Skibsted E. Quantum scattering at low energies. Journal of Functional Analysis. 2009;257(6):1828-1920. https://doi.org/10.1016/j.jfa.2009.05.026

Author

Derezinski, Jan ; Skibsted, Erik. / Quantum scattering at low energies. In: Journal of Functional Analysis. 2009 ; Vol. 257, No. 6. pp. 1828-1920.

Bibtex

@article{96b59db0149d11dfb95d000ea68e967b,
title = "Quantum scattering at low energies",
abstract = "For a class of negative slowly decaying potentials, including V(x):=−γ|x|−μ with 0<μ<2, we study the quantum mechanical scattering theory in the low-energy regime. Using appropriate modifiers of the Isozaki–Kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the Hamiltonian, including the energy 0. We show that the modified scattering matrices S(λ) are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces. These results are used to derive (oscillatory) asymptotics of the standard short-range and Dollard type S-matrices for the subclasses of potentials where both kinds of S-matrices are defined. For potentials whose leading part is −γ|x|−μ we show that the location of singularities of the kernel of S(λ) experiences an abrupt change from passing from positive energies λ to the limiting energy λ=0. This change corresponds to the behaviour of the classical orbits. Under stronger conditions one can extract the leading term of the asymptotics of the kernel of S(λ) at its singularities.",
author = "Jan Derezinski and Erik Skibsted",
year = "2009",
doi = "10.1016/j.jfa.2009.05.026",
language = "English",
volume = "257",
pages = "1828--1920",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Academic Press",
number = "6",

}

RIS

TY - JOUR

T1 - Quantum scattering at low energies

AU - Derezinski, Jan

AU - Skibsted, Erik

PY - 2009

Y1 - 2009

N2 - For a class of negative slowly decaying potentials, including V(x):=−γ|x|−μ with 0<μ<2, we study the quantum mechanical scattering theory in the low-energy regime. Using appropriate modifiers of the Isozaki–Kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the Hamiltonian, including the energy 0. We show that the modified scattering matrices S(λ) are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces. These results are used to derive (oscillatory) asymptotics of the standard short-range and Dollard type S-matrices for the subclasses of potentials where both kinds of S-matrices are defined. For potentials whose leading part is −γ|x|−μ we show that the location of singularities of the kernel of S(λ) experiences an abrupt change from passing from positive energies λ to the limiting energy λ=0. This change corresponds to the behaviour of the classical orbits. Under stronger conditions one can extract the leading term of the asymptotics of the kernel of S(λ) at its singularities.

AB - For a class of negative slowly decaying potentials, including V(x):=−γ|x|−μ with 0<μ<2, we study the quantum mechanical scattering theory in the low-energy regime. Using appropriate modifiers of the Isozaki–Kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the Hamiltonian, including the energy 0. We show that the modified scattering matrices S(λ) are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces. These results are used to derive (oscillatory) asymptotics of the standard short-range and Dollard type S-matrices for the subclasses of potentials where both kinds of S-matrices are defined. For potentials whose leading part is −γ|x|−μ we show that the location of singularities of the kernel of S(λ) experiences an abrupt change from passing from positive energies λ to the limiting energy λ=0. This change corresponds to the behaviour of the classical orbits. Under stronger conditions one can extract the leading term of the asymptotics of the kernel of S(λ) at its singularities.

U2 - 10.1016/j.jfa.2009.05.026

DO - 10.1016/j.jfa.2009.05.026

M3 - Journal article

VL - 257

SP - 1828

EP - 1920

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 6

ER -