A STURM SEPARATION THEOREM FOR A LINEAR 2 nth ORDER SELF-ADJOINT DIFFERENTIAL EQUATION

For the 2nth order equation, (1)nv(2n)+ qvO, with q continuous, we obtain a Sturm Separation theorem, involving n + 1 solutions of the equation, which is somewhat analogous to the classical result that the zeros of two linearly independent solutions of the second order equation separate each other.


Introduction
For the second order equation (1.1) with q(x) continuous on [a, oc), we have the well known separation theorem: Theorem 1.1 (Sturm Separation): Let yl(x) be a solution of equation (1.1) with two consecu- tive zeros a <_ b < c < cx, Yi(b) Yi(C) 0, and let Y2(X) be linearly independent of yl(x).Then there exists exactly one point d in the open interval (b,c) with y2(d)-0.
In this paper we obtain an analog of the above Sturm Separation theorem for the 2nth order equation (--1)nV(2n)+qv--0, x E [a, (1.2) with q continuous in [a, oc).There has been a considerable amount of work leading to numerous generalizations of Sturm separation and oscillation theorems to fourth order and 2n-order self- 1This research was partially supported by National Science Foundation grants DMS- 8813113 and DMS-8905202 to Florida Institute of Technology and DMS-88000839 and DMS-8905232 to Colorado School of Mines.Printed in the U.S.A. @1995 by North Atlantic Science Publishing Company C.T. FULTON, S. PRUESS, L. WU adjoint differential equations.A number of results, dealing with properties of zeros of two linearly independent solutions have been given by Leighton and Nehari [11].With regard to possible generalizations of the above separation theorem to the fourth order equation, Leighton   and Nehari [11, p. 329] wrote in 1958: "Simple examples show that the zeros of two solutions of [equation (1.2) above with n 2] do not necessarily separate each other in the same regular fashion as the zeros of two solutions of [equation (1.1) above].This is not surprising, since three zeros of a solution [of equation (1.2) with n 2] may be given arbitrarily, and between two consecutive zeros of one solution there may therefore lie three zeros of another solution.However, if account is taken of this and some other peculiarities of the fourth-order case, a considerable measure of regularity appears."The results of Leighton and Nehari generally deal with two solutions of the fourth order equation, as opposed to trying to deal with a full set of four linearly independent solutions.
In this paper we follow the analysis of Chapter 7 of K. Kreith [9], which leads us to a version of Sturm separation theorem for the 2nth order equation (1.2) which has a slightly different character than other known generalizations.Particularly, we show that for any given solution of equation (1.2) having two consecutive n-fold zeros there must exist another solution (from an n- dimensional subspace) which has an n-fold zero between the two given n-fold zeros.In the process we also develop some determinental identities concerning Wronskians of 2n functions and establish a matrix version of the Green's formula associated with the 2nth order equation.The key to understanding oscillation theory for 2nth order equations appears to lie in looking at solution spaces of dimension n and trying to deal as much as possible with a full set of linearly independent solutions instead of just two such solutions.The present approach helps to underline some useful analogies of the fourth and 2nth order equations to the standard second order equation.We also note that the matrix form of the Green's formula (equation (3.11) below) which utilizes two sets of n solutions, each being 'conjoined', does not seem to be widespread in the literature.The determinental identities in section 4 represent simplifications of identities obtained by Weyl-Kodaira and Everitt, and simplify further when the two sets of solutions are taken to be 'conjoined'; this simplification results in a "matrix form of Green's formula for equation (1.2) which seems to represent a stronger analogy to the second order equation than the usual scalar form.Some results which are related to the Sturm separation theorem of the present paper are the comparison theorems obtained by G. Ladas [10, pp.564-565] and K. Kreith [9, p. 71, Theorem  7.4].We also note that it is possible to give an independent proof of our Sturm separation theorem (Theorem 5.1 below) which relies on a Lagrange identity associated with a factorization of the 2nth order operator in (1.2) as a product of an nth order operator and its adjoint; the identities are from W.A. Coppel [1] and M.S.P. Eastham [2] and the independent proof is due to M.S.P. Eastham.It should perhaps also be mentioned that there are more abstract versions of the Sturm separation theorems which may possibly yield separation theorems for the even-order equation.
