In this thesis we study two applications of spectral theory. Suppose that T is a linear operator on a Hilbert space H. The spectrum of T is the set of all complex numbers z for which T ? z I is not invertible. The spectral analysis or spectral theory of a bounded linear operator is an essential problem in functional analysis. It studies the relations between an operator T and its resolvent (T ? zI)?1. It also studies the spectrum of T, the spectrum of (T ? zI)?1 and the relations between eigenvalus and eigenvectors of T. This theory has many applications in mathematics, physics, computer science, engineering and other sciences. For example we can observe some applications of this theory in number theory, graph theory, quantum physics and solving many differential equations. Here, we first study the application of spectral theory in solving a partial differential equation, named wave equation, in three dimension space. This equation is obtained via the simulation of two adjacent vibrating three dimensional wedges with a common edge. For this, we introduce a selfadjoint operator on a suitable Hilbert space H for the spatial part of the problem. Then by using the complex and sine Fourier transforms to our equation we reduce our problem in three dimensional space with singularities to a non-singular problem in one dimensional space. We solve this problem by the limiting absorption principle and the expansion in generalized eigenfunctions. These results which we obtain in one dimensional space can be extended to three dimensional space by using two dense suaces of H. After this, the initial boundary value problem will be solved by the spectral theorem for unbounded operators on Hilbert spaces. Finally, we introduce the special ltr"