Characteristic Vectors of Bordered Matrices with Infinite Dimensions I

Abstract

The statistical properties of the characteristic values of a matrix the elements of which show a normal (Gaussian) distribution are well known (cf. [6] Chapter XI) and have been derived, rather recently, in a particularly elegant fashion.¹ The present problem arose from the consideration of the properties of the wave functions of quantum mechanical systems which are assumed to be so complicated that statistical considerations can be applied to them. Since the physical problem has been given rather recently in some detail in another journal [3], it will not be reviewed here. Actually, the model which underlies the present calculations shows only a limited similarity to the model which is believed to be correct. Nevertheless, the calculation which follows may have some independent interest; it certainly provided the encouragement for a detailed investigation of the model which may reproduce some features of the actual behavior of atomic nuclei.