Edwards introduces a "theory of U-manifolds" and obtains an abstract version of separation theorem which yields the above separation theorem as a Corollary ([3, p. 53]), but he gives no similar application for higher order equations.Similarly, F. Neumann [13] and ([14, pp. 223-224]) uses a differential geometry approach which enables a proof of the second order separation theorem to be given, although, as yet, no similar separation theorem for the fourth or 2nth order equation has surfaced.
For odd order equations, separation theorems are of a different character.The type of separation theorem obtained in Theorem 5.1 below does not appear to be possible; there is evidently no analog of the concept of an "n-fold zero".Some oscillation and separation results for third order equations are given in Neuman [14,Sec. 10.4].
The Sturm separation theorem of the present paper has also recently been used by the authors in [6] to given an elementary proof of the equivalence of various definitions of oscillation at infinity.

The Sturm-Picone Identity for the Second Order Equation
The principal tool of this paper is a Sturm-Picone identity for the 2nth order equation obtained by K. Kreith [9, Chapters 7 and 8].Since our proof of the separation theorem for the 2nth order equation mimics the classical Picone-style proof of the separation theorem for the 2nd order equation, we give in this section the 2nd order Sturm-Picone identity and the classical proof.
Lemma 2.1 (Picone Identity): If u, v and pl u', qlv' are differentiable for x E I and v(x) 7 0 in I, then (2.3) Proof: This arises from straight forward differentiation.
But Case (it) gives an immediate contradiction since Po-q0-> 0 and Pl In Case (i) we are also lead to a contradiction since (i)implies vu'-uv' d(-) 0 or v 2 dx u(x)=_h'v(x) for all xE[,/] for some K#0.But then assumption.
v(a) v() 0 contrary to Lemma 2.3 (Sturm Separation Theorem): solution u(x) of equation (2.1).Let v(x) be any other solution of equation (2.1) which is linearly independent of u(x).Then v(x) has exactly one zero in the interior of the interval (a,/3).In other words, the zeros of any two linearly independent solutions of (2.1) interlace.
Let a < be two consecutive zeros of a nontrivial Since u and v are linearly Proof: Suppose, on the contrary, that v(x) : 0 for all x E (a,/3).independent it follows that v(a) # 0; for otherwise we would have W(u,v) =0, which implies that the Wronskian, Wx(u,v), is zero for all x and that u and v are therefore linearly dependent.For the same reason we know that v(/3) 0. But when ql Pl and q0 =-P0, equation (2.5) becomes j.,lu,-,i,,.-u.11ov' I ' o .
Hence u and v are linearly dependent on (a,/) contrary to assumption.El 3. A Matrix Version of Green's Formula The Green's formula for the second order equation involves the Wronskian of two solutions on the right hand sidc.The corresponding scalar form of Green's formula for even order equations involves the bilinear concomitant (or skew symmetric bilinear form) of two scalar solutions of the right hand side.But two solutions are not enough to form a Wronskian of 2n solutions of equation (1.2).The object of this section is to show how the Green's formula can be cast in a matrix form so that a 'matrix'-Wronskian of 2n solutions arises on the right hand side.Following the discussion of determinental identities in Section 4, it will then appear in Lemma 5.1 of Section 5 that the determinant of our 'matrix'-Wronskian coincides, for two 'conjoined' sets of n solutions, to the usual scalar Wronskian of 2n solutions; our matrix Green's formula is therefore more in line with the Green's formula for the second order equation.
For simplicity we restrict attention to the simplest form of the 2nth order equation, (-1)nu(2n) + pu 0 where p is continuous on some interval I [a, b].In terms of the vectors 1)n-lu(2n-1)(x _u (x) and _w(x) (.-1) equation (3.1)1 may be written as the first order system 1)'-u(2"-2)( where the matrices A, B, Cp, D are defined by The system form (3.1)2 is a Hamiltonian system since the above matrices satisfy the properties A-DT, B BT, C CT.
We shall also find it helpful to make use of the matrix form for two solutions of (3.1)2 namely and {Ul, u2,... Un} is any set of n solutions of equation (3.1)1.
