DOUGLAS BARKER SEARS: HISr WORK IN DIFFERENTIAL EQUATIONS

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1 DOUGLAS BARKER SEARS: HISr WORK IN DIFFERENTIAL EQUATIONS P.J. BROWNE, W.N. EVERITT AND 1.W. KNOWLES Dedicated to the memory of Douglas Sears 1. Douglas Sears: a life in brief. Douglas Sears was an accomplished mathematician with wide interests, both mathematical and otherwise. He was also an exceptional mentor to his many students and to not a small number of his younger colleagues. In his early years as a professional mathematician in South Africa he devoted his talents to a period of research work in the theory of generalised hypergeometric functions. In 1950 he left South Africa for a long visit to the University of Oxford where he trained under the great analyst E.C. Titchmarsh; it was under his influence that Sears took up research interests in differential equations; in particular he was influenced by the first edition of the Titchmarsh book on eigenfunction expansions associated with ordinary, linear differential equations and this subject thereafter dominated his research interests. Following his Oxford days he returned to South Africa to take up the Chair and Headship of the Department of Mathematics at the University of Cape Town, a position he held until the early sixties. During this period he travelled widely in the United States and also returned to Oxford. Thereafter he moved to the University in Nairobi, Kenya, and three years later to Australia for seven years. But he returned permanently to South Africa in 1971 as professor and Head of the Department of Mathematics at the University of the Witwatersrand, Johannesburg, a position he held for twelve years. Finally his teaching interests encouraged him to assist one of the then "black" Universities with its programme to educate teachers of mathematics for five years before his eventual retirement at the end of 19g7 at the age of 69. He married Teda De Moor in 1941 and, three years after her death in 1986, was married to Brunhilde Helm. He spent his last years with her in the beautiful surroundings of the Western Cape. His son, Michael, is a professional mathematician and until recently occupied the chairmanship of the Department of Mathematics at the University of the Witwatersrand, the same position as held by his father in earlier years.

2 2 P.J. BROWNE, W.N. EVERITT AND 1.W. KNOWLES 2. Work on differential equations. In this note we shall restrict attention to his published work in and around differential equations, leaving his other works (notably in the theory of theory of special functions) for comment by others. His main area of interest centred around what is today known as the spectral theory of linear operators generated by linear, second-order differential expressions, both in the ordinary and partial cases. Specifically, he was greatly interested in studying the theory of selfadjoint operators, as well as related questions involving eigenfunction expansions and other spectral properties including classical oscillation theory for Sturm-Liouville operators. Over the last century or so the formal Sturm-Liouville expression r defined by where I is an interval of the real line R, has been the object of much study, with the interest driven by applications from calculus of variations to quantum mechanics; here q is a real-valued coefficient defined on I satisfying the minimal condition q E L;,(I). In particular, a variety of differential operators can be generated from r Sy the imposition of boundary conditions at the end-points of I, leading to an appropriate, densely defined operator domain in the Hilbert function space L~ (I). Of these, the self-adjoint operators have the greatest physical interest, partly because of the central r61e that such operators play in the Hilbert space theory of quantum mechanics. Now such an operator becomes self-adjoint precisely when the correct number of boundary conditions are assigned at the end-points of I and in the case when I = [0, oo), for example in the early days of quantum mechanics, there was much confusion among the physicists (see [6, p. 1961) as to how to assign the "boundary condition at oo". This point was settled in a famous paper [25] of Hermann Weyl in which he showed that, depending on q, there are exactly two possibilities, which he named the "limit-point case" and "the limit-circle cbe", respectively. In the former case, which is where most of the quantum mechanical interest lies, it turns out that no boundary condition at oo is needed, and this greatly simplifies the subsequent theory in such applications. The task was thus reduced to proving that a given potential q was in the "limit-point case". The same paper of Weyl also characterized the limit-circle and limit-point cases according to whether all solutions of the differential equation ry = 0 were or were not in L2(I); one of Sears' first papers tackled this question. At that time only a few results of a specialized nature were known, mainly to the end that the potential q had to be bounded below by certain specific functions. In [17] he showed it was enough that where Q satisfies Q(x) 3 0 ( x E [0, oo) ) and does not grow too quickly at infinity. This result was a precursor of a whole industry of such results that appeared when the subject became an active area of research in the 1970s (see [q); one form of a necessary and sufficient condition (not unlike the result (2.2)) was eventually provided by Atkinson (another student of Titchmarsh!) in [I]. In the same paper [17], Sears also provided a similar condition for the analogous Schrodinger potential

