University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 12-1958 Spectral Theory of Self-Adjoint Ordinary Differential Operators Charles C. Oehring University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Mathematics Commons Recommended Citation Oehring, Charles C., "Spectral Theory of Self-Adjoint Ordinary Differential Operators. " PhD diss., University of Tennessee, 1958. https://trace.tennessee.edu/utk_graddiss/2951 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Charles C. Oehring entitled "Spectral Theory of Self-Adjoint Ordinary Differential Operators." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. F. A. Ficken, Major Professor We have read this dissertation and recommend its acceptance: Leo Simons, Walter Snyder, O. M. Harrod, D. D. Lillian Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) November 24, 1958 To the Graduat e Council: I am submitting herewith a thesis written by Charles c. Oehring entitled "Spectral Theory of Self-Adjoint Ordinar,y Differential Operators ." I recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Ma thematics. Major Professor We have read this thesis and recommend its acceptance: L��� � (f). 11. I�.). 'Q.IJ_Lf?s- ._. Accepted for the Council:· 4UJ Dean of the!'4:rL� uate SFOOJ. - SPECTRAL 'mEORY OF SELF-ADJOINT ORDINARY DIFFERENTIAL OPERATORS A 'IHESIS Submitted to The Graduate Council of �Ihe University of Tennessee in Partial Fulfillment o! the Requirements for the degree of Doctor of Philosophy' Charles Co Oehring December 1958 ACKNOWLEDGMENT 'lbe author wishes to express his thanks to Professor F. A. Ficken for his assistance in the preparation of this paper. 457.724 TABLE OP CONTEN'IS CHAPTER PAGE IN 'ffiODUCTION . • • • • • • • • • iv I. PRELIMINARIES • • . • 0 • • • • • • 1 Linear operators; spectral representation of self-adj oint operators • • • • • • • • o • • • • • • • • • • • 1 Generalized direct sums of Hilbert spaces • • • • • • • • 3 Differential Operators • • • • • • • • • • • • • • • • • 6 .n. THE �XPANSION 'IHEOREM AND PARS EVAL EQUALITr •••••••• 10 Lenuna • • . • • • • 25 Theorem • • • • • • • • • • 32 Corollary to �eorem 2.64 •• . 40 rn. BOUNDARY CONDITIONS • • • • • • • . • • • 42 EXTENSIOOS SYMME'IRIC IV. SPEC'IRA OF SELF-ADJOINT OF A OP ERA'IUR 49 Von Neumann's theory of symmetric extensions • • • • • • 49 �e �pee� of __a �el�-adjoint operator ••• • • • • • • 51 _ Spectra of self-adjoint extensions • • • • • • • • • • • 51 BIBLIOCB..APHY • • .. • • • • • • • • • • • • • • • • • . 56 IN 'lRODUCTION Many of the properties of the ordinary Fourier seri es expansion of a given function are shared by the orthogonal expansions in terms of eigenfunctions of a second order ordinary differential operator. Let p • p(x) and q • q(x) be real-valued functions such that p , p' , and q are continuous, and p(x) 0 on a .f'inite interval ? , a � x � b • Le� � be a C<?JUplex p�ameter. The classical Sturm­ 1 Liouville theory [9, section 27; 4, Chapter 7; 211 Chapter 1] is con- cerned with solutions or the differential equation � y , -(py•)' + Cf7 • which satisfy certain real boundary conditions whose form need not be given here. These solutions, the so-called eigenfunctions, exist only tor certain values of , the corresponding eigenvalues. The St .A urm- Liouville theory states that the eigenvalues constitute a countable set of rea1 numbers which cluster only at • The corresponding + oo eigenfUnctions constitute an orthogonal system on [a,b] whieh is com- . 2': plete in L (a,b) • . 'lhus the Parseval relation is also valid. now the possibilities a � and b • are allowed• It • - + oc or if the restriction p(x) 0 is required merely for a b , > <: x <: the complexity of the situation increases. In these cases the problem is called singular, in contrast to the regular case considered above. lwumbere in brackets refer to the bibliography at the end of this paper. To obtain satisfactory analogues of the completeness and Parseval theorems, i� now becomes necessar.y to replace the series expansion of an arbitrar.y 2 function in L (a,b) by an expansion in terms of a Stieltjes integral. 