Spectral theory for self-adjoint linear relation (SALR) on a Hilbert space and its application in homogenous abstract cauchy problem

Abstract

A spectral theory studies eigenvalues and eigenvectors of SALR on H. SALR on Hilbert space H is a linear relation satisfying A = A*. Many applications of SALR on quantum theory, such as the homogenous abstract Cauchy problem.If M is an operator that has an inverse then eigenvalues and eigenvectors are easily determined, but If M is an operator that does not have an inverse then eigenvalues and eigenvectors are quite difficult determined. One way that can be done is to use a linear relation. Furthermore, there are some properties of spectral theoryof linear operator that can not apply to SALR. This paper aims to give a spectral theory for SALR and its application in a homogenous abstract Cauchy problem.