3 DB SEARS: HIS WORK IN DIFFERENTIAL EQUATIONS 3 in R2. Here, the limit-pointllimit-circle dichotomy is no longer valid, and one seeks to characterize those potentials q for which the minimal Schrodinger operator is essentially self-adjoint; again the physical interest is that one does not have to worry about assigning a "boundary condition at infinity". While no complete essential self-adjointness theorem is known, most, if not all, potentials of physical interest have now been classified, many using criteria similar to this early work of Sears. Related work of Sears on L2-solutions of Sturm-Liouville equations appears in [8, 13,14, 181. He also proved in [15] a result on the discreteness of the spectrum when I = (0, b], when the end-point b is singular, generalizing a result of F'riedrichs I51 The paper [21] is worthy of special mention; this work was written jointly with E.C. Titchmarsh and followed on from the detection by Sears of an error in [22, Chapter IV], but corrected in the second edition [24]. Titchmarsh had failed to note the existence of an infinite2sequence of negative eigenvalues with limit point at -00. The analytical reason for the existence of Chese negative eigenvalues may well have escaped both Titchmarsh and Sears; it is due to the existence of an end-point in the limit-circle oscilla&ry case for the differential equation concerned. In any case the paper [21] produced all the correct formulae. This example came to light again when the SLEIGN2 computer program, see [2], was in preparation and there appeared a need for examples to illustrate the capacity of the program to compute eigenvalues in the limit-circle case. Thi~~equation was adopted for this purpose and named the Sears-Titchmarsh equation; details are given in [2, Section 6, Example Eigenfunction expansions, A substantial part of his early training with Titchmarsh brought him into contact with the theory of eigenfunction expansions for Sturm-Liouville operators. Indeed, he was intimately involved with the corrections for [22] wd the proofing of [23]; the works [16, 211 reflect this early interest. He later expanded on this theme and became interested in the (unitary) integral transforms that one can associate with a Sturm-Liouville equation analogous to the way in whi&the Fourier transform is associated with the case q = 0. Using a general approach to integral transforms due to Bochner (see [3]) he found a new approach to the derivation of the basic theorems (i.e. the Parseval relation and the expansion theorems) in [ll, 121. He later showed in [9, 101 that these ideas extend to unitary maps between appropriately defined measure spaces. In [4] it was shown that these ideas have perhaps their most natural home in the abstract spectral representation of a self-adjoint operator on a Hilbert space H by means of a unitary mapping from H to a space L2(p) for some matrix measure p. The paper [16] is an outstanding contribution to the inter-connection between the classical analytical approach of Titchmarsh to eigenfunction expansions and the underlying structure of abstract linear operator theory in Hilbert spaces. It is clear that this paper made a great impression on Titchmarsh when he came to write up his results on eigenfunction expansion theory for partial differential equations, [23], where in appropriate places there is mention of the links between classical and operator theoretic methods.

4 4 P.J. BROWNE, W.N. EVERITT AND 1.W. KNOWLES 4. Administration. Aside from his mathematical work, Douglas was a skilled and experienced administrator who thought a great deal about how people from his generation should best transmit their knowledge to the next generation. His inaugural lecture at Wits in 1974 [19] was interesting reading after a gap of some 25 years for one of us (IWK). His topic was the "New" Mathematics, but the difficulties that he discussed concerning the preparation of incoming university students seem to be timeless (and international). 5. Memories From PJB and IWK. Besides ourselves, Douglas had two other Ph. D. students (that we know of) : Stephen D. Wray, who graduated in 1974 from Flinders University, and John G. O'Hara, who graduated from the University of the Witwatersrand in His time in Australia was spent first at the University of Adelaide and then (circa 1967) at the then newly created Matthew Flinders University of South Australia, located in the rolling foothills just outside of Adelaide. The three of us that studied with him at Flinders recall a somewhat carefree time; this was after all the late sixties. Nonethsless, under his patient tutelage we learned not only mathematics, but how to be mathematicians. As an additional bonus not in the usual contract, over numerous lunches and dinners we were introduced to the delights that the wider world of academia offers: he was equally at home in discussions on politics ("vote them out"), music, and literature; in a very real sense he educated us in the widest sense of that word. He was a man of great charm, wit and ability. We miss him greatly From WNE. I first met Douglas Sears in the early 1950s; he was visiting Oxford again and we sat together at the Titchmarsh Seminar, held at 5 pm on Fridays during the Oxford terms (this was the inheritance from the time of G.H. Hardy in Oxford). At the time both of us were engaged in reading the formidable manuscript of Titchmarsh that led to the book [23]; I was impressed with the command that Douglas had of both classical and functional analysis, and if it had not been for these discussions with him I should not have been able to later read effectively the proof sheets of the book. In those yeas his mind was permanently restless with mathematics and I marvelled at his energy for, and devotion to the subject. I cannot now be certain how many times thereafter I met him; however in the 1970s and 1980s I was in South Africa on a number of visits and never failed either to see him or to speak with him over the telephone. Either to be with him or to speak with him was a form of oasis in the never-ending bustle of life. My final contact with him in respect of mathematics was my appointment as external examiner for the thesis of his last Ph.D. student John G. O'Hara. I invited Douglas on at least a half-dozen of occasions to be a principal speaker at the differential equation conferences held at the University of Dundee, Scotland

