Parametrizations of Titchmarsh s 0 m() 0 -Functions in the Limit Circle Case

Transcription

1 Parametrizations of Titchmarsh s 0 m() 0 -Functions in the Limit Circle Case Von der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinisch-Westfälischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation Vorgelegt von Mathematiker (Master of Science, University of Minnesota, USA) Charles Thomas Fulton aus Pasadena, Kalifornien Referent: Professor Dr. J. Walter Korreferent: Professor Dr. G. Hellwig Tag der mündlichen Prüfung: 6. Juli 973

2 Contents Introduction 3 2 Preliminaries 4 3 The Limit-circle 0 m () 0 -Functions and Associated Weyl Spaces 9 4 Boundary Functionals 20 5 The Characteristic Functions of K. Kodaira and the Rellich Initial Numers 24 6 The Weyl Spaces of Hermann Weyl 28 7 Titchmarsh s m ()- Functions in the Limit Circle Case 34 8 Examples 42 9 The Weyl Spaces of M.H. Stone 48 0 Biliography 52 2

3 Introduction We consider the di erential expression, = d2 + q (x) for x 2 [a; c), with a < c 5, () dx2 and assume that q(x) is real-valued and continuous in [a; c) and elongs to the limit circle case at c. Let L 2 [a; c) e the Hilert Space of complex-valued square integrale functions on [a; c) and let L 0 and L e the operators taking f into f de ned y, D (L 0 ) := ff 2 L 2 [a; c)j f and f 0 are asolutely continuous and f = 0 in the Neighorhood of a and of cg (2) D (L ) := ff 2 L 2 [a; c)j f and f 0 are asolutely continuous on compact susets of [a; c) and f 2 L 2 [a; c)g. (3) The self-adjoint extensions of L 0 which arise from separated oundary conditions at a and c have een characterized in the literature y a variety of di erent rands of oundary conditions at the (singular) point c. In contrast to the oundary conditions originally given y H. Weyl ([37, p.232, Equa. 4]), those given y M.H. Stone in his treatise of Linear Transformations in Hilert Space ([29, p.475,theorem 0.7]), which depend on solutions of the eigenvalue equation for = i, and those used y K. Kodaira in [6, p. 924, Equa..7 III ], which depend on an aritrary element of D(L ), E.C. Titchmarsh in his remarkale ook, Eigenfunction Expansions Associated with Second Order Di erential Equations, Part I, gave oundary conditions at the singular point which depend on a function of the eigenvalue parameter, m() (cf. [3, p.3,equa ]). In Titchmarsh s treatment each oundary condition at c is determined y a function of, m(), which is meromorphic and whose poles (which are real and simple) coincide with the eigenvalues of the corresponding self-adjoint extension of L 0. In this dissertation I show that Titchmarsh s limit-circle 0 m() 0 -functions (corresponding to a xed oundary condition at a) can e represented in the form, a () cos + () sin m () =, 2 [0; ), (4) c () cos + d () sin where a(), (), c(), and d() are entire functions of. Each value of corresponds to a di erent oundary condition at c and the spectrum of the operator associated with this oundary condition is then determined as the poles of the associated m() function (or the zeros of c() cos + d() sin ). Also, the connection etween the Titchmarsh oundary conditions associated with the m()-functions of (4) and the oundary conditions arising out of Hilert-Space 3

4 approach taken y M.H. Stone will e estalished. This permits the conclusion that the -parametrization in (4) yields all possile limit-circle m()-functions. The only paper which was found to contain a representation of the type (4) was the 950 paper of D.B. Sears and E.C. Titchmarsh, Some Eigenfunction Formulae [27], which was written to correct some erroneous statements made in the rst edition of Titchmarsh s ook (notaly Theorem 5.8, p.08, which asserts the existence of a continuous spectrum under limit circle conditions at ). There, for the case c =, and under additional assumptions on q(x) (to guarantee the occurrence of the limit circle case and to permit application of the Liouville Transformation), a representation of the type (4) is otained (p. 68, 3 lines efore Section 3). As will e seen in this dissertation these additional assumptions on q(x) are unnecessary for the representation (4). 2 Preliminaries Letting 2 (a; c) we consider rst the Hilert Space L 2 [a; ] of complex-valued square integrale functions on [a; ] and de ne operators L 0; and L ; taking f into f y, D (L 0; ) := ff 2 L 2 [a; ]j f and f 0 are asolutely continuous and f = 0 in a Neighorhood of a and of g 2 () D (L ; ) := ff 2 L 2 [a; )j f and f 0 asolutely continuous in [a; ] and f 2 L 2 [a; ]g. (2) For f, g 2 d (L ) we de ne, The formula, Z x a W x (f; g) := f (x) g 0 (x) f 0 (x) g (x) for x 2 [a; c). 2 (3) (f) g f (g) dt = W x (f; g) W a (f; g) for x 2 [a; c) (4) holds for all f; g in D(L ) and will e called Green s Formula. A self-adjoint extension of L 0; arising from separated oundary conditions at a and is necessarily one of the operators L ;, for some 2 [0; ) and 2 [0; ), de ned y, D L ; := U ; := ff 2 D(L ; )j Ra (f) = 0, R (f) = 0g, (5) This false theorem was recently repeated in the ook of E. Hille, Lectures in Ordinary Di erential Equations, cf. [3, p.53, Theorem 0.3.9]. 2 Primes denote derivatives with respect to x. 4

