Proceedings of the Edinburgh Mathematical Society (1992) 35, 337-348 < ON A PROBLEM OF R. G. D. RICHARDSON by PAUL BINDING and HANS VOLKMER (Received 12th March 1990) In 1913 Richardson published necessary and sufficient conditions for a system of three Sturm-Liouville equations, linked by three parameters, to possess eigenfunctions with arbitrarily many zeros. His work contains errors, but we give conditions of his type valid for k self-adjoint equations, with k parameters. 1980 Mathematics subject classification (1985 Revision): 34B25 1. Introduction In 1912 R. G. D. Richardson published a landmark paper [12] on coupled Sturm-Liouville problems of the form Y (l.i) for k = 2. Richardson assumed Dirichlet end conditions on finite intervals, and analytic coefficient functions pn>0, qm and rmn, and he obtained necessary and sufficient conditions on the rmn guaranteeing existence of solutions with specified numbers of zeros for the eigenfunctions ym. A year later he attempted [13] the more difficult case k = 2>. His methods involved detailed examination of eigencurves and surfaces defined by single equations in two and three parameters, and led to a unified oscillation theory for indefinite, as well as the better known left ("polar") and right definite cases. In many ways these investigations were far ahead of their time: for example, oscillation theory for indefinite problems really restarted in the early 1970's (cf. [6], [7] etc.) and a systematic study of the geometry of eigencurves and surfaces resurfaced with Turyn [14] in 1979, although specific properties of eigencurves can be found in, say, [10], [1], [9] and [8]. Richardson's geometric reasoning can easily be put into a more modern (and rigorous) analytical framework when k = 2, but there are difficulties when k = 3, as observed by Turyn (loc. cit). In fact we shall show that Richardson's necessary and sufficient conditions in [13] are both in error, but we shall examine special cases for /c = 3 where his reasoning does lead to correct conclusions. Our general results will be couched in terms of self-adjoint operators acting on k Hulbert spaces Hm, m=l,...,k, and the Sturm-Liouville setting (1.1) will be deduced as a special case. Necessary and sufficient conditions for the existence of eigenvalues can also be found in [15] and 337 338 P. BINDING AND H. VOLKMER relevant references therein, but under special "definiteness" conditions. Our methods here are rather different, since, following Richardson, we presuppose no definiteness conditions. Our plan is as follows. In Section 2 we shall discuss a general necessary condition and compare it with "left" and "right" definiteness, In Section 3 we specialise this to the Sturm-Liouville case and we compare it with Richardson's fe=3 condition, which we show is not in fact necessary. In Section 4 we establish a sufficient condition by a degree theoretic argument. Finally Section 5 is devoted to comparisons between our sufficient conditions, our necessary conditions, left definiteness (which although related is not directly comparable with our conditions) and Richardson's sufficient conditions. We conclude with an example to show that his k = 3 condition is not in fact sufficient. 