الفهرس 
C-semi-GroupsOnly 14 pages are availabe for public view
 |  | from | 111 |  |  |
| | | from | 111 | | |
Abstract
This thesis is mainly concerned with the perturbation of some partial differential operators ”perturbation means perturbation of Laplace (elliptic)
.operators by a potential function”. To study this perturbation the relation between forms corresponding to these operators and Co-semi-groups |cfI0,18]. The idea of admissible (regular) potentials, which is given by [Voigt [cf.30] is also used in studying this perturbation.
| The thesis consists of an introduction, four chapters and a list of Rferences, together with English and Arabic summaries. Those can be Summarized as follows:
jpThe introduction:
It includes a quick hint for the purpose of the thesis and its contents. Finally a historical review for what has been achieved to study the ferturbalion of these partial differential operators is reported.
Chapter I:
This chapter can be considered as a background for the basic material used in this thesis.
Chapter II:
This chapter studies the perturbation of the generator of a given [positive C„-semi-group by using the concept of admissible (regular), Htentials, which was introduced by Voigt [30] in Lp( Q ), 1 ^p<«>,
Flere (Q, T^n ) is a measure space. hapter III: This chapter includes three sections Voigt ([30] proposition [5.8]) fave the perturbation of -A by a positive potential. This result is Generalized to any closed, positive, densely defined, self - adjoint operator mm § 3.1 (cf 3.1.2) and the perturbation of a given regular Dirichlet form is handled, which is given by [26]. Also ([30] proposition 5.7) gave the perturbation of -A by a negative potential. This result is generalized to any
> i r
self - adjoint operator associated with a regular Dirichlet form (cf. theorem 3.2.1) and the perturbation of a given regular Dirichlet form is dealt, which is studied by [26]. In § 3.3 the results of § 1.1 and § 1.2 are applied to the
Schrodinger s form ( h +1 + -1 _ ), where
h(f,g) = £ < 3, f, d; g>
i=l
and
V(f)= I|vf| |f|2dM<«>, n
k e fi, with V| as a measurable function Vt: fi-»sJl. Theorems (3.3.5), |3.3.6)and (3.3.7)are due to Stollmann[26]. Corollaries (3.2.7), (3.3.8), theorems (3.3.8) and (3.3.9) did not occur in literatures (as far as we
Know).
r Chapter IV:
This chapter studies the perturbation of coercive forms using the fconcept of admissible (regular) potentials. In § 4.1 the definition of feoercive forms, which is given by [18] and the coercivity of the in-order Mifferential forms are given. In § 4.2 the concept of admissible potentials is applied to study the perturbation of coercive forms. The results of this Schapter (4.1.4), theorems (4.1.9), (4.1.10), corollray (4.1.11), theorem H4.2.1) and proposition (4.2.3) did not occur in literatures (as far as we know) in Sobolev spaces.
References:
The references used in preparing this thesis were organized according to the alphabetical order of the researchers.