Andrei Badanin | Searcharxiv

arXiv2005

The Lyapunov function for Schrödinger operators with a periodic 2x2 matrix potential

We consider the Schrödinger 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.

math.SP↗

arXiv2005

Spectral asymptotics for periodic fourth order operators

We consider the operator ${d^4dt^4}+V$ on the real line with a real periodic potential $V$. 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 above the bottom of the spectrum). We also show that the periodic and anti-periodic spectrum together determine the spectrum of our operator. Finally, we show that for small potentials $V\ne 0$ the spectrum in the lowest band has multiplicity 4 and the bottom of the spectrum is a resonance, and not a periodic (or anti-periodic) eigenvalue.

math.SP↗