On Justification of the Asymptotics of Eigenfunctions of the Absolutely Continuous Spectrum in the Problem of Three One-Dimensional Short-Range Quantum Particles with Repulsion

2019 ◽  
Vol 238 (5) ◽  
pp. 566-590 ◽  
Author(s):  
I. V. Baibulov ◽  
A. M. Budylin ◽  
S. B. Levin
Keyword(s):  
Continuous Spectrum ◽  
Short Range ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Quantum Particles

On the absolutely continuous spectrum of a vector-matrix Dirac system

1994 ◽  
Vol 124 (2) ◽  
pp. 253-262 ◽  
Author(s):  
Stephen L. Clark
Keyword(s):  
Asymptotic Behaviour ◽  
Continuous Spectrum ◽  
Long Range ◽  
Short Range ◽  
Oscillatory Potentials ◽  
Energy Functional ◽  
Dirac System ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
Vector Matrix

A Dirac system is considered which has a matrix-valued long-range, short-range and oscillatory potentials. The system has one singular endpoint at infinity. Additional conditions on the potential are given which guarantee particular asymptotic behaviour of an energy functional associated with a certain set of solutions. This asymptotic behaviour guarantees the existence of a purely absolutely continuous spectrum outside a gap containing the origin.

scholarly journals Absolutely continuous spectrum of Dirac operators with square-integrable potentials

2014 ◽  
Vol 144 (3) ◽  
pp. 533-555
Author(s):  
Daniel Hughes ◽  
Karl Michael Schmidt
Keyword(s):  
Spectral Function ◽  
Continuous Spectrum ◽  
Scattering Problem ◽  
Dirac Operators ◽  
Mass Term ◽  
Absolutely Continuous Spectrum ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
The One

We show that the absolutely continuous part of the spectral function of the one-dimensional Dirac operator on a half-line with a constant mass term and a real, square-integrable potential is strictly increasing throughout the essential spectrum (−∞, −1] ∪ [1, ∞). The proof is based on estimates for the transmission coefficient for the full-line scattering problem with a truncated potential and a subsequent limiting procedure for the spectral function. Furthermore, we show that the absolutely continuous spectrum persists when an angular momentum term is added, thus also establishing the result for spherically symmetric Dirac operators in higher dimensions.

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