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Semi-Classical Behavior of the Spectral Function Ivana Alexandrova
 

Summary: Semi-Classical Behavior of the Spectral Function
Ivana Alexandrova
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3,
Tel.: 1-416-946-0318, Fax: 1-416-978-4107, email: alexandr@math.toronto.edu
February 25, 2005
Abstract
We study the semi-classical behavior of the spectral function of the Schršodinger
operator with short range potential. We prove that the spectral function is a semi-
classical Fourier integral operator quantizing the forward and backward Hamiltonian
flow relations of the system. Under a certain geometric condition we explicitly compute
the phase in an oscillatory integral representation of the spectral function.
1 Introduction
We study the structure of the spectral function associated with the semi-classical Schršodinger
operator with short range potential on Rn
. We prove that the appropriately cut-off spectral
function is a semi-classical Fourier integral operator associated to the union of the backward
and the forward Hamiltonian flow relations of the principle symbol of the operator. We also
show how this allows us, under a certain geometric assumption, to compute the phase in an
oscillatory integral representation of the spectral function.
Our result is motivated by the following theorem by Vainberg. In [7, Theorem XII.5]

  

Source: Alexandrova, Ivana - Department of Mathematics and Statistics, State University of New York at Albany

 

Collections: Mathematics