2. A necessary condition Let Tm, Vmn, n=l,...,k, be self-adjoint operators on separable Hilbert spaces Hm, Tm being bounded below with compact resolvents and Vmn being bounded, and let Wm(X) = Tm-Vm(X), where Vm(X) = £*=i K J , , m=l,...,k. Here and below we assume A = [A1...Ak]r6lR*. Let the eigenvalues of Wm(k) be listed as counted by multiplicity. With Um as the unit sphere of Hm, the multiparameter eigenvalue problem Wm(k)um = 0, umeUmnD(Tm), m = l,...,fc, (2.1) can be studied via the equations ptr(A) = O, m=l,...,k. (2.2) Indeed any solution of (2.2) yields nontrivial nullspaces for each Wm(k), and conversely any X satisfying (2.1) also satisfies (2.2) for some index i=(il,...,ik). For examples of this setting, we refer, for instance, to [15]. We use lower case letters to denote quadratic forms, e.g., tm(u) = (u, Tmu), and V(u) will be the kxk matrix with (m,ri) entry vmn(um) and with /nth row vm(um). Our study of the equations (2.2) depends mainly on the following representation of the functions p£ given by the minimax principle pL-W = min{max{t m (u)-v m (u)A: U el7 m n£}:£c=D(r m ),dim£ = im}. For any subspace G of Hm, we write (2.3) ON A PROBLEM OF R. G. D. RICHARDSON 339 It is convenient to make the assumption (A) For each m and X, the maximum of the spectrum of Vm{X) is not an eigenvalue of finite multiplicity. Then we define for any fixed finite i and by virtue of (A) we have om{l) = sap{vm{u)kueUm}. (2.4) We shall need three lemmas involving single equations, i.e. with m fixed. Lemma 2.1. Let i = im be fixed and let the sequence Xj e Uk satisfy p'm(XJ) ^ 0 for all j , with ||A'||-+oo and XJ/\\lJ\\^X*. Then am(X*)^0. Proof. For any i dimensional subspace F of D(Tm), (2.3) gives ; for some uJeUmnF. Assuming, without loss of generality, that uJ-+ueUmnF, J divide (2.5) by \\X \\ and proceed to the limit to give for some ueUmnF. (2.5) we may By definition of om(X*), this suffices for the result. Lemma 2.2. As for Lemma 2.1, but with the inequalities reversed. Proof. By assumption, pi,(A)gpjn(A)^O. Then (2.3) gives (2.5) (with the inequality reversed) for some sequence uJe Um n D(Tm). We again divide by ||>f|| and obtain Since tm{uj) is bounded below, we see that, for every given e > 0, vm(uJ)>l* > — e whenever j is sufficiently large. The result now follows from (2.4). • Lemma 2.3. Let i(j) be a sequence of positive integers converging to infinity and let p^{Xj) = 0for all j . Then \\XJ\\-*oo. Proof. For each i, p'm(XJ)^0 as ;->oo. Thus if k' has an accumulation point X* in Uk, 340 P. BINDING AND H. VOLKMER then, by continuity of p'm, it follows that pjn(A*)^O for all i. This contradicts the fact that p'm(X*)~*oo as i-*oo. • W e are now ready for the basic necessary condition. Theorem 2.4. Fix m and positive integers in for n = l,...,k, n + m. If, for an infinite set ofim, (2.2) admits eigenvalues X = X(m\ then there is ft^O such that M) if (2.6) n=\,...,k, Proof. By Lemma 2.3, the eigenvalues k(m) satisfy ||A<m)||-KX> as i m -*oo. Let ft be an accumulation point of tthe sequence A(m)/||A(m)||. Then an(fi) = 0 follows readily from Lemmas 2.1 and 2.2. Moreover pjn(llm))^pi£(ttm)) = O so Lemma 2.2 gives 0^<x m (//). • Corollary 2.5. / / there is a fixed index i so that (2.2) admits eigenvalues of any index jj>i then for any nonempty subset X of {l,...,k} there is / i r # 0 so that (2.6) holds for H = f^ ifme'L and n ^ Z . Here a n d below we use the componentwise order on Uk. 3. Differential equations With reference to (1.1), suppose l/p m and qm are L l with pm positive, and define bm-\, -(pmy')' + qm where BC represents separated boundary conditions. Then define = -(pmy')' + qmy, (Vmny)(xm) = rmn(xm)y(xm) so [11, §§ 18, 19] the hypotheses placed o n Tm and Vmn in Section 2 are satisfied provided the rmn are L ^ . It is well known (cf. [15, Theorem 3.5.1]) that a solution of (2.1) has index i if and only if the