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Abstract
SUMMARY
The objective of this thesis is to study the concept of separation of differential operators for some differential expressions with operator potentials in Banach spaces, Hilbert spaces and weighted Hilbert spaces. Also we study the existence and uniqueness of the solutions for some differential equations in these spaces, which is a related topic to the separation property.
This thesis consists of four chapters:
In Chapter 1, we are concerned with a survey about the concept of separation of differential expressions and some of the basic definitions and main theorems, that is the necessary background, which will be useful in the next chapters.
In Chapter 2, we have studied the separation property of the Sturm-Lioüville differential operator, as well as the separation of a Schrödinger-type operator with an operator potential in Banach spaces. Moreover, we have studied the separation of a non-linear Schrödinger-type operator with an operator potential in Banach spaces.
In Chapter 3, we have studied the separation of the Sturm-Lioüville differential operator and of a higher order ordinary differential operator, as well as the separation of the Schrödinger operator with an operator potential in Hilbert spaces. Moreover, we have studied the existence and uniqueness of the solution of the Schrödinger equation in Hilbert spaces.
In Chapter 4, we are concerned with studying the separation of a Schrödinger-type operator and the separation of the general second order elliptic differential operator with an operator potential in weighted Hilbert spaces. Moreover, we have studied the existence and uniqueness of the solution of a second order differential equation in weighted Hilbert spaces.
We wish to point out that the results in Chapter 2 and Chapter 3 have been written in the form of four papers:
1- Separation of the Sturm-Lioüville differential operator with an operator potential.
This paper has been accepted for publication in “Applied Mathematics and Computation”. (U.S.A) Elsevier Science (North-Holland) [38]
2- Separation of the Schrödinger operator with an operator potential in the Hilbert spaces.
This paper has been accepted for publication in “Applicable Analysis: An International Journal”. (U.S.A) Taylor & Francis Ltd. (U.K) [39]
3- Separation for Schrödinger-type Operators with operator potentials in Banach spaces.
This paper has been accepted for publication in “Applicable Analysis: An International Journal”. (U.S.A) Taylor & Francis Ltd. (U.K) [49]
4- Separation for higher order ordinary differential operators with operator potentials in Hilbert spaces.
This paper has been submitted for publication in “Journal of Inequalities in Pure and Applied Mathematics”. [51]
The results in Chapter 4 have been written in the form of two papers:
5- Separation of the general second order elliptic differential operator with an operator potential in the weighted Hilbert spaces.
This paper has been accepted for publication in “Applied Mathematics and Computation”. (U.S.A) Elsevier Science (North-Holland) [40]
6- On the separation of elliptic differential operators with operator potentials in weighted Hilbert spaces.
This paper has been accepted for publication in “Panamerican Mathematical Journal”. (U.S.A) [50]