5 where, R a (f) := cos f (a) + sin f 0 (a) (6) R (f) := cos f () + sin f 0 () (7) Because of the analogy to the singular limit circle case it will e desirale to summarize rie y Titchmarsh s approach to the expansion theorem associated with the self-adjoint operator L ;. In order to prove the expansion theorem for the operator L ; (; 2 [0; ) xed) Titchmarsh introduces a function `; () associated with this operator as follows: For each 2 C (C = the complex numers) let a; (x) ; a; (x) e the fundamental system of the eigenvalue equation, u = u, (8) de ned y the initial conditions, a; (a) a; (a) sin 0 a; (a) 0 = a; (a) cos Then we have: cos for all 2 C. 3 (9) sin W x a; ; a; = for all 2 C and all x 2 [a; c). (0) a; (x) = a; (x) and a; (x) = a; (x) 4 () For xed x 2 (a; c), a; (x), 0 a; (x), a; (x), and 0 a; (x) are entire functions of. (2) R a (f) = W a a;; f for all f 2 D (L ; ) and all 2 C. (3) R a a; = 0 for all 2 C. (4) W a a;; 0 a; = sin ( 0 ) for all ; 0 2 C. 0 W a a;; a; 0 = sin ( 0 ) for all ; 0 2 C. (5) 0 For Im 6= 0, a solution of (8) satisfying the oundary conditions at is, ; (x) := a; (x) + `; () a; (x) (6) 3 We will indicate the dependence of a; and a; on y writing a; and a; when this is desirale for clarity. 4 Bars denote complex conjugates. 5

6 where `; () is de ned (for Im 6= 0) y, `; () := a; () cos + 0 a; () sin a; () cos + 0 a; () sin. (7) Since ; (x) and ; 0 (x) satisfy the oundary condition (7) at it follows that W ; ; ; 0 = 0 for all ; 0 ; Im 6= 0, Im 0 6= 0. 5 (8) The function `; () is Titchmarsh s 0 m () 0 -function for the operator L ; the expansion theorem follows as in [3, Chapter 2], with m () = `; Thus if n, n = ; 2; 3;, are the poles of `; (), and if and (). r ; ;n := Re s = n `; (), (9) then the eigenfunctions of the operator L ; are (cf. [3, Equa and Theorem 2.7 (i) and (ii)]): q ; n (x) := r ; ;n a; n (x). (20) Moreover, it follows from Titchmarsh s proof of the expansion theorem that these eigenfunctions form a complete orthonormal set for L 2 [a; ]. Since it will e needed for later reference we now give a rief summary of Titchmarsh s discussion of Weyl s limit-point/limit-circle Alternative (cf. [3, p ]): For each, Im 6= 0, A ; (z) := a; () z + 0 a; () a; () z + 0 a; () (2) is linear fractional transformation (invertile ecause of (0) ) which maps the real z-axis onto the circle C; whose center and radius are respectively, where = Im. If w 2 C and z = % ; := W W a; ; a; a; ; a; r; := W = a; ; a; 2 jj R a j a; (x)j 2 dx (22) A ; (w), the following algeraic equation 6 holds for all 5 In the regular case equation (8) corresponds to Titchmarsh [3, p. 26, Lemma 2.3]. 6 For = i compare with Hellwig [, p.230, Equa. (5) and (4)], and with [36, p.228, Equa 29]. 6

7 , Im 6= 0 : a; () + w a; () 2 Im z = W a; + w ; ; a; + w a; 2i Z = a; (x) n w 2 dx % ; 2 = a Z a r; o 2 a; (x) + w a; (x) 2 dx + Im w. (23) From this equation the following result can e oserved: Theorem. (Weyl Alternative) (i) For each, Im 6= 0, we have either () lim!c r; = 0 or (2) lim!c r; =: rc; > 0. (ii) 7 If case (2) occurs for a single, Im 6= 0, then it occurs for all, Im 6= 0. From (22) and (23) it follows that in case () (the limit point case) there is only one solution of (8) for Im 6= 0 which is in L 2 [a; c), and in case (2) (the limit circle case) two linearly independent solutions of (8) in L 2 [a; c). In the limit circle case the circle C converges (for each, Im 6= 0) to a limit circle C ; c;, whose center and radius are, respectively % c; := W c a; ; a; W c a; ; a; where Remark. Since rc; := = W c a; ; a; 2 jj R a j a; (x)j 2 dx, (24) W c (f; g) := lim x!c W x (f; g) for f; g in D (L ). it is clear that for each, Im 6= 0, `; () = A ; (cot ) for all 2 [0; ), (25) f ;; () :! `; 7 Titchmarsh [3, p.43-44]. In the limit circle case there are two linearly independent solutions of (.8) in L 2 [a; c) for real as well as complex although this does not follow from Titchmarsh s argument. Compare Hellwig [, p.223, Theorem ]. () 7

8 is a one-to-one mapping from [0; ) onto the circle C ;. In the present paper we apply a theorem of Hartman ([9, p. 273, Theorem. and.2]) to de ne for each 2 C a fundamental system c; (x) ; c; (x) of (8) y "initial conditions" at the singular point c which have the properties (0)-(5) (and notaly the property (3)) with c in the place of a, and with a Rellich-type oundary condition at c in place of (7). Making use of the two fundamental systems a; (x) ; a; (x) and c; (x) ; c; (x) we the de ne (De nition 2 elow) an 0 m () 0 -function associated with each oundary condition at c. Titchmarsh s proof of the expansion theorem ([3, p. 3, Theorem 2.7 (i) and (ii)]) for the corresponding operator will apply with our de nition of the m () function. We then estalish the equivalence of our de nition with Titchmarsh s de nition afterwards in Section 6. The advantage of our approach to the de nition is that we otain the representation (0.4) almost automatically. In the following sections a Weyl-Space will refer to any space U of the form, U = ff 2 D (L ) j R a (f) = 0, R c (f) = 0g, where R c (f) = 0 is a oundary condition at c such that U is the domain of a self-adjoint extension of L 0. It should perhaps e mentioned that Titchmarsh s ook has een regarded with some suspicion as far as the limit circle case is concerned. Titchmarsh s simultaneous treatment of the limit circle case and the discrete limit point case in Chapter 2 ([30] and [3]) is, in particular, a little confusing since the 0 m () 0 functions in the two di erent cases are, of course, di erent ojects. This confusion is depicted for example in the following words of D.B. Sears (cf. [26, p.49]): "In the limit point case, m () = lim! ` () 8 exists as an analytic function, regular in either half plane. Titchmarsh uses this function also in the limit circle case, and thus requires no distinction in his theorems etween the two possiilities. This seems to require some further discussion. Levinson appears to aandon the limit circle case in his second paper. While the function k (t) is not unique in this case, the whole theory may in fact e put in a form in which the type of the equation is immaterial, except for a few details." In the aove cited paper Sears modi es the approach of N. Levinson in [7] and shows that the limit circle case can e treated y making use of Levinson s 8 This is the function `; () de ned in (.7). 8

