REGULARIZED TRACE FOR EIGEN - FUNCTION AND EIGEN- VAL UES OF STURM-LIOUVILLE OPERAORS WITH DIFFERENI .FORMS OF BOUNDARY CONDITIONS

Abstract

It is well known that the summation of the diagonal elements m a square matrix is equal to the summation of the eigenvalues of linear operator in finite dimensional space. In other words, the trace of a matrix is . equal to the spectral trace in n-dimensional space. It is worth mentioning that this theorem is satisfied also in the case of unclear operators which are defined in Hillbert space [24]. Thus we might ask the following question :Is this theorem applicable in case of unbounded operators ? ; especially in the case of differential operators the trace of matrices and spectral trace are not exist. For this reason we define the so­ called "Regularized trace".