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On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrödinger operator

R. Fabbri (2002)

Bollettino dell'Unione Matematica Italiana

In this paper we study the Lyapunov exponent β ⁢ E for the one-dimensional Schrödinger operator with a quasi-periodic potential. Let Γ ⊂ R k be the set of frequency vectors whose components are rationally independent. Let Γ ⊂ R k , and consider the complement in Γ ⁢ C r ⁢ T k of the set D where exponential dichotomy holds. We show that β = 0 is generic in this complement. The methods and techniques used are based on the concepts of rotation number and exponential dichotomy.

On the meromorphic solutions of a certain type of nonlinear difference-differential equation

Sujoy Majumder, Lata Mahato (2023)

Mathematica Bohemica

The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation f n ( z ) + P d ( z , f ) = p 1 ( z ) e α 1 ( z ) + p 2 ( z ) e α 2 ( z ) , where P d ( z , f ) is a difference-differential polynomial in f ( z ) of degree d ≤ n - 1 with small functions of f ( z ) as its coefficients, p 1 , p 2 are nonzero rational functions and α 1 , α 2 are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation.

On the method of Esclangon

Ján Andres, Tomáš Turský (1996)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

On the necessary and sufficient conditions for the convergence of the difference schemes for the general boundary value problem for the linear systems of ordinary differential equations

Malkhaz Ashordia (2021)

Mathematica Bohemica

We consider the numerical solvability of the general linear boundary value problem for the systems of linear ordinary differential equations. Along with the continuous boundary value problem we consider the sequence of the general discrete boundary value problems, i.e. the corresponding general difference schemes. We establish the effective necessary and sufficient (and effective sufficient) conditions for the convergence of the schemes. Moreover, we consider the stability of the solutions of general...

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