Quantum scattering at low energies

Jan Derezinski; Erik Skibsted

Quantum scattering at low energies

Research output: Working paper/Preprint › Working paper › Research

Abstract

For a class of negative slowly decaying potentials, including $V(x):=-\gamma|x|^{-\mu}$ with $0<\mu<2$ , we study the quantum mechanical scattering theory in the low-energy regime. Using modifiers of the Isozaki--Kitada type we show that scattering theory is well behaved on the {\it whole} continuous spectrum of the Hamiltonian, including the energy $0$ . We show that the $S$ --matrices are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the 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 $-\gamma|x|^{-\mu}$ we show that the location of singularities of the kernel of $S(\lambda)$ experiences an abrupt change from passing from positive energies $\lambda$ to the limiting energy $\lambda=0$ . This change corresponds to the behaviour of the classical orbits. Under stronger conditions we extract the leading term of the asymptotics of the kernel of $S(\lambda)$ at its singularities; this leading term defines a Fourier integral operator in the sense of Hörmander [Hö]4.

Original language	English
Place of publication	Århus
Publisher	Department of Mathematical Sciences , University of Aarhus
Number of pages	94
Publication status	Published - 2007

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title = "Quantum scattering at low energies",

abstract = "For a class of negative slowly decaying potentials, including with , we study the quantum mechanical scattering theory in the low-energy regime. Using modifiers of the Isozaki--Kitada type we show that scattering theory is well behaved on the {\it whole} continuous spectrum of the Hamiltonian, including the energy . We show that the --matrices are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the 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 --matrices for the subclasses of potentials where both kinds of -matrices are defined. For potentials whose leading part is we show that the location of singularities of the kernel of experiences an abrupt change from passing from positive energies to the limiting energy . This change corresponds to the behaviour of the classical orbits. Under stronger conditions we extract the leading term of the asymptotics of the kernel of at its singularities; this leading term defines a Fourier integral operator in the sense of H{\"o}rmander [H{\"o}]4.",

author = "Jan Derezinski and Erik Skibsted",

year = "2007",

language = "English",

publisher = "Department of Mathematical Sciences , University of Aarhus",

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T1 - Quantum scattering at low energies

AU - Derezinski, Jan

AU - Skibsted, Erik

PY - 2007

Y1 - 2007

N2 - For a class of negative slowly decaying potentials, including with , we study the quantum mechanical scattering theory in the low-energy regime. Using modifiers of the Isozaki--Kitada type we show that scattering theory is well behaved on the {\it whole} continuous spectrum of the Hamiltonian, including the energy . We show that the --matrices are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the 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 --matrices for the subclasses of potentials where both kinds of -matrices are defined. For potentials whose leading part is we show that the location of singularities of the kernel of experiences an abrupt change from passing from positive energies to the limiting energy . This change corresponds to the behaviour of the classical orbits. Under stronger conditions we extract the leading term of the asymptotics of the kernel of at its singularities; this leading term defines a Fourier integral operator in the sense of Hörmander [Hö]4.

AB - For a class of negative slowly decaying potentials, including with , we study the quantum mechanical scattering theory in the low-energy regime. Using modifiers of the Isozaki--Kitada type we show that scattering theory is well behaved on the {\it whole} continuous spectrum of the Hamiltonian, including the energy . We show that the --matrices are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the 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 --matrices for the subclasses of potentials where both kinds of -matrices are defined. For potentials whose leading part is we show that the location of singularities of the kernel of experiences an abrupt change from passing from positive energies to the limiting energy . This change corresponds to the behaviour of the classical orbits. Under stronger conditions we extract the leading term of the asymptotics of the kernel of at its singularities; this leading term defines a Fourier integral operator in the sense of Hörmander [Hö]4.

M3 - Working paper

BT - Quantum scattering at low energies

PB - Department of Mathematical Sciences , University of Aarhus

CY - Århus

ER -

Quantum scattering at low energies

Abstract

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