scholarly journals Instability of the absolutely continuous spectrum of ordinary differential operators under local perturbations

1989 ◽  
Vol 142 (2) ◽  
pp. 591-604 ◽  
Author(s):  
Rafael René del Río Castillo
Keyword(s):  
Continuous Spectrum ◽  
Differential Operators ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
Local Perturbations ◽  
Ordinary Differential Operators

scholarly journals ON THE ABSOLUTELY CONTINUOUS SPECTRUM IN A MODEL OF AN IRREVERSIBLE QUANTUM GRAPH

2005 ◽  
Vol 92 (1) ◽  
pp. 251-272 ◽  
Author(s):  
SERGEY N. NABOKO ◽  
MICHAEL SOLOMYAK
Keyword(s):  
Quantum System ◽  
Continuous Spectrum ◽  
Differential Operators ◽  
Jacobi Matrix ◽  
Mathematical Expression ◽  
Real Parameter ◽  
Explicit Description ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
Additional Branch

A family $\mathbf{A}_\alpha$ of differential operators depending on a real parameter $\alpha \ge 0$ is considered. This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely continuous spectrum $\sigma_{a.c.}$ of the operator $\mathbf{A}_\alpha$ and its multiplicity for all values of the parameter. The spectrum of $\mathbf{A}_0$ is purely absolutely continuous and admits an explicit description. It turns out that for $\alpha < \sqrt 2$ one has $\sigma_{a.c.}(\mathbf{A}_\alpha) = \sigma_{a.c.}(\mathbf{A}_0)$, including the multiplicity. For $\alpha \ge \sqrt2$ an additional branch of the absolutely continuous spectrum arises; its source is an auxiliary Jacobi matrix which is related to the operator $\mathbf{A}_\alpha$. This birth of an extra branch of the absolutely continuous spectrum is the exact mathematical expression of the effect that was interpreted by Smilansky as irreversibility.

SPECTRAL THEORY OF HIGHER ORDER DIFFERENTIAL OPERATORS

2005 ◽  
Vol 92 (1) ◽  
pp. 139-160 ◽  
Author(s):  
HORST BEHNCKE
Keyword(s):  
Continuous Spectrum ◽  
Spectral Theory ◽  
General Class ◽  
Differential Operators ◽  
Higher Order ◽  
Asymptotic Integration ◽  
Singular Continuous Spectrum ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
Decay Properties

The absolutely continuous spectrum of a very general class of differential operators of order $2n$ is determined, for operators whose coefficients satisfy conditions that combine smoothness and decay properties. The main methods are asymptotic integration and the analysis of the associated $M$-matrix. The form of the solutions precludes the absence of a singular continuous spectrum.

Sign in / Sign up

Export Citation Format

Share Document