9 spectral functions. 9 He must however, take his limit-circle 0 m () 0 function to e m () = lim n! ` n () where f n g is a suitale sequence, in order to otain a well-de ned spectral function k (t), cf. [7, p. 50, Equa. 4.4]. He then remarks later (p.57), "Naturally the spectrum in the limit circle case depends on the sequence through which!." As we shall see this paper the limit-circle m ()-functions (and corresponding spectra) do indeed depend on the choice of the sequence f n g and we shall give two examples showing that di erent choices of the sequence do produce di erent limit functions. Moreover, we show that the class of all limit functions for all possile choices of the sequence f n g can e indexed y a real parameter and represented in the form (4). I am indeted to Professor Dr. J. Walter for insisting that di erent choices of the sequence f n g would yield di erent m ()-functions. I am also grateful for his suggestion that I investigate example of Section 7 elow to see that this is the case. I thank also Professor Dr. G. Hellwig and his Assistants for many lively discussions in his Colloquium on the Spectral Theory of Singular Elliptic Di erential Operators. 3 The Limit-circle 0 m () 0 -Functions and Associated Weyl Spaces Let fu (x) ; v (x)g e real-valued fundamental system of (8) for = 0 satisfying and put Y (x) := u (x) u 0 (x) W (u; v) = () v (x) v 0 (x) for x 2 [a; c). (2) For each f 2 D (L ) we de ne a vector-valued function Sf for x 2 [a; c) y, (Sf) (x) := (Y (x)) f (x) Wx (f; v) f 0 =. (3) (x) W x (f; u) Because we assume q (x) elongs to the limit circle case at a, the solutions u (x) and v (x) are square integrale over [a; c) and also elong to D (L ). It follows 9 A treatment of the limit circle case employing the notion of the spectral function was later included in the ook of Coddington and Levinson, cf [4, p ]. This approach was also included in the second edition of Titchmarsh s ook, cf. [3, Chapter VI]. Levinson s spectral function is Titchmarsh s function 0 k () 0, cf. [3, p.38, Equa ]. 9

10 from (2.4) that oth components of Sf can e continuously extended to e de ned at the singular point c. 0 (Sf) (c) := lim x!c (Sf) (x) = Wc (f; v) W c (f; u) for all f 2 D (L ). (4) For two elements f; g 2 D (L ) we de ne the Wronskian of Sf and Sg y, fw x (Sf; Sg) := (Sf) (x) (Sg) 2 (x) (Sf) 2 (x) (Sg) (x) for x 2 [a; c], Be- where (Sf) (x) and (Sf) 2 (x) are the rst and second components of Sf. cause of () we have the algeraic identity, W x (Sf; Sg) := f W x (Sf; Sg) for x 2 [a; c), and ecause of the existence of the limits in (4) we can take the limit as x! c on oth sides to otain, W c (f; g) = f W c (Sf; Sg) for all f; g 2 D (L ). (5) We now de ne a two-parameter family of Weyl -Spaces arising from separated oundary conditions as follows: U ; 2 := ff 2 D (L ) jr a (f) = 0 and R c (f) = 0g, (6) where ; 2 [0; ), and where R c (f) is de ned y, R c (f) := (Sf) (c) cos + (Sf) 2 (c) sin for all f 2 D (L ). 2 (7) The operator taking f into f which has U ; 2 as its domain will e denoted L ;. The symmetry of L ; follows easily since R c (f) = R c (g) = 0 implies f W c (Sf; Sg) = 0 and therefore W c (f; g) = 0 y (5). Also L ; is a real operator, that is L ; f for all f 2 U ; 2, since U ; 2 is closed under complex conjugation. The self-adjointness of L ; will e deducile from the completeness of its eigenfunctions. In order to treat the spectral theory of the operator L ; (for xed ; ) in a manner analogous to the discussion of the operator L ; in section, we now de ned a fundamental system of (8) for all 2 C y initial conditions at the 0 This is not possile in the limit point case for this reason the following discussion does not apply when the limit point case occurs at c. Compare K. Kodaira [6, p.924, Equa. (.5) with =0]. 2 Boundary conditions at the singular point were rst written in this form y F. Rellich in [22, p. 354, Equa. (9)]. Compare also K. Jörgens [5, Corollary to Theorem 4, p. 9.0] and J. Weidmann [36, p. 272]. 0

11 singular point c. This can e done as follows: We write the equation (8) in the form, dz dx = 0 u z with z := ( q) 0 u 0, (8) and introduce the change of variales 3, y := (Y (x)) z, (9) otaining dy = B (x) y, where dx (0) uv v B (x) := u 2 for x 2 [a; c). uv () Remark 2. The mapping S on D (L ) de nes for each 2 C ( = 0 included) a one-to-one correspondence etween the solution spaces of (8) and (0). Theorem 2. Let q(x) e continuous in [a; c), a < c 5, and elong to the limit circle case at c. We have: (i) If y (x) is a solution of (0) for some 2 C, then y (c) := lim x!c y (x) exists. Moreover, for all x 0 2 [a; c) and all x 2 [a; c) we have, Z x jy (x 0 )j exp jj u 2 + v 2 ds x 0 and 5 jy (x)j Z x 5 jy (x 0 )j exp jj u 2 + v 2 ds x 0 (2) jy (x) y (c)j 5 jy (x 0 )j jj M (x) exp fjj M (x 0 )g (3) where M (x) := R c x u2 + v 2 ds, and where the asolute value of the complex vector y (x) is de ned as usual, that is, (ii) To each vector jy (x)j := z z 2 solution y (x) of (0) such that q jy ; (x)j 2 + jy 2; (x)j 2. in CC there exists (for every 2 C) a unique y (c) := lim x!c y (x) = z 3 Compare Hartman [9, p.330, Equa 2.28 with q = q 0 ]. z 2.