corresponding solution ym of (1.1) has im— 1 zeros in ]a m , bm[. This leads us to the following condition which is of Richardson's type, and which is a special case of Corollary 2.5. By virtue of (2.4), the functions am now become am(k) = ess sup | X Krmn(xmY am^xm^ bm\. (3.1) Corollary 3.1. / / (7.7) admits solutions with abritrarily many zeros for each ym, then there exist p m # 0 , m = 1,...,k, such that, for all m,n=l,...,k, <rn(Cm)=0 if m*n, (3.2) ON A PROBLEM OF R. G. D. RICHARDSON eJLDZO if m = n. 341 (3.3) For comparison, we state Richardson's necessary condition (forfe= 3 and with analytic Pm> 4m> r™) m t n e following form. Let R(x) be the 3 x 3 matrix with (m, n)th entry rmn(xm) in (1.1). (RNC) There exists vectors £m ( = — [A (m y m) v (m) ] T in his notation) such that, for all m,n = 1,2,3, )m£0 for all xn if (3.4) and (R(x)nm = 0 for some xm. (3.5) In our notation (2.4), (3.1) this means ff.(f")^0 if m#n (3.6) n (3.7) and for all m. It is evident that our conditions in Corollary 3.1 imply RNC except for the left-hand inequality in (3.7), and the following example shows that this infimal condition is not necessary. Example 3.2. Let pm = 1, qm = 0 for m = 1,2,3, and let l—Xi — *! 0 R(x) = - x 2 0 l-x2 0 0 1 We solve (1.1) over the interval [0,1] with Dirichlet end conditions; since the coefficients are analytic, this is a problem of Richardson's type. We claim that solutions of (1.1) exist for each index i 2: (1,1,1). This is a consequence of Theorem 4.1 below, but may also be seen as follows. For m = 3, we have the (right definite) problem —y'i = ^y^, which evidently has solutions of any index i 3 ^ 1. For m= 1,2 we have a left definite system [15, p. 43], since the cofactor matrix C(x) of l-Xj -Xjl -x 2 1-xJ satisfies C(x) OHO- 342 P. BINDING AND H. VOLKMER and Tm is positive definite for m= 1,2. Moreover the determinant of -x2 l-x2 is not identically zero. Thus solutions Ax, X2 exist of any index ( I 1 , I 2 ) ^ ( 1 , 1) —see, e.g., [15, Theorem 3.5.2]. It follows that (3.2), (3.3) must hold: indeed it is enough to choose the ftm as the unit coordinate vectors in (R3. On the other hand, (3.4) requires and this forces £3 = [0 0 <x]r. Further, (3.5) requires [0 0 1] £? to vanish for some x 3 , so <x = 0 and thus RNC fails if one excludes the triviality £3 = 0. We conclude this section by comparing the above conditions with the standard ones of right definiteness (RD) and left definiteness (LD). RD means that V(u) has determinant of one sign for all choices of the um. This implies that, for each subset £ (including 0) of {1,..., k}, there is vE # 0 such that an(^)^0^im(^,Um)(^m(v^) (3.8) for all meZ and w£Z. See (2.4), (3.1) for the notation and [2, Theorem 9.7.1] for a stronger result. Since [15, Theorem 3.5.2] RD implies that eigenvalues exist for all indices i, Corollary 2.5 shows that (2.6) must hold for some pL for all Z#0. A direct proof that RD implies (2.6) is not obvious in general, but for fe^3, simple geometric arguments based on perturbing vs will suffice. In similar fashion one can show that RD implies (3.7) and hence RNC. Analogous reasoning holds for LD, which consists of a definiteness condition on the Tm, and an "ellipticity" condition which requires existence of a unit ioeUk such that C(u)eo>0, where C(u) is the cofactor matrix of V(u). Rotating axes so that a> = [0...0 l ] r , we may use [3, Theorem 6.3] to show that ellipticity implies the existence for each £ # 0 of v1, with v£ = 0, such that (3.8) holds. Thus the above implications of RD also hold for LD. 4. A