Jacobi matrices together with Sturm-Liouville operators and have already been developed in parallel for many years. However not much in terms of spectral theory has been done in the discrete setting compared to the continuous version especially in higher order operators. The main objective of this study is to compute the deficiency indices of Fourth order difference operator.
Inhaltsverzeichnis (Table of Contents)
- Abstract
- Introduction
- Basic Concepts
- Definition 0.1.
- Definition 0.2.
- Definition 0.3.
- Definition 0.4.
- Hamiltonian System
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
The primary objective of this study is to determine the deficiency indices of a fourth-order difference operator, which is a discrete analogue of a Sturm-Liouville operator. The study focuses on the operator generated by a specific fourth-order difference equation defined on a weighted Hilbert space.
- Deficiency indices of fourth-order difference operators
- Spectral theory in the discrete setting
- Asymptotic summation and Levinson-Benzaid-Lutz theorem
- Discrete Hamiltonian systems
- Self-adjoint extensions of symmetric operators
Zusammenfassung der Kapitel (Chapter Summaries)
- Abstract: This section provides a concise overview of the study's objective and scope, highlighting the focus on computing the deficiency indices of a fourth-order difference operator.
- Introduction: This chapter introduces the research problem, emphasizing the lack of extensive spectral theory development for higher-order difference operators compared to their continuous counterparts. It outlines the study's aim of investigating the deficiency indices of a fourth-order difference operator generated by a specific difference equation defined on a weighted Hilbert space.
- Basic Concepts: This section defines fundamental concepts essential for understanding the subsequent analysis. It introduces forward and backward difference operators, describes symmetric operators and their extensions, and defines the deficiency indices of an operator.
- Hamiltonian System: This chapter discusses the transformation of the fourth-order difference equation into a first-order Hamiltonian system. It explains the use of quasi-differences and introduces the spectral parameter to solve the equation.
Schlüsselwörter (Keywords)
This study focuses on the deficiency indices, fourth-order difference operators, discrete spectral theory, asymptotic summation, Levinson-Benzaid-Lutz theorem, Hamiltonian systems, and self-adjoint extensions.
Frequently Asked Questions
What is the main objective of this mathematical study?
The primary goal is to compute the deficiency indices of a fourth-order difference operator in a discrete setting.
What is a fourth-order difference operator?
It is the discrete analogue of a continuous fourth-order Sturm-Liouville operator, defined on a weighted Hilbert space.
What are deficiency indices in spectral theory?
Deficiency indices are parameters used to determine the existence and nature of self-adjoint extensions of symmetric operators.
How is the Hamiltonian system relevant to this study?
The study transforms the fourth-order difference equation into a first-order Hamiltonian system to facilitate the analysis using quasi-differences.
Which theorems are used for the asymptotic analysis?
The study utilizes the Levinson-Benzaid-Lutz theorem for asymptotic summation to determine the behavior of the operator.
- Citar trabajo
- Evans Mogoi (Autor), 2019, Asymptotic Condition and Deficiency Indices of Difference Operators, Múnich, GRIN Verlag, https://www.grin.com/document/496168
