Some remarks about the limit point and limit circle theory

Abstract

LetL be a formally selfadjoint differential operator andp a real-valued function, both ona≤x<∞. The deficiency indices are the numbers of solutions ofLu=λpu for Im λ>0 and for Im λ<0 which have a certain regularity atx=∞. (A) Ifp(x)≥0 this regularity means that the integral ofp(x)│u│² converges at infinity. (B) Ifp changes its sign for arbitrarily large values ofx butL 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 whenL is of odd order. It is then indicated how the method carriers over to (B).