12 Proof. : For 6= 0 the assumptions of [9, p. 273, Theorem. and.2], are ful lled since, Z c a jb (x)j dx = jj Z c a u 2 + v 2 dx, which is nite ecause u and v are in L 2 [a; c). (The inequality (2) comes from Hartman [9, Equa. (.9), p.274], and the inequality (3) comes from [9, Equa. (.4), p.275].) For = 0 the solutions of (0) are constant vectors and we have equality in (2) and (3). Using Theorem 2 we now make the following de nition: De nition. Let 2 [0; ). For each 2 C, let y () (x) and y(2) (x) e the unique solutions of (0) de ned y, 0 y () ; B (c) y (2) ; (c) 0 A sin cos A (4) y () 2; (c) y (2) 2; (c) cos sin and let c; (x) and c; (x) denote the solutions of (8) de ned y, c; (x) 0 c; (x) c; (x) 0 c; (x) 0 A B := Y y () ; (c) y () 2; (c) We have the following facts analogous to (0)-(5): c; ; c; = Wx f y () = W x ; y(2) y (2) ; (c) y (2) 2; (c) C A (5) for all 2 C and all x 2 [a; c]. (6) c; (x) = c; (x) and c; (x) = c; (x) (7) For xed x 2 (a; c), c; (x), 0 c; (x), c; (x), and 0 c; (x) are entire functions of. 4 (8) Rc (f) = W c r c;; f for all f 2 D (L ) and all 2 C. (9) 4 The functions y () (x) and y(2) (x) are continuous in x and on (a; c) C. This can e proved using easy estimates, making use of (2.2) and the analyticity in for xed x. Also for xed the functions c; (x) and c; (x) are in nitely di erentiale for x 2 (a; c). This follows from [5, Corollary 4, p.283]. 2

13 R c r c; = 0 for all 2 C. 5 (20) W c c; ; 0 c; 0 = sin ( 0 W c c; ; 0 c; 0 = sin ( 0 ) for all ; 0 2 C. ) for all ; 0 2 C. (2) The only statement which is not self-evident is (8). Even in the case c <, the analyticity of the solutions y () (x) and y(2) (x) in cannot e concluded from usual theorems asserting analyticity in parameters ([5, p. 284, Corollary 5]) ecause the elements of the matrix B (x) may diverge as x! c. The analyticity of the functions c; (x), 0 c; (x), c; (x), and 0 c; (x) in can, however, e deduced from the analyticity in of the functions a; (x), 0 a; (x), a; (x), and 0 a; (x) as follows: We de ne four functions of y, w ; () := W x a; ; c; p ; () := W x c; ; a; p ; 2 () := W x a; ; c; v ; () := W x ( c; ; a; ) (22) where the right hand side is, of course, independent of x. Evaluating the aove Wronskian at x = c and applying (5) and (4) we have, w ; () = S a; (c) cos S a; (c) sin 2 = Rc a; p ; () = (S a; ) (c) cos + (S a; ) 2 (c) sin = R c ( a; ) p ; 2 () = S a; (c) sin S a; (c) cos 2 v ; () = (S a; ) (c) sin + (S a; ) 2 (c) cos. (23) The following algeraic facts are readily veri ed: p () p 2 () w () v () = for all 2 C. (24) 5 In [5, p.472, Lemma 42] (which applies with 0 = when the limit circle case occurs at oth endpoints) the existence of a solution (x; ) of (.8) satisfying a given oundary condition which is real-valued and in nitely di erentiale in x and for x 2 (a; c) and 2 ( ; ) is proved. The function r c; (x) de ned in (2.5) satis es the oundary condition (2.7) and has all the properties asserted in Lemma 42 (cf. (2.8) and footnote), eing in addition analytic in. The de nitions in (2.4) and (2.5) thus produce the solution desired in Lemma 42 immediately, while the proof of this lemma is consideraly more complicated, rings a good deal of functional analysis and spectral theory into play, and yields slightly weaker information (as far as the limit circle case is concerned). 3

14 for all 2 C. where 0 c; (x) A p () c; (x) v () c;0 (x) c;0 (x) c;0 (x) c;0 (x) A = sin cos A A = C w () 0 a; (x) A (25) p 2 () a; (x) cos sin B D u (x) v (x) u (x) v (x) A := cos v (a) + sin v 0 (a) (26) (27) B := cos u (a) sin u 0 (a) C := cos v 0 (a) sin v (a) D := cos u 0 (a) + sin u (a) Oservations:. For xed x 2 (a; c), the analyticity of (S a; ) (x) and (S a; ) (x) in follows immediately from the analyticity in of a; (x), 0 a; (x), a; (x) and 0 a; (x) ecause of (9). 2. From (3) it is clear that (x) = Sa; (c) for all 2 C, and lim x!c S a; lim (S a;) (x) = (S a; ) (c) for all 2 C, x!c and ecause of the estimate in (3) it follows that the convergence is uniform on compact -sets. Thus y the Weierstrass Theorem the limit functions S a; (c) and (Sa; ) (c) are entire functions of. 3. The analyticity of the functions w (), p (), p 2 (), and v () follows immediately from the analyticity of S a; (c) and (Sa; ) (c) in ecause of the algeraic relations in (23). 4. For xed x 2 (a; c) the analyticity of c; (x), 0 c; (x), c; (x) and 0 c; (x) in follows from the analyticity of the functions w (), p (), p 2 (), and v (), and the analyticity in of a; (x), 0 a; (x), a; (x) and 0 a; (x) ecause of the algeraic relations in (25). 4

