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

Abstract

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 lies the problem of computing the eigenvalues of these higher order operators when the coefficients are unbounded. In this thesis, using asymptotic integration, we have investigated the asymptotics 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 endeavuored 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 was in the range (2, 2) ≤ defT ≤ (4, 4) under different growth and decay conditions of coefficients. In addition, the absolutely continuous spectrum is either half or full line of spectral multiplicity 1 or 2 depending on the integrability of p −1 2 1.

Contents

Acknowledgements iii

Dedication iv

Abstract v

Table of Contents vi

Index of Notations viii

Chapter 1 INTRODUCTION 1

1.1 Background of the study 1

1.2 Basic Concepts 4

1.3 Statement of the problem 12

1.4 Objective of the study 13

1.5 Significance of the study 13

1.6 Research methodology 14

Chapter 2 LITERATURE REVIEW 15

Chapter 3 ASYMPTOTIC INTEGRATION 18

3.1 Hamiltonian system 18

3.2 Asymptotic Integration 22

3.3 Eigenvalues of C(x) 23

3.4 Dichotomy condition 26

3.5 Diagonalisation 29

Chapter 4 DEFICIENCY INDICES AND SPECTRA 33

4.1 Deficiency Index. 33

Chapter 5 CONCLUSION AND RECOMMENDATION 41

5.1 Conclusion 41

5.2 Recommendation 42

References 43

CHAPTER 1. INTRODUCTION

T is said to be bounded if there exists a positive real number C such that for all x ∈ D(T), _ Tx _≤ C _ x _. If this number does not exist then the operator is unbounded. Example of unbounded operators are some differential operators defined on the space of polynomials of degree n. In mathematics, more specifically functional analysis and operator theory, the concept of unbounded operators provides an abstract framework for dealing with differential operators and unbounded observables in quantum mechanics. The domain of an operator is a linear subspace, not necessarily the whole space. In contrast to bounded operators, unbounded operators defined on a given space do not form an algebra, not even a linear space, because each one is defined on its own domain. Here, the space where T is defined, is a L2 w([0,∞)) Hilbert space. The theory of unbounded operators developed in the late 1920’s and early 1930’s was part of developing a rigorous mathematical framework for quantum mechanics. This theory as developed by John Von Neumann and Marshall Stone [6], is very important in this research. For instance, in [12], Naimark has used the results of Von Neumann on unbounded operators which were solved using graphs to extend his research on linear differential operators. This approach entails substantial simplifications and its applications to theory of differential equations which yield a unified approach to diverse problems arising in differential equations and their corresponding operators. In the case of unbounded operators, the most important aspects considered are the domains and extension problems. For the Hilbert space adjoint operator T∗ of a linear operator T to exist,

T must be densely defined in H and D(T) ⊂ D(T∗). It is a well known fact that a self-adjoint linear operator is symmetric, but the converse is not generally true in the unbounded case. Generally, properties of an operator depends largely on the domain and may change under extensions and restrictions. It is shown In [9] that an unbounded linear operator satisfying the relation, : < Tx,y >=< x,Ty > cannot be defined on all of H. Higher order differential operators generated by (1.1) above, as defined on a Hilbert space, have received much attention, though there still lies the problem of computing the eigenvalues of these higher order differential operators when the coefficients are unbounded. Because of this, we have investigated the deficiency indices of minimal differential operators and the location of absolutely continuous spectrum of self-adjoint extension operators of the minimal differential operators generated by (1.1) on L2 w([0,∞)) using asymptotic integration.

1.2 Basic Concepts

Definition 1.2.1

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. Definition 1.2.2 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. Definition 1.2.3 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.

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