For the sake of the generalized Picone identity to be proved in Lemma 5.3 below, we also introduce a second set of solutions of the above equation with another potential function" (-1)nv (2n) + qv O, and A, B, C q, D are the same as above with q replacing p.
Proof: Differentiating and employing (3.2)3 we find A T zTBT).z2VT1 .(CVATZ2) -xWx(Vl,V2) making use of the symmetry of B and C. It follows that the Wronskian of V and V 2 is constant for any two solutions ]Vl/and]Vz2o/ ofequation (3.2)3.
The above 'Wronskian of two pairs of solutions of equation (3.2)1 is an n x n matrix, while the ordinary Wronskian of two scalar solutions of the second order equation (1.1) is a scalar; but both are constant quantities associated with solutions of the same equation.For the nth order equation the Wronskian is Wx(Vl, ?92)V lv V iv2, so comparing with the above we can view V1, V 2 as analogs of vl, v 2 and Z1, Z 2 as analogs of Vl, v.In some sense, Z1, Z 2 are the "derivatives" (or perhaps the "nth derivatives") of V1, V 2.
(3.8) k=l But the bilinear concomitant of two scalar solutions is not an appropriate analog of the Wronskian because it involves only two of a possible set of 2n linearly independent solutions of the 2nth order equation.Taking Vi [@i-1)l, 1<_ i, j <_ n, (3.9)1 and (i-1)] VII vj j n + 1,...,2n, 1 <: _ n, in Lemma 3.1, we obtain the following analog of the second-order Green's formula (3.6)1 which involves up to 2n linearly independent solutions of the 2nth order equation.First we observe that by elementary algebra the n x n Wronskian matrix defined in Lemma 3.1 can be written in terms of the bilinear concomitants in (3.8) as, [Vl,Vn+l] [Vl, V2n Wx(VI, VII Differentiation of (3.10) yields the Green's formula in a matrix form by making use of the scalar form (3.

Determinental Identities
The purpose of this section is to relate the n n matrix Wronskian Wx(VI, VII arising in (3.5) and (3.10) and the matrix version of Green's formula (3.11)2 to the usual scalar determinant of 2n functions.Since the identities of this section hold for arbitrary functions with sufficiently many derivatives, we state all results in terms of arbitrary functions since all proofs are of a pure- ly algebraic and combinatorial character.Applications of these identities when the functions are solutions of the differential equations (3.2)1-(3.2)awill be utilized in Section 5.
(4.2) k=l The bilinear concomitant (4.2) (or the constant bilinear concomitant in (3.8) associated with the Green's formula (3.7) in differential equations) is a special case of a 'skew-symmetric bilinear form' which arises as an invariant of a symplectic group (see H. Weyl [15, Chapt.VII).
To establish proofs of the identities below we need the following elementary definitions and lemmas: Definition 4.1: A permutation of (1, 2, n) is a reordering of the integers in the form (il, i2,"-, in) where each integer occurs once and only once.
Definition 4.2: If for a pair of numbers in a permutation (not necessarily consecutive), the first number is larger than the second one, then we say this pair forms an inverse order.The total number of inverse orders in a permutation is called the inverse order number of the permutation.The inverse order number of a permutation (il,i2,...,in) is denoted as v(il,i2,...,in).Definition 4.3: A permutation which has even (odd) number of inverse orders is called an even (odd) permutation.Definition 4.4: An interchange of any two numbers of a permutation is called a transposition.Lemma 4.1: transposition.
Proof: This theorem is well known But we give a short outline of the proof since some of the steps will lead to a reformulation of this identity which will enable us to establish its connection to the Wronskian introduced in (3.5).
Step 1: Putting the expression (4.2) for the bilinear concomitant in (4.3) and performing the necessary multiplications the right-hand side of (4.3) may be written as n 2n 2n ,ki-n 2nn!
where kj is the summation index in (4.2) for fl fi2j_ Step 2: Each of the conditions and f2 fi2j" (k I 1, 2n kl, k 2 1, 2n k2,..., k n 1, 2n kn).If (4.6)1 or (4.6)2 holds, there are four terms associated with permutations of the four functions having equal derivatives which exactly cancel in the inner sum in (4.5).Moreover, the inner sum can be partitioned into sets of four terms of this type and thus the whole inner sum vanishes.