15 Remark 2.2 For xed x 2 (a; c) the analyticity of a; (x) and 0 a; (x) in can e proved y showing that the Cauchy-Riemann equations hold (cf. F. John [4, p.29]). A similar argument can e used to show that the solutions y () (x) and y (2) (x) of (4) are analytic for 6= 0, and then Theorem 2 can e used to show that = 0 is a removale singularity. This gives an alternate proof for analyticity in of c; (x), 0 c; (x), c; (x) and 0 c; (x). Remark 2.3 The fact that w (), p (), p 2 (), and v () have only real (and simple) zeros can e proved in the same manner as in the regular case, cf. Titchmarsh [3, p.-2]. We note only that in the present (singular) case Titchmarsh s proof of the simplicity of the zeros of w ; () must e slightly modi ed. One has to make use (for cos 6= 0) of the equation fw c S a; ; S a; 0 = cos hw ; 0 S a; instead of [3, p.2, line 3], W a; ; a; 0 = cos which applies only for < c. (c) w; () S a; 0 h w ; 0 a; () w ; () a; 0 () 2 (c) i Remark 2.4 Titchmarsh s proof ([3, p.2, two lines efore Theorem.9]) that the zeros of the function w ; () (de nition [3, p.7, Equa..6.]) are ounded elow does not carry over to the singular case ecause there is no analogue of Lemma (7) in the singular case. As a corollary of Theorem 2 and the aove discussion we have the following result which gives a formula for the computation of the zeros of w ; () : Corollary 2. (i) If = 0 is not a zero of w ; (), the zeros of w ; () are the solutions of the following real equation in : i, = R c a Ra (u sin v cos ) [u (x) sin v (x) cos ] a; (x) dx (28) (ii) If = 0 is a zero of w ; (), then the remaining zeros of w ; () are the real solutions of the equation, Z c a [u (x) sin v (x) cos ] a; (x) dx = 0. (29) 5

16 Proof. Applying Green s Formula to a; (x) and c;0 (x) and using (9), (23), (2.3), and (26) we otain, ( 0) Z c a a; (x) [u (x) sin v (x) cos ] dx = W a a;0; c;0 + w ; (), (30) from which statements (i) and (ii) follow. Remark 2.5 Similar formulas for the zeros of p (), p 2 (), and v () can e otained in the same way. In order to prove the expansion theorem for the operator L ; (; 2 [0; ) xed) ala Titchmarsh, we now introduce an 0 m () 0 -function associated with this operator as follows: De nition 2. For each ; 2 [0; ) we de ne for all not zeros of w ; (): From (25) it follows immediately that m ; () := p; () w ; () and6 (3) ; (x) := a; (x) + m ; () a; (x). (32) ; (x) = w ; () c; (x) for all not zeros of w ; (). (33) Thus from (20) it is clear that ; (x) satis es the oundary condition at c and from (2) we have the following analogue of (2.8) at c: W ; c ; ; = 0 for all ; 0 with Im 6= 0 and Im 0 6= 0. (34) 0 The expansion theorem for the operator L ; can now e proved as in [3, p. 28-4], if one identi es Titchmarsh s 0 m () 0 and 0 (x; ) 0 functions with the functions m ; () and ; (x) de ned in (3) and (32). We note only that equation (34) corresponds to Titchmarsh s Lemma 2.3, p.26, on which the rest of Titchmarsh s discussion hinges. Following Titchmarsh one applies Green s Formula to ; (x) and ; (x) for Im 6= 0 and Im 0 0 6= 0 and makes use of (34) to otain as a rst result (cf. [3, p.28, Equa. 2.5.]): Z c a ; (x) ; (x) dx = m; () m ; (35) 6 Then numerator and denominator cannot simultaneously e zero since this would imply the linear dependence of a; (x) and a; (x), cf. Equation (22). 6

17 Titchmarsh s technique can then e applied to otain the eigenfunctions from (35) and the expansion theorem for the operator L ; follows from [3, Theorem 2.7 (i) and (ii), p.3]. Thus if n, n = ; 2; 3;, are zeros of w ; () and if then the functions, r ; n := Res m ; () = p; ( n) w 0 ( n) > 0 = n (36) ; n (x) := p r ; n a; (x), (37) are the eigenfunctions of L ; and form a complete orthonormal set in L 2 [a; c). Since the eigenvalues of the operator L ; are the zeros of w ; (), they can e computed from (28), or (29), as the case may e. Remark 2.6 The self-adjointness of the operator L ; can e inferred from Titchmarsh s equations (2.6.2), (2.6.3), together with the equation, R c ( ;f ) = 0, 7 (38) all of which hold for for aritrary f 2 L 2 [a; c) and aritrary, Im 6= 0. Alternatively, the self-adjointness of L ; can e regarded as a consequence of the fact that it is a (closed) symmetric operator whose eigenfunctions constitute (ecause of Titchmarsh s proof) a complete orthonormal set for L 2 [a; c), cf. Hellwig [, p.84, Theorem 6]. Remark 2.7 The existence of in nitely many zeros of w ; () can e regarded as a consequence of the expansion theorem. For if w ; () were to have only nitely many zeros, there would e only nitely many eigenfunctions ; n (x), which then could not span the Hilert Space L 2 [a; c). From (3) and (23) we have the following representation of the limit-circle 0 m () 0 -functions, m ; () = (S a;) (c) cos + (S a; ) 2 (c) sin S a; (c) cos + S, (39) a; (c) sin with 2 [0; ), which is of the type (4), since the functions (S a; )(c) and (S a; )(c) are analytic in. From (23) we also have the following corollary: Corollary Fix 2 [0; ). Then for each real 0 there exists exactly one 2 [0; ) such that 0 is in the spectrum of L ;. Thus if f ; n, n = ; 2; 3; g is the spectrum of L ; it follows that [ [ ; n = < for each xed. (40) 2[0;) n= 7 This equation states that for aritrary f 2 L 2 [a; c), ;f (x) := ( L ; ) f (x) satis es the -oundary condition at c. In Titchmarsh s terminology the analogous statement would e, lim x! W x ( ; (; 0 ; f)) = 0 for all ; 0 with Im 6= 0 andim 0 6= 0, a fact which Titchmarsh does not mention ut which follows from Lemma

