Evgeny Korotyaev

Zigzag nanoribbons in external electric fields

We consider the Schrodinger operator on nanoribbons (tight-binding models) in an external electric potential V on the plane. The corresponding electric field is perpendicular to the axis of the nanoribbon. If V = 0, then the spectrum of the Schrodinger operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. If we switch on a weak electric potential V → 0, then there are two cases: (1) this eigenvalue splits into the small spectral band. We determine the asymptotics of the spectral bands for small fields. (2) the unperturbed eigenvalue remains the flat band. We describe all potentials when the unperturbed eigenvalue remains the flat band and when one splits into the small band of the continuous spectrum.

Inverse resonance scattering for Jacobi operators

The Jacobi operator (Jf)n = an−1fn−1 +anfn+1 + bnfn on ℤ with real finitely supported sequences (an − 1)n∈ℤ and (bn)n∈ℤ is considered. The inverse problem for two mappings (including their characterization): (an, bn, n ∈ ℤ) → {the zeros of the reflection coefficient} and (an, bn, n ∈ ℤ) → {the eigenvalues and the resonances} is solved. All Jacobi operators with the same eigenvalues and resonances are also described.

Conformal spectral theory for the monodromy matrix

For any N x N monodromy matrix we define the Lyapunov function which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the Hill operator. The Lyapunov function has (real or complex) branch points, which we call resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy. We show that the endpoints of each gap are periodic (anti-periodic) eigenvalues or resonances (real branch points). Moreover, the following results are obtained: 1) We define the quasimomentum as an analytic function on the Riemann surface of the Lyapunov function; various properties and estimates of the quasimomentum are obtained. 2) We construct the conformal mapping with imaginary part given by the Lyapunov exponent, and we obtain various properties of this conformal mapping, which are similar to the case of the Hill operator. 3) We determine various new trace formulae for potentials and the Lyapunov exponent. 4) We obtain a priori estimates of gap lengths in terms of the Dirichlet integral. We apply these results to the Schrodinger operators and to first order periodic systems on the real line with a matrix-valued complex self-adjoint periodic potential.

Borg-type uniqueness theorems for periodic Jacobi operators with matrix-valued coefficients

We give a simple proof of Borg-type uniqueness theorems for periodic Jacobi operators with matrix-valued coefficients.

Resonances for periodic Jacobi operators with finitely supported perturbations

We describe the resonances and the eigenvalues of a periodic Jacobi operator with finitely supported perturbations. In the case of small diagonal perturbations we determine their asymptotics.

Periodic Jacobi operator with finitely supported perturbation on the half-lattice

Maths2010 British Mathematical Colloquium and British Applied Mathematics Colloquium Apr 6, 2010- Apr 9, 2010, Edinburgh, UK

Zigzag nanoribbons in external electric and magnetic fields

We consider the Schrodinger operators on zigzag nanoribbons (tight-binding models) in external magnetic and electric fields. If these fields are absent, then the spectrum of the Schrodinger operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. We describe all magnetic and electric fields for which the unperturbed flat band remains the flat band and when one splits into the small band of the continuous spectrum. Also we determine spectral asymptotics for small fields and solve inverse spectral problem.

Global Estimates of Resonances for 1D Dirac Operators

We discuss resonances for 1D massless Dirac operators with compactly supported potentials on the real line. We estimate the sum of the negative power of all resonances in terms of the norm of the potential and the diameter of its support.

Trace formulae for Schrödinger operators with complex-valued potentials on cubic lattices

We consider a class of Schrödinger operators with complex decaying potentials on the lattice. Using some classical results from complex analysis we obtain some trace formulae and use them to estimate zeros of the Fredholm determinant in terms of the potential.

Resonances for Euler–Bernoulli operator on the half-line

Abstract We consider resonances for fourth order differential operators on the half-line with compactly supported coefficients. We determine asymptotics of a counting function of resonances in complex discs at large radius, describe the forbidden domain for resonances and obtain trace formulas in terms of resonances. We apply these results to the Euler–Bernoulli operator on the half-line. The coefficients of this operator are positive and constants outside a finite interval. We show that this operator does not have any eigenvalues and resonances iff its coefficients are constants on the whole half-line.