In this thesis, using asymptotic integration, we have investigated the asymptotic of the eigensolutions and the deficiency indices of fourth order differential operators with unbounded coefficients as well as the location of absolutely continuous spectrum of self-adjoint extension operators.
We have mainly endeavored to compute eigenvalues of fourth order differential operators when the coefficients are unbounded, determine the deficiency indices of such differential operator and the location of the absolutely continuous spectrum of the self-adjoint extension operator together with their spectral multiplicity. Results obtained for deficiency indices were in the range (2, 2) ≤ defT ≤ (4, 4) under different growth and decay conditions of co-efficients.
The concept of unbounded operators provides an abstract framework for dealing with differential operators and unbounded observable such as in quantum mechanics. The theory of unbounded operators was developed by John Von Neumann in the late 1920s and early 1930s in an effort to solve problems related to quantum mechanics and other physical observables. This has provided the background on which other scholars have developed their work in differential operators. Higher order differential operators as defined on Hilbert spaces have received much attention though there still lays the problem of computing the eigenvalues of these higher order operators when the coefficients are unbounded.
Table of Contents
Chapter 1 INTRODUCTION
1.1 Background of the study
1.2 Basic Concepts
1.3 Statement of the problem
1.4 Objective of the study
1.5 Significance of the study
1.6 Research methodology
Chapter 2 LITERATURE REVIEW
Chapter 3 ASYMPTOTIC INTEGRATION
3.1 Hamiltonian system
3.2 Asymptotic Integration
3.3 Eigenvalues of C(x)
3.4 Dichotomy condition
3.5 Diagonalisation
Chapter 4 DEFICIENCY INDICES AND SPECTRA
4.1 Deficiency Index
Chapter 5 CONCLUSION AND RECOMMENDATION
5.1 Conclusion
5.2 Recommendation
Research Objectives and Themes
This thesis investigates the deficiency indices of fourth-order minimal differential operators with unbounded coefficients and identifies the location of the absolutely continuous spectrum of their self-adjoint extensions using asymptotic integration techniques.
- Computation of eigenvalues for fourth-order differential operators under unbounded conditions.
- Determination of deficiency indices based on varying asymptotic criteria.
- Analysis of the absolutely continuous spectrum and associated spectral multiplicity for self-adjoint extension operators.
- Application of Levinson’s theorem and M-matrix theory to connect spectral properties with eigensolutions.
Excerpt from the Book
1.2 Basic Concepts
A linear operator T : X → Y is said to be bounded if there exists C ≥ 0 such that || Tx || ≤ C || x || for all x ∈ X. If the positive real number C does not exist, then the operator T is said to be unbounded.
A linear operator T from one topological vector space, X, to another one, Y, is said to be densely defined if the domain of T is a dense subset of X. For example, consider the space C([0, 1]; R) of real valued continuous functions defined on the unit interval. Let C1([0, 1]; R) denote the subspace consisting of all continuously differentiable functions. Equip C([0, 1]; R) with the supremum norm ||.||∞; this makes C([0, 1]; R) into a real Banach space. The differential operator D, that is, D(f) = f' for f ∈ C([0, 1], R) is given by: D(f) = C1([0, 1]; R). so D can only be defined on C1([0, 1]; R) and hence D is densely defined.
Let H be a Hilbert space and T be a densely defined operator from H into itself. If T* is a Hilbert adjoint of T such that T ⊂ T*, then T is called a symmetric operator, that is to say, for each x and y in the domain of T we have < Tx, y > = < x, Ty >. If T = T*, then T is self-adjoint operator and if T is symmetric with its second adjoint T** essentially self-adjoint, then T = T** and T is said to be essentially self-adjoint.
Summary of Chapters
Chapter 1 INTRODUCTION: This chapter defines the mathematical framework, including unbounded differential operators on Hilbert spaces, and establishes the research objectives and methodology.
Chapter 2 LITERATURE REVIEW: This chapter reviews the historical development of unbounded operator theory and highlights previous work on the spectral properties of higher-order operators.
Chapter 3 ASYMPTOTIC INTEGRATION: This chapter details the conversion of differential equations into Hamiltonian systems and applies Levinson's theorem to estimate eigenvalues and establish dichotomy conditions.
Chapter 4 DEFICIENCY INDICES AND SPECTRA: This chapter presents the explicit calculation of deficiency indices for the minimal operator and describes the location and spectral multiplicity of the absolutely continuous spectrum.
Chapter 5 CONCLUSION AND RECOMMENDATION: This chapter summarizes the findings regarding deficiency indices and spectral multiplicity and suggests directions for further research in higher-order operator theory.
Keywords
Unbounded operators, Differential operators, Hilbert space, Asymptotic integration, Levinson’s theorem, Eigenvalues, Deficiency indices, Self-adjoint extension, Spectral multiplicity, Hamiltonian systems, M-matrix, Continuous spectrum, Functional analysis, Quantum mechanics, Dichotomy condition.
Frequently Asked Questions
What is the core subject of this research?
The research focuses on the spectral analysis of fourth-order differential operators, specifically addressing the challenges posed when the operator coefficients are unbounded.
What are the central thematic fields?
The work integrates functional analysis, operator theory, and asymptotic integration to determine the properties of differential operators in weighted Hilbert spaces.
What is the primary objective of the study?
The goal is to compute deficiency indices and locate the absolutely continuous spectrum of self-adjoint extensions of minimal differential operators with unbounded coefficients.
Which scientific methodology is utilized?
The study employs asymptotic integration, primarily based on Levinson's theorem, along with M-matrix theory, to approximate solutions and characterize the spectrum.
What is covered in the main body of the work?
The main body covers the theoretical conversion of fourth-order equations into first-order Hamiltonian systems, the application of dichotomy conditions, and the derivation of results for deficiency indices.
Which keywords characterize this thesis?
Key terms include unbounded operators, Hilbert space, asymptotic integration, Levinson’s theorem, deficiency indices, and spectral multiplicity.
How is the M-matrix used in this context?
The M-matrix serves as an ideal tool to connect the spectral properties of the self-adjoint extension of an operator with the asymptotics of its eigenfunctions.
What significance do the deficiency indices hold?
Deficiency indices provide essential quantitative information about the spectra of self-adjoint extensions and characterize the number of linearly independent square-integrable solutions.
- Citation du texte
- Rodgers Achiando (Auteur), 2016, Deficiency Indices and Spectra of Fourth Order Differential Operators with Unbounded Coeffcients on a Hilbert space, Munich, GRIN Verlag, https://www.grin.com/document/376969
