On the difference equations with periodic coefficients. (English) Zbl 1012.39008

Buslaev, V.; Fedotov, A.

Adv. Theor. Math. Phys. 5, No. 6, 1105-1168 (2001).

Summary: We study entire solutions of the difference equation \[ \psi(z+h)= M(z)\psi(z),\;z\in\mathbb{C},\;\psi(z) \in\mathbb{C}^2. \] In this equation, \(h\) is a fixed positive parameter, and \(M:\mathbb{C}\mapsto SL(2,\mathbb{C})\) is a given matrix function. We assume that \(M(z)\) is a \(2\pi\)-periodic trigonometric polynomial. The main aim is to construct the minimal entire solutions, i.e. the solutions with the minimal possible growth simultaneously as for \(z\to-i \infty\) so for \(z\to+i \infty\).
We show that the monodromy matrices corresponding to the bases made of the minimal solutions are trigonometric polynomials of the same order as the matrix \(M\). This property relates the spectral analysis of the one-dimensional difference Schrödinger equations with the potentials being trigonometric polynomials to an analysis of a finite dimensional dynamical system.

MSC:

39A11	Stability of difference equations (MSC2000)
39A12	Discrete version of topics in analysis
37J45	Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)

Keywords:

asymptotics; difference equations; periodic coefficients; minimal entire solutions; Schrödinger equations; finite dimensional dynamical system