Mathematics – Spectral Theory
Scientific paper
2005-04-10
Mathematics
Spectral Theory
Scientific paper
A family $A_\alpha$ 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 $A_\alpha$ and its multiplicity for all values of the parameter. The spectrum of $A_0$ is purely a.c. and admits an explicit description. It turns out that for $\alpha<\sqrt 2$ one has $\sigma_{a.c.}(A_\alpha)= \sigma_{a.c.}(A_0)$, including the multiplicity. For $\alpha\ge\sqrt2$ an additional branch of absolutely continuous spectrum arises, its source is an auxiliary Jacobi matrix which is related to the operator $A_\alpha$. This birth of an extra-branch of a.c. spectrum is the exact mathematical expression of the effect which was interpreted by Smilansky as irreversibility.
Naboko Sergey N.
Solomyak Michael
No associations
LandOfFree
On the absolutely continuous spectrum in a model of irreversible quantum graph does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the absolutely continuous spectrum in a model of irreversible quantum graph, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the absolutely continuous spectrum in a model of irreversible quantum graph will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-64173