Evgeny Korotyaev

Spectral asymptotics for periodic fourth-order operators

We consider the operator d 4 dt4 +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 6= 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.

Trace formulae and high energy asymptotics for Stark operator

Abstract In , we consider the unperturbed Stark operator H0 (i.e., the Schrödinger operator with a linear potential) and its perturbation H = H 0 + Vby an infinitely smooth compactly supported potential V. The large energy asymptotic expansion for the modified perturbation determinant for the pair (H 0, H) is obtained and explicit formulae for the coefficients in this expansion are given. By a standard procedure, this expansion yields trace formulae of the Buslaev–Faddeev type.

Effective Masses for Zigzag Nanotubes in Magnetic Fields

We consider the Schrödinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon nanotubes in magnetic field. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We obtain identities and a priori estimates in terms of effective masses and gap lengths.

The inverse problem for perturbed harmonic oscillator on the half-line with a dirichlet boundary condition

Abstract.We consider the perturbed harmonic oscillator

Inverse Problem for the Discrete 1D Schrödinger Operator with Small Periodic Potentials

T_{D}\psi=-\psi^{\prime\prime}+x^{2}\psi+q(x)\psi, \psi(0)=0,

A priori estimates for the Hill and Dirac operators

in

Sharp asymptotics of the quasimomentum

L^{2}(\mathbb{R}_{+})