18 Proof. : The spectrum of L ; coincides with the set of zeros of w ; (). Given 0 2 < we note from (23) that 0 is a zero of w ; () if and only if cot = S a; 0 2 (c) S (4) a; 0 (c). Since S a; (c) and 0 S a; (c) are not oth zero (otherwise W 0 2 c a; 0, a; 0 = 0 y (5)) (4) de nes a unique value of 2 [0; ). (We assume the convention Arc cot () = 0, that is, = 0 when the right hand side in (4) is in nite.) Remark 2.8 The expansion theorem for the operator L ; can also e otained y making use of left-handed m ()-functions which are de ned y 9, Putting 2 () := P ; 2 () w ; () : (3)0 m ; it is clear (cf. Equa. (25)) that ; 2; (x) := c; (x) + m; 2 () a; (x) ; (32) 0 ; 2; (x) = w ; () a; (x) for all not (33)0 Analo- zeros of w ; (), and therefore satis es the oundary condition at a. gous to (34) we then have, W ; a 2; ; ; 2; = 0 for all ; 0 not zeros (34) 0 0 of w ; (). Titchmarsh s discussion of the expansion theorem then requires only slight modi cation to show that the eigenfunctions of L ; are q ; 2;n (x) := r ; 2;n c; n (x) (37) 0 where n are the zeros of w ; () and where, 9 Because of (2.8) and footnote, the functions c; (x) and c; (x) are suitale for use in [5, p. 364, Theorem 8]. If, in this theorem, T is the self-adjoint operator L ; and (t; ) and 2 (t; ) are functions c; (t) and c; (t) respectively, then the left-handed m ()-function of (3) 0 (or, as the case may e, of (39) 00 elow) is the function () = + () de ned in this theorem. If, on the other hand, (t; ) and 2 (t; ) are chosen to e functions a; (t) and a; (t) respectively, then the function () = + () is the right-handed m ()-function of (3). Representations of the type (0.4) for the function () = + () do not seem to occur in the ook of Dundford and Schwartz. In this ook the spectral theory and expansion theorem for eigenvalue prolems involving the limit circle case at oth end points is accomplished in [5, Theorem 2, p.33 (n=2)], y appealing to the theory of Compact Operators. This approach seems to have the disadvantage of not yielding information on the dependence of the eigenvalues on the choice of oundary conditions. Compare, for example, the aove mentioned theorem with our Corollary 2.. 8

19 r ; 2;n := Res m; 2 () = p; w 0 ( n) < 0: (36) 0 = n 2 ( n) Evaluating the wronskians in (22) at x = 0 we have the following representation of the left-handed m ()-functions: 2 () = c; (a) cos + 0 c; (a) sin c; (a) cos + 0 ; 2 [0; ): (39)0 c; (a) sin m ; Note: The eigenfunctions ; 2;n (x) in (37)0 may di er in sign from those in (37) ut we have in any case equality y taking the square, and equating the squares yields the equation, p ( n ) p 2 ( n ) =, a fact which can e veri ed y looking at equation (24). Remark 2.9 For the interval (a; c) the aove theory can e applied if the limit circle case occurs at a as well as at c. One has only to apply Theorem 2 to rede ne the fundamental system a; (x), a; (x) y initial conditions at the singular point a alá De nition, and to replace the oundary condition (2.6) y, R a (f) := (Sf) (a) cos + (Sf) 2 (a) sin ; (:6) 0 where Sf is de ned as in (3). (The de nitions of D (L 0 ) and D (L ) also require ovious modi cation.) The fundamental system a; (x), a; (x) then has the properties (0)-(5) as efore with the aove oundary condition. The functions in (22) can e de ned as efore and equations (24)-(27) remain valid, with the exception that (27) must e replaced y, 0 a;0 A sin cos u (x) = : (27) 0 cos sin v (x) a;0 (x) The right-handed m ()-functions are as in (39) and the formula (39) 0 for the left-handed m ()-functions changes to, m ; 2 () = (S c;) (a) cos + S 0 c; (a) sin 2 S c; (a) cos + S (39)00 c; (a) sin where 2 [0; ). Remark 2.0 From the aove theory and from Remark 2.8 and Remark 2.9 it is clear that there is no need to reak the interval into two parts (Compare 2 9

20 Titchmarsh [3, p.42, Section 2.8]) in order to otain 0 m () 0 -functions for singular prolems which involve the limit circle case at oth endpoints. In fact as long as the limit circle case occurs at at least one end of the interval, Theorem 2 can e applied at that end of the interval to de ne a fundamental system at that end point and then the 0 m () 0 -functions (right-handed or left-handed as the case may e) can e de ned in terms of this fundamental system. Remark 2. It will e oserved that the assumptions made on q (x) (cf. Equation (.)) include the possiility that lim x!c q (x) exists. Thus for c <, the spectral theory of the regular Sturm- Liouville prolem on the closed interval [a; c] is contained in the aove discussion. Remark 2.2 If for 2 (a; c), functions ; (x) and ; (x) are de ned as in (4) and (5) y putting in place of c, then the aove discussion with ; (x) and ; (x) in place of c; (x) and c; (x) yields the spectral theory of the Sturm-Liouville operators, A ;, ; 2 [0; ), de ned y, D (A ; ) := ff 2 D (L ; ) jr a (f) = 0 and R (f) = 0g,20 (42) where R (f) is de ned y, R (f) := (Sf) () cos + (Sf) 2 () sin (:7)0 for all f 2 D (L ; ). If wc ; () denotes the function de ned in (22) and w ; () the function with ; (x) and ; (x) in place of c; (x) and c; (x) it will e noted from (23) and (3) that lim!c w; () = wc ; () uniformly on compact sets in the -plane. It follows (y application of Rouche s Theorem) that each zero of wc ; () is a limit of zeros of w ; () as! c. This means that each point of the spectrum of L ; can e approximated y eigenvalues of the regular Sturm-Liouville operator A ; as! c. 4 Boundary Functionals In their encyclopedic volumes on Linear Operators N. Dunford and J.T. Schwartz take a Hilert-Space approach to the spectral theory associated with nth.-order di erential equations, an approach which was rst taken y M.H. Stone for 20 There is a one-to-one correspondence etween the and parameters such that A ; = L ;. Compare Corollary 4., Lemma 6., and Remark 6.3 elow. 20

