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J. Phys. A: Math. Gen. 32 (1999) L537L546. Printed in the UK PII: S0305-4470(99)07975-5 LETTER TO THE EDITOR
 

Summary: J. Phys. A: Math. Gen. 32 (1999) L537L546. Printed in the UK PII: S0305-4470(99)07975-5
LETTER TO THE EDITOR
Smooth adiabatic evolutions with leaky power tails
J E Avron and A Elgart
Department of Physics, Technion, 32000 Haifa, Israel
E-mail: avron@physics.technion.ac.il and elgart@physics.technion.ac.il
Received 20 September 1999, in final form 28 October 1999
Tosio Kato--in memoriam
Abstract. Adiabatic evolutions with a gap condition have, under a range of circumstances,
exponentially small tails that describe the leaking out of the spectral subspace. In general, adiabatic
evolutions without a gap condition do not seem to have this feature. This is a known fact for
eigenvalue crossing. We show that this is also the case for eigenvalues at the threshold of the
continuous spectrum by considering the Friedrichs model.
1. Introduction
Adiabatic theorems describe the solutions of initial-value problems where the Hamiltonian
generating the evolution depends slowly on time. In quantum mechanics the description is
in terms of spectral information of the instantaneous Hamiltonian. A few basic references on
various types of adiabatic theorems are [4,7,10,12,15,16].
To formulate the problem more precisely it is convenient to replace the physical time t by
the scaled time s = t/. One is then concerned with the solution of the initial value problem

  

Source: Avron, Joseph - Physics Department, Technion, Israel Institute of Technology

 

Collections: Physics