5 DB SEARS: HIS WORK IN DIFFERENTIAL EQUATIONS 5 in the 1970s; alas he never accepted. I am grateful to life that it gave me the opportunity to meet, talk and be influenced by this remarkable man and mathematician. 1. ATKINSON, F.V., A class of limit-point criteria, In: Spectral Theory of Diflerential Operators, pp , Mathematics Studies Vol. 55, (Ian W. Knowles and Roger T. Lewis, eds.) Amsterdam: North,Holland, BAILEY, P.B., EVERITT, W.N. AND ZETTL, ANTON., Computing eigenvalues of Sturm-Liouville problems, Results in Mathematics 20 (1991), BOCHNER, S. AND CHANDRASEKHARAN, K., Fourier If.ansforms, Princeton, BROWNE, P.J., A Class of Linear Operators, Ph. D. Thesis, Flinders University, South Australia, FRIEDRICHS, K.O., Studies and Essays Presented to R. Courant (1948), KATO, T., Fundamental properties of Hamiltonian operators of Schrodinger type, IIFons. Amer. Math. Soc. 70 (1951), KAUFFMAN, ROBERT M., READ, THOMAS T. AND ZETTL, ANTON., The Deficiency Index Problem for Powers of Ordinary Differential Expressions, Lecture Notes in Mathematics, Volume 621, Springer-Verlag, Berlin, SEARS, D.B., Integrable square solutions of a partial differential equation, But. Inst. Polatehn. Iagi (N.S.) 4(8) (1958) no. 3-4, , Integral transforms over certain function spaces, 11, Quart. J. Math. Oxford Ser. (2) 8 (1957) , Integral transforms over certain function spaces, I, Quart. J. Math. Oxford Ser. (2) 8 (1957), , Integral transforms and eigenfunction theory, 11, Qud. J. Math. Oxford Ser. (2) 6 (1955), , Integral transforms and eigenfunction theory, Quart. J. Math. Oxford Ser. (2) 5 (1954), ' 13., Some properties of a differential equation, 11, J. London Math. SOC. 29 (1954), , Some properties of a differential equation, J. London Math. Soc. 27 (1952), , On the spectrum of a certain differential equation, J. London Math. Soc. 26 (1951), , An expansion in eigenfunctions, Proc. London Math. Soc. (2) 63 (1951), , Note on the uniqueness of the Green's functions associated with certain differential equations, Canadian J. Math. 2 (1950), , On the solutions of a linear second order differential equation which are of integrable square, J. London Math. Soc. 24 (1949), , The New Mathematics, Witwatersrand University Press, Johannesburg, 1974.

6 P.J. BROWNE, W.N. EVERITT AND 1.W. KNOWLES 20., Sturm-Liouville Theory, (unpublished notes), SEARS, D.B. AND TITCHMARSH, E.C., Some eigenfunction formulae, Quart. J. Math. Oxford Ser. (2) 1 (1950), TITCHMARSH, E. C., Eigenfunctions Expansions associated with Second- Order Diflerential Equations, Oxford, , Eigenfunctions Expansions associated with Second-Order Diflerential Equations, Volume 2, Oxford, , Eigenfunctions Expansions associated with Second-Order Differential Equations, Oxford (second edition), WEYL, H., ~ber gewohnliche DifFerentialgleichungen mit Singularitaten und die zugehorigen Entwicklungen willkiirlicher Funktionen, Math. Ann. 68 (1910), DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF SASKATCHEWAN, SASKATOON, SASKATCHEWAN, S7N 5E6 CANADA. DEPARTMENT OF MATHEMATICS AND STATISTICS, THE UNIVERSITY OF BIRM- INGHAM, BIRMINGHAM, UK. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ALABAM AT BIRMINGHAM, BIRMINGHAM, AL USA. iwkavorteb. math. uab. edu Received 16 May 1997