'lhe singtil_ar·case has been treated exhaustively by Weyl [23J24], by Stone [20] who uses the general theory of unb ounded symmetric operators, by . Tltchmarsh [21] who uses function theoretic methods, by Kodaira [10] who combines and simplifies the ideas of Weyl and Stone, by Yosida_ [2$], and [13Jlh] as limi.ting b7- Levineon. who obtains results cases of theorems valid for compact subintervals of (a,b) • '!he basic facts in the regular ease extend verbatim to a formally se�f��djoint �iff�rent�al operator L (see Chapter I) of arbitrar.y order n , with complex coefficients defined on a compact interval, provided the th coefficient of the n order derivative does not vanish on that interval [�J.. Chapter 7]. Proofs in this case require no�ing more than the �ilbert­ ' Schmidt theory of integral equations . When the problem is singular, general results have been obtained nly rece tly Glazman [.5] has general­ _ o n . ized several important_r�sults of Weyl and Stone conce�ning the nature or the boundary conditions when the coefficients are real. Kodaira [11] has also discussed boundar,y conditions in the case of real coefficients and an of and proved the alogues the completeness. Parseval relations. Coddington [2] and Lev.i.nson [15] have considered these questions when the co�f.fieients are complex and haTe obtained the expansion and Parseval theorems in two important cases. In these cases they also prove the in- verse transform theorem which is the analogue of the Riesz..;Fiseher theo­ rem in the !heory of ordinary Fourier series. One of the mai.n results or the present paper is a proof, in Chap- vi ter II, of the expansion, Parseval, and inverse transform theorems, by use of the �eor.y of generalized direct sums of Hilbert spaces, which theory - .�· � is due to von Ne These theorems are most elegantly sta e in a umann [22]. t d - . .. ro:m g1v n them by Kodaira Let be Hermiti n non­ � [10]. (fjk) an a , decreasing n by n matrix whose elements are Lebesgue-Btieltjes measures -· - - on the real A axis, (see Chapter • Let and be n-vactor ./'). II) "X w . --- valued i th co fUnctions or A vi th mponent �i (A) and t.Ui (A) res­ pect1Tel7. If we introduce the inner product n - (w,'X.) • { I... Wj(A.)I (}I.)d/' (�), j,k•1 \ jk it is easy to see that the set -"1- of those uJ for which (w,w) <:. oo becomes Hilbert space in this inner product. The expansion, Parsev l a a 9 and inverse transform theorems may then be stated as the following theorem, which will be proved in ·- Chapter. IIo If L is a formally self-adjoint differential operator of order n , and if H is self- j int extension of L in the Hilbert space a ad o - . L 2 (a, b) , then corresponding to each system of linearly independent sol u- ti?�• ( x, i\) j • 11 o o. or Ly·• Ay on (a, b) , ·there exists sj 9 n, a spectral matri x (/'jk) , with the properties described aboveg· su�h . that the as s ci ted Hilbert space is unitarily equivalent L 2 (a,b) o o a -rL to Thus if u L2'(a,b) and is its image in 1 the Parseval £ w JL equality vii 2 is valid. 1he fact that to each w in JL.. corresponds a u E, t (a,b) is the content of the inverse transform theorem. Finally the expansion theorem, the analogue of the completeness theorem in the regular case, states that u and W are related by the specific formulas and n u(x) • f.._ Bj(·x,A) uJk{l,)d i'A) 1 j,Lk-1 ft where the fi rst integral converges in the norm of _n_ and the second in the norm of L2(a,b) • Chapter I contains definitions and facts pertinent to later chap- ters. In Chapter III a generali zation, with a different proof, is given of a theorem of Coddington [-2, 'lheorem 3] which characterizes the self- adjoint extensions of L , when such extensions exist, in terms of cer- tain boundary conditions. !t is shown in Chapter that a theorem of Hartman an ·- . IV d Wintner [7] .concerning the eigenvalues of L in one of the eases considered by Weyl is aetua1ly valid in a much more general setting . viii the o the the a Since printing f text of this paper uthor has noted the appearance in the Canadian Journal of Mathematics, Vol. I, No. 3 (1958), of by F. Brauer which the gen n ppo -���-�6, a paper in ei functio expansion problem is treated using the method of generalized direct sums of Hilbert spaces. Brauer's approach to the problem is very similar to the one used e present paperJ in th -- and in .faet he proves the analogues of the main re- sul of Theorem 2.64 tor the eigeDTalue problem Lu • where . ts -- - 7\ Mu 1 L and M o i s�lf-adjoint differential operators and is semi- - are.