On the Deficiency Indices of Powers of Real 2nth -Order Symmetric Differential Expressions

Abstract

In [4] we proved that for the symmetric second-order differential expression My=-(py'y+qy(1.1) on [0, oo) with p and q real-valued and p > 0, the well-known Levinson criterion for M to be in the limit-point case also implies that any power of M (and hence any real polynomial expression in M) is also in the limit-point case. In this paper we consider the analogous problem for the general 2/ith-order symmetric expressionMy= Z (-\yXp n -j/ J) ) u) (1.2) ;=o on [0, oo) with the p n -. } real-valued for./ = 0, 1, ..,, n and p 0 > 0.The deficiency index d{M) of the real symmetric expression M can be defined as the number of linearly independent L 2 (0, oo) solutions of My = Ayfor any non-real number L This integer d(M) is independent of X (as long as im A # 0) and it was shown by Glazman in [7] that n < d(M) ^ 2/i and also that every value in this range is possible. When d(M) = n we say that M is in the limit-point case or that M is limit-point.If the coefficients of M are sufficiently differentiate we can form powers M k in the natural way: M l =M, M k y = M k~l (My), fc = 2, 3, .... These powers M 2 , M 3 , ... are again real symmetric differential expressions of type (1.2)-(see [3; p. 1289]). It is known that powers of Jimit-point expressions are not necessarily limit-point-(see [11] and [13]). Our aim in this paper is to find conditions on the coefficients pj which are sufficient for M k , if it exists, to be in the limitpoint case. We have two main results. The first shows that Hinton's condition in [9] for (1.2) to be limit-point (which almost reduces to Levinson's condition in the second-order case (1.1)) is sufficient not only for M to be limit-point as Hinton proves but for M k to be limit-point for any k = 1, 2, 3, .... Our second main result shows that Hartman's " interval type " sufficient condition in [8] for (1.1) to be limit-point, along with its extension by Brown and Evans [1] to the higher order case (1.2), is sufficient not only for M to be limit-point but also for M k to be limitpoint for any k = 1, 2, 3,

A limit-point criterion for 2n-th order symmetric differential expressions

“…We consider on / = [a, oo), aeU, the general 2n-th order symmetric differential expression j=0 j=0 (1.1) with real-valued coefficients pj, qj. For general n, limit-point criteria have been established by many authors, for example Hinton [7], Evans and Zettl [2], [3] (see also [10]), Naimark [11] and Fedorjuk [4] for real M, and Frentzen [5] for general M. In this present paper we derive a limit-point criterion that is not covered by the known criteria. We presume the "boundary coefficients" p o {t) and p n {t) to have as dominating terms real powers of t with positive coefficients and admit for the coefficients pj(t) (/ = l,...,n-1), qj(t) (/ = 0,...,n -1) a rate of growth depending on the growth of both of the "boundary coefficients" p 0 , p n .…”

“…In order to satisfy condition (2.1) we have to determine the norm-squares \\Nf\\l, \\Mf\\l By reason of the identity \\Nf\\ 2 2 = (Nf,Nf)=(f,N 2 f), where (,) denotes the inner product in L 2 (/), these norm-squares are closely related to the square N 2 (we shall choose the coefficients of M, N so smooth that these squares are defined in the classical manner). Now the square of a symmetric differential expression is again a symmetric differential expression.…”

Real-parameter square-integrable solutions and the spectrum of differential operators

“…We comment on our approach using knowledge of the number of square-integrable solutions for real values of the spectral parameter λ to obtain information about the spectrum. This approach contrasts with the commonly used methods based on asymptotic approximations of solutions and on perturbation theory, see [1][2][3][4][5][6][7][8]11,[13][14][15][17][18][19][20][21][22]29] and the references in these books and papers.…”

The essential self-adjointness of differential operators

“…We shall show that \\(l 2t + i)(tJ/4>)-v\\<s. \U -1)(/ 2 « + 0*11 ^ II(«A -1)(*2« + 0*IU' because ifr = 1 in ft 1 = life + 0*IU> because O^^^l = ll(^2t + 0 * -u||/ n i because t > = 0 outside ft 0 c ft 1 (5) as the term for r = 0 has been extracted to give us the term «/rZ 2 , 4>. Now,…”