sufficient condition The purpose of this section is to prove the following result for the operators Tm and Vmn of Section 2. We recall assumption (A) and the subsequent definition of am. Theorem 4.1. Suppose that (i) the imth eigenvalue of Tm, i.e. p|^(0), os positive for (ii) Vmn^0ifm,n = l,..., m=l,...,k. ON A PROBLEM O F R. G. D. RICHARDSON 343 (iii) for some ft>0, (Tm(p) > 0 for all m = 1,..., k. Then (2.1) has a solution k>0 of any given index j ^ i . Remarks, (a) Theorem 4.1 applies to Example 3.2 with /i = [l 1 1] T . (b) By (iii), there are im-dimensional subspaces Em of D(Tm) such that, for some /i>0, k K(u)/i>0 forall u e £ : = X (EmnUm). (4.1) m= 1 We also note by (i) and the minimax principle (2.3) that there is eeE such that L0, m=\ k. (4.2) (c) If we drop our assumption (A) on the spectra of the Vm(l) then Theorem 4.1 remains true if we replace assumption (iii) by (4.1). We shall need two lemmas, the first involving the set D = {XeUk:X>0 and t(n)> V{u)(X-p) for some u e £ } where t = [tl...tk]T. Lemma 4.2. D is a nonempty open bounded subset of Uk. Proof. Openness of D is elementary. For any ueE, (ii) and (4.1) give vmn(um)^0 whenever m^n and V(u)p>0, for some //>0. By a well known result, cf. [15, Lemma 5.5.1], K(u)" 1 exists with all elements nonnegative. In particular He):=V{e)-H(e)eD. (4.3) Moreover, for any k e D there is u e E such that 0<k<fi+V(u)-it(u). These bounds are uniform because K(u)"1t(u) is continuous on the compact set £. • Our second lemma concerns the function g = [0,1] x R*-»R't given by um e Um nFm, Re (em, um) = 0}: F,<=D(TJ, dim Fm = im}. (4.4) Standard arguments using the semiboundedness of tm(um) show that g is continuous. Lemma 4 3 . For each <xe[0,1], the map g(a, •) does not vanish on the boundary 3D ofD. Proof. Elementary considerations show that if A e dD then for some m either 344 P. BINDING AND H. VOLKMER (a) Am = 0 or (b) max{tm(um)-vm(um)(k-p):umeEmnUm}=0. (4.5) (a) For any im-dimensional subspace Fm of D(Tm), choose umeFmnUm so that (u)^pt(0), Re(e m ,uJ = O and Re(T m e m ,uJ^O-see (i). In the notation of (4.4), then, ) by (4.2). Thus from (ii) andAm = 0 we have It follows that gm(oc, X) ^ p ^ (b) For any ume t/ m n Em such that Re(em,um) = 0, we have by (4.1) and (4.5) - v m « ) A = tm{u*m) - andsog m (a,A)<0. D We are now ready to prove Theorem 4.1. Since V(e) is invertible (as above) and vanishes only at A(e)eD by (4.3) we have by Lemma 4.2 that the degree deg(g(a,)• , D,0) is well defined and nonzero for <x = l. By Lemma 4.3, this remains true for a = 0. Now (2.3) shows that so there must be a solution to (2.2) in D. For j ^ i we note that (i)-(iii) hold with i replaced by j , so solutions also exist for such indices j. • 5. Discussion We shall now compare the conditions of Theorem 4.1 with various others, specialising for convenience to (1.1) with continuous rmn. Let us start with uniform left definiteness (ULD) where it is required that the operators Tm should be positive definite and C(x)a»0 for all xe X [am,hm] m=l for some to e Uk, where C(x) denotes the cofactor matrix of R(x). Continuity forces the components of C(x)ca to have positive lower bounds, hence the "uniformity". It can be shown [4, Lemma 2.1] that, after a nonsingular change of X coordinates, ULD implies (i)u ^ ( 0 ) > 0 for all m = l,...,k ON A PROBLEM O F R. G. D. RICHARDSON 345 for all x m e[a m ,fc m ] if m*n (»)« rmn(xm)<0 (i»)u rmm(xm)>0 for all x m e[a m ,fc m ] if m=l,...,k (iv)u c mn (x)>0 for all x m e [ a m , f c j , m,n= l,...,k. It should be noted that these four conditions