Two-body threshold spectral analysis, the critical case

Erik Skibsted; Xue Ping Wang

Two-body threshold spectral analysis, the critical case

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

Abstract

We study in dimension $d\geq2$ low-energy spectral and scattering asymptotics for two-body $d$-dimensional Schrödinger operators with a radially symmetric potential falling off like $-\gamma r^{-2},\;\gamma>0$. We consider angular momentum sectors, labelled by $l=0,1,\dots$, for which $\gamma>(l+d/2 -1)^2$. In each such sector the reduced Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift.

Original language	English
Place of publication	Århus
Publisher	Aarhus University, Department of Mathematical Sciences
Number of pages	25
Publication status	Published - 2010

Access to Document

http://www.imf.au.dk/publs?publid=873

Cite this

@techreport{909483c556804b3e8bc38919eec8d4fd,

title = "Two-body threshold spectral analysis, the critical case",

abstract = "We study in dimension $d\geq2$ low-energy spectral and scattering asymptotics for two-body $d$-dimensional Schr{\"o}dinger operators with a radially symmetric potential falling off like $-\gamma r^{-2},\;\gamma>0$. We consider angular momentum sectors, labelled by $l=0,1,\dots$, for which $\gamma>(l+d/2 -1)^2$. In each such sector the reduced Schr{\"o}dinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift.",

author = "Erik Skibsted and Wang, {Xue Ping}",

year = "2010",

language = "English",

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

address = "Denmark",

type = "WorkingPaper",

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

}

TY - UNPB

T1 - Two-body threshold spectral analysis, the critical case

AU - Skibsted, Erik

AU - Wang, Xue Ping

PY - 2010

Y1 - 2010

N2 - We study in dimension $d\geq2$ low-energy spectral and scattering asymptotics for two-body $d$-dimensional Schrödinger operators with a radially symmetric potential falling off like $-\gamma r^{-2},\;\gamma>0$. We consider angular momentum sectors, labelled by $l=0,1,\dots$, for which $\gamma>(l+d/2 -1)^2$. In each such sector the reduced Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift.

AB - We study in dimension $d\geq2$ low-energy spectral and scattering asymptotics for two-body $d$-dimensional Schrödinger operators with a radially symmetric potential falling off like $-\gamma r^{-2},\;\gamma>0$. We consider angular momentum sectors, labelled by $l=0,1,\dots$, for which $\gamma>(l+d/2 -1)^2$. In each such sector the reduced Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift.

M3 - Working paper

BT - Two-body threshold spectral analysis, the critical case

PB - Aarhus University, Department of Mathematical Sciences

CY - Århus

ER -