Vladimir Buslaev | Searcharxiv

arXiv2008

The scattering matrix and associated formulas in Hamiltonian mechanics

We survey the basic notions of scattering theory in Hamiltonian mechanics with a particular attention to the analogies with scattering theory in quantum mechanics. We discuss the scattering symplectomorphism, which is analogous to the scattering matrix. We prove identities which relate the Calabi invariant of the scattering symplectomorphism to the total time delay and the regularised phase space volume. These identities are analogous to the Birman-Krein formula and the Eisenbud-Wigner formula in quantum scattering theory.

math.SP↗

arXiv2002

On the difference equations with periodic coefficients

In this paper, we study entire solutions of the difference equation $ψ(z+h)=M(z)ψ(z)$, $z\in{\mathbb C}$, $ψ(z)\in {\mathbb C}^2$. In this equation, $h$ is a fixed positive parameter and $M: {\mathbb C}\to SL(2,{\mathbb C})$ is a given matrix function. We assume that $M(z)$ is a $2π$-periodic trigonometric polynomial. We construct the minimal entire solutions, i.e. entire solutions with the minimal possible growth simultaneously as for im$z\to+\infty$ so for im$z\to-\infty$. We show that the monodromy matrices corresponding to the minimal entire solutions are trigonometric polynomials of the same order as $M$. This property relates the spectral analysis of difference Schrödinger equations with trigonometric polynomial coefficients to an analysis of finite dimensional dynamical systems.

math-ph↗