Mathematics – Spectral Theory
Scientific paper
2005-08-29
Mathematics
Spectral Theory
Scientific paper
We consider the Schr\"odinger operator on the real line with a 2x2 matrix valued 1-periodic potential. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic on a two sheeted Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove the existence of real as well as non-real resonances for specific potentials. We determine the asymptotics of the periodic and anti-periodic spectrum and of the resonances at high energy. We show that there exist two type of gaps: 1) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, 2) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We also show that periodic and anti-periodic spectrum together determine the spectrum of the matrix Hill operator.
Badanin Andrei
Brüning Jochen
Korotyaev Evgeny
No associations
LandOfFree
The Lyapunov function for Schrödinger operators with a periodic 2x2 matrix potential 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 The Lyapunov function for Schrödinger operators with a periodic 2x2 matrix potential, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Lyapunov function for Schrödinger operators with a periodic 2x2 matrix potential will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-405232