21 second-order di erential equations in [29, Chapter 0]. The notion of 0 oundary values for 02 used y Dunford and Schwartz in [5, Chapter XIII], will e particularly helpful to elucidate the connection etween the oundary conditions in (7) and the oundary conditions given y H. Weyl, M.H. Stone, E.C. Titchmarsh, and K. Kodaira. For this reason we give in this section the de nitions and results which will e needed. For results not proved here reference will e made to the ook of Dunford and Schwartz. Also we will use the terminology 0 oundary functional 0 instead of 0 oundary value 0. De nition Let e as in (). (i) A oundary functional for is a continuous linear functional on the (complete) Hilert Space ( D (L ) ; kk ) which vanishes on D (L 0 ). Here kk denotes the graph norm, i.e. Z c kfk = jfj 2 dx + a Z c a =2 jfj 2 dx A oundary functional for, (), is called 0 real 0 if f = (f) for all f 2 D (L ). (ii) A oundary functional 0 for at c 0 is a oundary functional for, (), such that, f 2 D (L ) and f = 0 in a Nd: of c implies (f) = 0. Boundary functionals 0 for at a 0 are similarly de ned. (iii) If () is a oundary functional for at c then (f) = 0, f 2 D (L ) is called a oundary condition 0 for at c 0. This oundary condition is called real if () is a real oundary functional. (iv) Two oundary functionals for ; () and 2 (), are called 0 equivalent 0 if they are linearly dependent. Remark 3. A oundary functional for necessarily vanishes on D L 0 since it vanishes on D (L 0 ) and is continuous in the graph norm. From the decomposition, D (L ) = D L 0 D + D (where D + and D are the positive and negative de ciency spaces, cf. [5, p , De nition 9 and Lemma 0]) it follows that the space M of oundary 2 The notion of oundary values is due to J.W. Calkin, [3]. 2

22 functionals for can e identi ed as the 4-dimensional space of linear functionals on the 4-dimensional Hilert Space ( D + D ; kk ). Lemma 3. (i) 23 If a is a regular endpoint, then a () is a oundary functional for at a if and only if a (f) = z f (a) + z 2 f 0 (a) for some z, z 2 in C. () (ii) 24 If c is a singular end point, then c () is a oundary functional for at c if and only if there exists a function w (x) 2 D (L ) such that c (f) = W c (w; f) for all f 2 D (L ). (2) (iii) If M is the space of oundary functionals for and M a, M c the spaces of oundary functionals for at a and c respectively, then M a and M c are two-dimensional spaces 25 and M = M a M c 26 (3) Remark 3.2 For each f 2 D (L ) let ;c (f) := (Sf) (c) and 2;c (f) := (Sf) 2 (c). From (2) and (3.4) it is clear that ;c () and 2;c () are oundary functionals for at c. Also ecause of (), z ;c () + z 2 2;c () = 0 implies z = z 2 = 0. Hence ;c () and 2;c () are linearly independent oundary functionals for at c and therefore span the space M c. We note also that they are real oundary functionals since the functions u (x) and v (x) in () were taken to e real-valued. From Lemma 3. (i) and (iii) and from Remark 3.2 it is clear that any oundary functional for ; (), can e represented in the form, (f) = z f (a) + z 2 f 0 (a) + z 3 (Sf) (c) + z 4 (Sf) 2 (c), (4) where z, z 2, z 3, and z 4 are complex numers. De nition (i) If in the representation (4), jz j 2 + jz 2 j 2 > 0 and jz 3 j 2 + jz 4 j 2 > 0, the oundary functional () is called a mixed oundary functional. 23 Dunford and Schwarz [5, p.30, Corollory 23.] 24 iid. p. 302, Theorem 27, and p. 303, Corollary iid. p iid. p. 298, Theorem 9. 22

23 (ii) Two oundary functionals () and 2 () are called separated if neither is a mixed oundary functional. The corresponding oundary conditions are then also called separated oundary conditions. The following theorem gives a necessary and su cient condition for two oundary functionals for at c to e equivalent. Theorem 3. Let w (x) and w 2 (x) e functions in D (L ) such that (f) := W c (w ; f) and 2 (f) := W c (w 2 ; f) are nontrivial (that is, nonzero) oundary functionals for at c (i.e. z = z 2 = 0 and jz 3 j 2 + jz 4 j 2 > 0 in (4)). Then: The oundary functionals () and 2 () are equivalent if and only if In this event we have, W c (w ; w 2 ) = 0. W c (w ; f) = KW c (w 2 ; f) for all f 2 D (L ) with K 6= 0 where (5) 8 >< K = >: (Sw ) (c) (Sw 2) (c) = (Sw) 2 (c) (Sw 2) 2 (c), if (Sw 2) (c) 6= 0 and (Sw 2 ) 2 (c) 6= 0 (Sw ) (c) (Sw 2) (c), if (Sw 2) 2 (c) = 0 (Sw ) 2 (c) (Sw 2) 2 (c), if (Sw 2) (c) = 0 Proof. Applying (5) we have for all f 2 D (L ): Oservations: (f) = (Sw ) (c) (Sf) 2 (c) (Sw ) 2 (c) (Sf) (c) 2 (f) = (Sw 2 ) (c) (Sf) 2 (c) (Sw 2 ) 2 (c) (Sf) (c) W c (w ; w 2 ) = (Sw ) (c) (Sw 2 ) 2 (c) (Sw ) 2 (c) (Sw 2 ) (c). If W c (w ; w 2 ) = 0, the null spaces of () and 2 () are identical and they are therefore linearly dependent. 23

