Some remarks about the limit point and limit circle theory

Abstract

LetL be a formally selfadjoint differential operator and p a real-valued function, both on a≤x<∞. The deficiency indices are the numbers of solutions of Lu=λpu for Im λ>0 and for Im λ<0 which have a certain regularity at x=∞. (A) If p(x)≥0 this regularity means that the integral of p(x)│u│² converges at infinity. (B) If p changes its sign for arbitrarily large values of x but L has a positive definite Dirichlet integral it is natural to relate the regularity to this integral. Weyl’s classical study of the deficiency indices is reviewed for (A) with the help of elementary theory of quadratic forms. Individual bounds are found for the deficiency indices also when L is of odd order. It is then indicated how the method carriers over to (B).