Deficiency Indices and Spectra of Fourth Order Differential Operators with Unbounded Coeffcients on a Hilbert space

Inhaltsverzeichnis (Table of Contents)

• INTRODUCTION
  • Background of the study
  • Basic Concepts
  • Statement of the problem
  • Objective of the study
  • Significance of the study
  • Research methodology
• LITERATURE REVIEW
• ASYMPTOTIC INTEGRATION
  • Hamiltonian system
  • Asymptotic Integration
  • Eigenvalues of C(x)
  • Dichotomy condition
  • Diagonalisation
• DEFICIENCY INDICES AND SPECTRA
  • Deficiency Index
• CONCLUSION AND RECOMMENDATION
  • Conclusion
  • Recommendation

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This thesis explores the asymptotic behavior of eigensolutions and deficiency indices of fourth-order differential operators with unbounded coefficients. The primary objective is to determine the eigenvalues, deficiency indices, and the location of the absolutely continuous spectrum of self-adjoint extension operators associated with these operators.

• Asymptotic analysis of eigensolutions
• Deficiency indices of fourth-order differential operators
• Absolutely continuous spectrum of self-adjoint extensions
• Impact of unbounded coefficients on operator properties
• Application of asymptotic integration techniques

Zusammenfassung der Kapitel (Chapter Summaries)

• Chapter 1: INTRODUCTION This chapter introduces the concept of unbounded operators and their relevance in quantum mechanics and other fields. It outlines the problem, objectives, significance, and research methodology of the study.
• Chapter 2: LITERATURE REVIEW This chapter reviews existing literature related to the topic, providing a foundation for the research undertaken in the thesis.
• Chapter 3: ASYMPTOTIC INTEGRATION This chapter delves into the concepts of Hamiltonian systems, asymptotic integration, eigenvalues, and the dichotomy condition. It also discusses the process of diagonalization within this framework.
• Chapter 4: DEFICIENCY INDICES AND SPECTRA This chapter focuses on the concept of deficiency indices and their relationship to the spectral properties of the studied differential operators.

Schlüsselwörter (Keywords)

The key focus of this thesis lies in the analysis of fourth-order differential operators with unbounded coefficients, exploring their asymptotic behavior, deficiency indices, and spectral properties. Key concepts include asymptotic integration, deficiency indices, absolutely continuous spectrum, self-adjoint extensions, and Hamiltonian systems.