Remark: For n-2, for example, we find 4 vectors are ruled out by condition (4.6)1 and 4 by condition (4.6)2 so that there are only 16-8-8 admissible k-vectors.
Step 3: It is clear that the total number of 'admissible' k-vectors which give a nonzero contribution for the inner sum in (4.5) is 2nn!.
Step 2 in the above proof leads us in a natural way to the following corollary.
[3 Remark: The identity (4.12) with the first sum on the right replaced by det(Wx(Fi, Fii)) found an application in the spectral theory of the fourth order differential operator given by Everitt.In [4, p. 151, Equation (8.2)] Everitt gives a version of (4.12) for n 2 which he uses in connection with solutions of the fourth order self-adjoint equation which are defined by initial conditions at each endpoint of the interval; the normalization of those solutions by initial condi- tions causes his solutions at x a to satisfy [1, 2] 0 and at x b to satisfy [X1, X2] 0 which makes them 'conjoined' (Definition 5.1 below), so that the one term arising in the second sum in (4.12) is zero.
From Everitt's theorem and the above corollaries of the Weyl-Kodaira theorem, we have the following simple corollary: where the sum over the permutations (il,i2,...,i2n) is subject to the restrictions listed in Corollary 4.1.Of course, the right-hand side in (4.16) can be replaced by the right-hand sides in (4.9) or (4.12).

A Sturm Separation Theorem for 2nth Order Equations
For the basic lemmas needed to prove the separation theorem we rely on the development given by K. Kreith [9, Chap.7].Our lemmas 3.1, 5.2 and 5.3 correspond to lemmas 7.1, 7.2 and 7.3 of [9]; similarly, lemma 7.4 of [9] contains the ideas which led us to Theorem 5.1, although the formulation as a separation theorem seems heretofore to have been overlooked.
We first need to address the annoying fact that the bilinear concomitants appearing in the se- cond sum in equation (4.12) which involve [fi, fj] with 1 < i, j _< n or n + 1 < i, j < 2n, need not be zero for arbitrary choices of linearly independent functions {fl, f2,'", f2n}" For the second order equation the Wronskian of two solutions is either zero of nonzero on an interval I according as the solutions are linearly independent on I or not.This property does not extend directly to the 2nth order equation because the matrix Wronskian (in (3.5)) of a solution V of with itself is not there is result equation (3.2) 3 necessarily However, analogous zero.
an Z associated with 2n linearly independent solutions, provided they are normalized so that two sets of n solutions each have matrix Wronskians with themselves equal to zero.We therefore make the following definition:  )._ (:_ ) (5.1) [Vn, Vl]  [Vn, Vn]   Remark: We note that the n solutions {Vl,...,Vn} which make linearly independent.
I VI conjoined may be Z We can now show that if the determinant of the matrix Wronskian of two 'conjoined' solutions of (3.2)3 is zero, then the corresponding 2n solutions of (3.2)1 are linearly dependent.
Lemma 5.1: Let and be two conjoined solutions of equation (3.2)3 on [a,b] where z, z,ij VI, VII are defined in (3.9) and Zi, ZII correspond to them as above.Then for the matrix Wronskian of (3.10) we have det(Wx(Vi, Vii))=O on [a,b] if and only if {Vl,...,V2n} are linearly dependent on [a,b].
Proof: Since V I and VII are conjoined we have that [vi, vj](x) 0 whenever 1 < i, j < n or n + 1 < i, j < 2n.Apply Corollary 4.2 with {vi} in place of {fi} and it follows that the second sum in equation (4.12) vanishes, since every term must involve at least one factor of this type; all terms in which the factors v and vj belong to the sets {Vl,...,Vn} and {Vn+l,...,V2n} respectively, are isolated in the first term of (4.12).Hence, by Corollaries 4.2 and 4.3 we have Wx(Vl,..., V2n) det(Wx(Vi, VII)).
V! Note: For conjoined pairs of solutions of equation (3.2)3 the n x n Wronskian matrix in (3.5)   and (3.10) is evidently a fairly true analog of the Wronskian of the second order equation.