are not quite sufficient for the existence of eigenvalues. If (v)u det R(x) / 0 for some x then solutions of any index are guaranteed [15, Theorem 3.5.2] but otherwise there may be no eigenvalues. As a trivial example, one could take: Example 5.1. k = 2,pm=l,qm = 0, -[-: -:} The two equations (1.1) then force X1 —12 t o t a ^ e opposite signs for any index. It follows that there are no eigenvalues under Dirichlet boundary conditions. We now show that if we strengthen (v)u to (v);, det R(x) > 0 for some x then (i)u, (ii)u, (iv)u and (\)'u together imply our sufficiency conditions (i), (ii), (iii) of Theorem 4.1. First, (i)u strengthens (i) by requiring i = l and (ii)u strengthens (ii) by requiring strict inequality (and hence a negative upper bound). Moreover, a well known result for real matrices [15, Lemma 5.5.1] shows that (iv)u, (v)^ imply (iii) which in the present context becomes (iii)s R(x)ji > 0 for some x and some fi > 0. Hence we can state that, apart from (v)^, which must be compatible with the initial change of k coordinates, our sufficient condition of Theorem 4.1 is weaker than ULD. On the other hand the following example satisfies the conditions of Theorem 4.1 but drastically fails ULD. Example 5.2. As for Example 5.1 but with sin^l on [am,6m] = [0,7t] and /< = [—1 1] T . It is impossible to satisfy any of (ii)u, (iii)u or (iv)u even after a change of coordinates. Let us also point out that (i)u, (ii)u, (iii)u and (v)u are sufficient for the existence of 346 P. BINDING AND H. VOLKMER eigenvalues of arbitrary index if k^3. This is trivial if k = l. Iffc= 2 then (ii)u and (iii)u imply (iv)u so we have a uniform left definite problem (1.1). Iffe= 3 then (i)u, (ii)u and (iii)u imply local definiteness of (1.1) in the sense of [15, Theorem 3.3.2] which together with (v)u gives the existence of eigenvalues of any index. If k>3, however, then (i)u, (ii)u, (iii)u and (v)u do not guarantee existence of eigenvalues. Weaker forms of (ii)u, (iii)u are r m n (xJg0 for all xm if rmm(xm)^0 for some xm, m=\,...,k, which constitute Richardson's "normal form" for (1.1) (cf. [13, p. 301]) if the necessary conditions of Corollary 3.1 are satisfied with linearly indpendent pm. In this case the fim can be taken as X coordinate vectors after a nonsingular transformation. We should point out that this normal form cannot be guaranteed, since it may be impossible to choose linearly independent fim: cf. Example 5.2. Let us turn now to Richardson's Theorem [13, p. 299] which is for analytic coefficients and Dirichlet end conditions. Richardson assumes that nonzero solutions of (1.1) with / = 0 and ym(am) = 0 have im— 1 internal zeros: this is easily seen to imply (i) of Theorem 4.1. He also assumes (3.4) and the following strengthening of (3.5): (R{x)$m)m takes both signs for xm e [am, fcm]. (5.1) He then claims the existence of a solution to (1.1) so that each ym has im— 1 internal zeros, i.e., a solution of index i. The following example shows that Richardson's conditions are not sufficient as stated. Example 5.3. Let k = 3, pm = — q2 = — <fo = 1, <h = 0, 0 R{x) = n + x2 -K-x-, Xj -1 0 —1 with Dirichlet conditions on [am, bm] = — - , - , 0 -1 m= 1,2,3. We choose i = l since each Tm is positive definite, and with [0 1 1] T , [1 0 7i] r and [ - 1 0 7i]T as §m we may check (3.4) and (5.1), so Richardson's conditions are satisfied. We shall prove by contradiction that no solution of (1.1) exists with index 1. First we claim that m= 1 in (1.1) forces A3£-l. This follows because the eigenvalue A3 = A3(A2) of -/;=0*2*1-- (5-2) ON A PROBLEM O F R. G. D. RICHARDSON 347 corresponding to it = 1, is a convex function having asymptotic slopes min max as A2-» + oo: see, e.g. [5, Section 2]. Also the transformation Xj-* — x, shows that A3 is even. Hence A 3 (i 2 )^A 3 (0)= - 1 for all A2. On the other hand, m = 2 and 3 in (1.1) give so integration by parts yields */2 */2 0 = A, J J -*/2 -K/2 Since the integrand does not vanish, we see that Ax =0. Hence = - ^ 3 . and this contradicts (5.2). We conclude by returning to Richardson's "normal form", which exists when the £m are linearly independent, as is the case in Example 5.3. In "normal form", Richardson's necessary conditions of the rmn become {u)N (i.e. (ii) of Theorem 4.1) and ("ORN rmm(xm) = 0 for some xm, while his sufficiency conditions are (ii)N and (i»)Rs r mm takes both signs on [ a m , b j . In "normal form", Example 5.3 then satisfies (i) and (ii) of Theorem 4.1, and also (iii)RS. Thus (iii)RS cannot be substituted for (iii) in Theorem 4.1. Acknowledgements. Binding thanks L. Turyn for much discussion and correspondence in the early 1980's on problems related to Richardson's. Both authors thank NSERC of Canada for supporting Volkmer's 1989 visit to the University of Calgary, during which this research was carried out. REFERENCES 1. F. M. ARSCOTT, Periodic Differential Equations (Pergamon Press, London, 1964). 348 P. BINDING AND H. VOLKMER 2. F. V. ATKINSON, Multiparameter spectral theory, Bull. Amer. Math. Soc. 74 (1968), 1-27. 3. P. A. BINDING, Multiparameter definiteness conditions II, Proc. Roy. Soc. Edinburgh 93A (1982), 47-61. Erratum: ibid 103A, 359. 4. P. A. BINDING and P. J. BROWNE, Classification of eigentuples for uniformly elliptic multiparameter problems, J. Math. Anal. Appl. 139 (1989), 268-281. 5. P. A. BINDING and P. J. BROWNE, Eigencurves for two-parameter self-adjoint ordinary differential equations of even order, J. Differential Equations 79 (1989), 289-303. 6. J. EISENFELD, On the number of interior zeros of a one-parameter family of solutions to a second order differential equation satisfying a boundary condition at one endpoint, J. Differential Equations 11 (1982), 202-206. 7. M. FAIERMAN, An oscillation theorem for a one-parameter ordinary differential equation of the second order, /. Differential Equations 11 (1972), 10-37. 8. M. FAIERMAN, A note on Klein's oscillation theorem for periodic boundary conditions, Canad. Math. Bull. 17 (1975), 749-755. 9. W. S. LOUD, Stability regions for Hill's equation, /. Differential Equations 19 (1975), 226-241. 10. J. MEIXNER and F. W. SCHAFKE, Mathieusche Funktionen und Sphdroidfunktionen (Springer, Berlin, 1954). 11. M. A. NAIMARK, Linear Differential Operators //(translated by E. R. Dawson, Ungar, New York, 1968). 12. R. G. D. RICHARDSON, Theorems of oscillation for two linear differential equations of the second order with two parameters, Trans. Amer. Math. Soc. 13 (1912), 22-34. 13. R. G. D. RICHARDSON, Uber die notwendigen und hinreichenden Bedingungen fur das Bestehen eines Kleinschen Oszillations Theorems. Math. Ann. 73 (1912/13), 289-304. Erratum: ibid. 74(1913), 312. 14. L. TURYN, Sturm-Liouville problems with several parameters, J. Differential Equations 38 (1980), 239-259. 15. H. VOLKMER, Multiparameter Eigenvalue Problems and Expansion Theorems (SpringerVerlag, 1988). DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF CALGARY CALGARY, ALBERTA CANADA, T2N 1N4 DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF WISCONSIN—MILWAUKEE P.O. Box 413 MILWAUKEE, WISCONSIN 53201 U.S.A.