Spectral Theory of Unbounded Self-adjoint Operators in Hilbert spaces

In this work we present a derivation of the spectral theorem of unbounded spectral operators in a Hilbert space. The spectral theorem has several applications, most notably in the theory of quantum mechanics. The theorem allows a self-adjoint linear operator on a Hilbert space to be represented in terms of simpler operators, projections. The focus of this work are the self-adjoint operators in Hilbert spaces. The work begins with the introduction of vector and inner product spaces and the definition of the complete inner product space, the Hilbert space. Three classes of bounded linear operators relevant for this work are introduced: self-adjoint, unitary and projection operators. The unbounded self-adjoint operator and its properties are also discussed. For the derivation of the spectral theorem, the basic spectral properties of operators in Hilbert space are presented. The spectral theorem is first derived for bounded operators. With the definition of basic spectral properties and the introduction of the spectral family, the spectral theorem for bounded self-adjoint operators is presented with a proof. Using Weckens lemma, the spectral theorem can be written for the special class of unitary operators. Using the spectral theorem for unitary operators, we can write the spectral theorem of unbounded self-adjoint operators. Using the Cayley transform, the unbounded self-adjoint operator is rewritten in terms of bounded unitary operators and the spectral theorem is presented in the most general form. In the last section of the thesis, the application of the above results in quantum mechanics is briefly discussed