REGULAR AND PATHOLOGICAL EIGENVALUE BEHAVIOR FOR THE EQUATION LAMBDA-U''=VU ON THE SEMIAXIS

We study the behavior of the eigenvalue distribution function n(lambda ) for the equation -lambda u(u) = Vu on R+, u(0) = 0. Typically, n(lam bda)=O(lambda(-1/2)) and, moreover, lambda(1/2)n(lambda) --> pi(-1) in tegral root V dx as lambda --> 0. We obtain criteria for each of these relations. Examples are presented where lim(lambda-->0)lambda 1/2n(la mbda) does exist, but is not equal to n(-1) integral root V dx. (C) 19 97 Academic Press.