The study of difference equations has avast history especially in relation to Sturm-Lioville equations and their discrete counterparts, Jacobi matrices, which have been analyzed using similar and related methods.However much is needed to be done in terms of spectral theory in the discrete setting. Therefore, in this study, we approximate the eigenvalues and establish the dichotomy condition of a Fourth Order Difference equation with Unbounded Co -efficients on a Hilbert Space.
Inhaltsverzeichnis (Table of Contents)
- INTRODUCTION
- DICHOTOMY CONDITION
- DIAGONALISATION
- ACKNOWLEDGMENT
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
The study aims to approximate the eigenvalues and establish the dichotomy condition for a fourth-order difference equation with unbounded coefficients on a Hilbert space. This is achieved through spectral analysis and the use of various techniques, including the transformation of the problem into a Levinson-Benzaid-Lutz form.
- Spectral theory in the discrete setting
- Dichotomy condition of a fourth order difference equation
- Approximations of eigenvalues of the characteristic polynomial
- Diagonalization of the system to convert it into Levinson-Benzaid-Lutz form
- The influence of unbounded coefficients on the solutions of difference equations
Zusammenfassung der Kapitel (Chapter Summaries)
- INTRODUCTION This chapter introduces the problem and outlines the research objective. It defines the fourth-order difference equation and the Hilbert space setting. The chapter also describes the forward and backward difference operators and the construction of the characteristic polynomial.
- DICHOTOMY CONDITION This chapter focuses on proving the dichotomy condition for the eigenvalues of the difference operator. It utilizes a theorem on asymptotically constant difference equations and introduces the concept of z-uniform dichotomy.
- DIAGONALISATION This chapter explores the process of diagonalizing the system to transform it into the Levinson-Benzaid-Lutz form. It involves computing the eigenvectors and utilizing an approach based on quasi-differences. The chapter also discusses the second diagonalization and the necessary smoothness and decay conditions.
Schlüsselwörter (Keywords)
The main keywords and focus topics of this study include difference operators, Jacobi matrices, Sturm-Liouville operators, spectral analysis, dichotomy condition, eigenvalues, Hilbert space, unbounded coefficients, Levinson-Benzaid-Lutz form, and diagonalization.
- Quote paper
- MSC (Pure mathematics) Evans Mogoi (Author), 2019, Dichotomy Condition of Difference Operators, Munich, GRIN Verlag, https://www.grin.com/document/491591
