In this thesis I look at the theory of compact operators in a general Hilbert space, as well as the inverse of the Hamiltonian operator in the specific case of L2[a,b]. I show that this inverse is a compact, positive, and bounded linear operator. Also the eigenfunctions of this operator form a basis for the space of continuous functions as a subspace of L2[a,b]. A numerical method is proposed to solve for these eigenfunctions when the Hamiltonian is considered as an operator on Rn. The paper finishes with a discussion of examples of Schrödinger equations and the solutions.

The interplay between generalizations of Riezs' famous representation theorem and Radon-Nikodým type theorems has a long history. This paper will explore certain aspects of the theory of bounded linear operators on continuous function spaces, Radon-Nikodým type properties, and their connections.

With appropriate smoothness and decay conditions, it has been shown that the deficiency index and spectral properties of unbounded differential operators are superpositions of the contributions from the individual clusters. The difference operators with almost constant coefficients are limit point at infinity and the absolutely continuous spectrum of their selfadjoint extensions coincide with that of the limiting selfadjoint extension operators.

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This thesis consists of eight chapters. Chapter 1 contains the preliminaries: the background, notation and results needed for this work. In Chapter 2 we study the problem of when P, the set of analytic polynomials, is dense in the Hardy space H^t(G) or the Bergman space L^t_nG, where G is a bounded domain and t ∈ [1,∞). Characterizations of special domains are also given. In Chapter 3 we generalize the definition of a Carleson measure to an arbitrary simply connected domain. Let G be a bounded simply connected domain with harmonic measure ω. We say a positive measure τ on G is a Carleson measure if there exists a positive constant c such that for each t ∈ [1, ∞) and each polynomial p we have ⎮⎮p⎮⎮_L¹(τ)≤ c ⎮⎮p⎮⎮ _Lᵗ(ω), We characterize all Carleson measures on a normal domain-definition: a domain G where P is dense in H¹(G). It turns out that P is dense in Hᵗ(G) for all t when G is normal. In Chapter 4 we describe some special simply connected domains and describe how they are related to each other via various types of polynomial approximation. In Chapter 5 we study the various equivalence classes of subnormal operators under the relations of unitary equivalence, similarity and quasi similarity under the assumption that G is a normal domain. In Chapter 6 we characterize the Carleson measures on a finitely connected domain. We are able to push our techniques in the latter setting to characterize those subnormal operators similar to the shift on the closure of R(K) in L²(σ) when R(K) is a hypo dirichlet algebra. In Chapter 7 we illustrate our results by looking at their implications when G' is a crescent. Several interesting function theory problems are studied. In Chapter 8 we study arc length and harmonic measures. Let G be a Dirichlet domain with a countable number of boundary components. Let ω be the harmonic measure of G. We show that if J is a rectifiable curve and E ⊂ ∂G ∩ J is a subset with ω(E) > 0, then E has positive length. / Ph. D.

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