In this study, the author has investigated the absolutely continuous spectrum of a fourth order self-adjoint extension operator of minimal operator generated by difference equation defined on a weighted Hilbert space with the weight function w(t) > 0, t ∈ N where p(t), q(t), r(t) and m(t) are real-valued functions.
The author has applied the M-matrix theory as developed in Hinton and Shaw in order to compute the spectral multiplicity and the location of the absolutely continuous spectrum of self-adjoint extension operator. These results have been an extension of some known spectral results of fourth order differential operators to difference setting. Similarly, they have extended results found in Jacobi matrices.
In this thesis, chapter 1 is about introduction and some preliminary results including literature review, objectives, methodology and basic definitions. In chapter 2, the author has given the results on the computation of the eigenvalues, dichotomy conditions and some results on singular continuous spectrum. Chapter 3 contains the main results in deficiency indices, absolutely continuous spectrum and the spectral multiplicity. Finally, the author has summarized his results in chapter 4 and also highlighted areas of further research.
Inhaltsverzeichnis (Table of Contents)
- Introduction
- Background
- Basic Concepts
- Literature Review
- Statement of the problem
- Objectives of the study
- Significance of the study
- Research Methodology
- Difference Operators
- Hamiltonian System
- Asymptotic Summation
- Bounded Coefficient
- Unbounded Coefficients.
- Dichotomy Condition.
- Diagonalisation
- Deficiency Indices and Spectrum
- Introduction
- Spectrum of Difference operators
- Chapterwise Summary
- Conclusion
- Recomendations.
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This study investigates the absolutely continuous spectrum of a fourth order self-adjoint extension operator of a minimal operator generated by a difference equation. The research explores the spectral analysis of this operator with unbounded coefficients, applying asymptotic summation and M-matrix theory.
- Spectral analysis of fourth order self-adjoint extension operators
- Application of asymptotic summation and M-matrix theory
- Unbounded coefficients in difference equations
- Absolutely continuous spectrum and deficiency indices
- Extension of spectral results from differential operators to difference settings
Zusammenfassung der Kapitel (Chapter Summaries)
- Chapter 1: Introduction
- Provides background information on Sturm-Liouville operators and Jacobi matrices, highlighting the development and relevance of difference equations.
- Introduces the difference equation under investigation, detailing its form, coefficients, and the forward difference operator used.
- Explains the research methodology, emphasizing the application of asymptotic summation and Levinson-Benzaid-Lutz theorem.
- Outlines the objectives and significance of the study, emphasizing the extension of existing spectral results to difference operators.
- Introduces basic concepts such as symmetric operators, self-adjoint operators, spectrum, resolvent operator, eigenvalues, and continuous spectrum.
- Chapter 2: Difference Operators
- Focuses on the computation of eigenvalues, dichotomy conditions, and singular continuous spectrum.
- Explores the concept of Hamiltonian system, its relationship with difference operators, and its role in spectral analysis.
- Discusses asymptotic summation, a crucial technique for analyzing the behavior of solutions to difference equations.
- Examines the impact of bounded and unbounded coefficients on the spectrum of difference operators.
- Chapter 3: Deficiency Indices and Spectrum
- Delves into the fundamental concepts of deficiency indices, which are essential for understanding the spectrum of self-adjoint operators.
- Presents key results regarding the absolutely continuous spectrum and spectral multiplicity of the investigated difference operator.
Schlüsselwörter (Keywords)
The main keywords and focus topics of this text include: fourth order difference operators, self-adjoint extensions, asymptotic summation, absolutely continuous spectrum, deficiency indices, spectral multiplicity, unbounded coefficients, M-matrix theory, Hamiltonian system, Levinson-Benzaid-Lutz theorem.
- Quote paper
- Evans Mogoi (Author), 2015, Absolutely continuous spectrum of fourth order difference operators with unbounded coefficients on a Hilbert space, Munich, GRIN Verlag, https://www.grin.com/document/375444