24 2. Now suppose () and 2 () are equivalent oundary functionals and that W c (w ; w 2 ) 6= 0. The w 2 (x) is in the null space of 2 (), ut not in the null space of (), which contradicts the assumption that () and 2 () are linearly dependent. To otain the value of K in (5) the aove expressions for (f) and 2 (f) can e sustituted on oth sides of (5) and the value of K determined in terms of (Sw j ) i, i = ; 2 and j = ; 2, y making use of the fact that (Sw j ) and (Sw j ) 2, j = ; 2 are not oth zero for j = ; 2. 5 The Characteristic Functions of K. Kodaira and the Rellich Initial Numers In his famous paper on the spectral theory on the spectral theory of secondorder di erential equations [6] K.Kodaira de nes 0 m () functions in the limit circle case which are of the same type introduced in section 2. He does not, however, give a parametrization of his characteristic functions of the type (.4). In this section we show that each of Kodaira s characteristic functions is one of the functions we have de ned in (3). Kodaira s oundary conditions at the singular point c (cf. [6, p.924, Equa..7 III ]) are, W c (w c ; f) = 0, () where w c (x) is an aritrary real-valued function in D (L ) satisfying a condition of nontriviality [6, p.924, Equa..6]) which can e expressed y, (Sw c ) 2 (c) + (Sw c) 2 2 (c) 6= 0. (2) For the interval [a; c) the corresponding Weyl Spaces are U ;wc 3 := ff 2 D (L ) j R a (f) = 0; W c (w c ; f) = 0g, 2 [0; ). (3) Kodaira associates with this Weyl Space a characteristic function, which is de ned [6, p.925, Equa. (.9) and p.934, Section 4] y, f c () := W c w c ; a; W c w c ;. (4) a; It will e noted that the oundary functional Rc () of (3.7) is of the type () with w c (x) = cos v (x) + sin u (x) and that with this choice of w c (x) Kodaira s characteristic function f c () is precisely the function m ; () in (3.39). Because of the aritrariness which Kodaira allows in the choice of the function w c (x) it is not exactly clear that 24

25 every Weyl Space of the type (3) is of the type (3.6) for some value of. see that this is the case we prove the following theorem: To Theorem 4. (i) The oundary functionals, Rc () := (S ()) (c) cos + (S ()) 2 (c) sin and Rc wc () := W c (w c ; ) with w c 2 D (L ) and real-valued are equivalent if and only if (Swc ) = Arc cot 2 (c) (Here Arc cot() = 0.) (5) (Sw c ) (c) (ii) If is as in (5) then W c (w c ; f) = KR c (f) for all f 2 D (L ) withk 6= 0 (6) where 8 >< K = >: (Sw c) (c) sin = (Swc) 2 (c) cos, if 6= 0, =2 (Sw c ) 2 (c), if = 0 (Sw c ) (c), if = =2 Moreover, f c () = m ; () for all not zeros of w ; (). (7) Proof. Applying Theorem 3 and (5) we oserve that W c (w c, u sin v cos ) = 0 if and only if cot = (Sw c) 2 (c) (Sw c ) (c). Since w c (x) is real-valued (Sw c ) (c) and (Sw c ) 2 (c) are real numers and they cannot oth e zero ecause of the nontriviality condition (2). Therefore (5) de nes a unique value of 2 [0; ). The formula for K in (6) follows at once from (5) with w (x) = w c (x) and w 2 (x) = cos v (x) + sin u (x). To see (7) we oserve from (4) and (6) that, f c () = KR c ( a; ) KR c a; = R c ( a; ) R c a;, which, comparing with (3.39), is readily seen to e m ; (). We now make use of Theorem 4 to estalish the connection etween the oundary functionals of (3.7) and the Rellich Initial Numers. In [22] F. Rellich 25

26 showed [22, p.354, Equa. 9] that the oundary conditions of H.Weyl could e written in the form, f cos + f 2 sin = 0, 2 [0; ), (8) where f and f 2 are initial numers associated with each f 2 D (L ) which are de ned as follows: Let e a real numer and (x),! (x) a real-valued fundamental system of (8) for = with W,! = 6= (9) For aritrary f 2 D (L ), f is a solution of the inhomogeneous equation, u u = g (x) := f (x) f (x), and is therefore uniquely representale in the form, f (x) = f (x) + f 2! (x) +! (x) (x) Z c x Z c x (y) [f (y) f (y)] dy! (y) [f (y) f (y)] dy (0) with f and f 2 in C: The numers f and f 2 which are uniquely de ned y (0) for each f 2 D (L ) are called the initial numers for f and c with respect to, and w. The Weyl Spaces associated with the oundary conditions in (8) are U ; 4 := ff 2 D (L ) jr a (f) = 0, f cos + f 2 sin = 0g, () with 2 [0; ) and 2 [0; ): Calculating the wronskians, W c (f; w ) and W c f; from (0) and making use of the fact that W (x) and (x) and f f are in L 2 [a; c), it is readily shown that f = W c (f; w ) and f 2 = W c f; 29 (2) Comparing (2) with (3.3) it will e noticed that if = 0 and if the fundamental system f 0 (x) ;! 0 (x)g is taken to e the funcamental system fu (x), v (x)g chosen in (3.), then f = (Sf) (c) and f 2 = (Sf) 2 (c), 28 In [22] Rellich assumes that (.8) is nonoscillatory for real at the singular point (which means p.349, Satz 2 that the self-adjoint operators associated with are ounded elow), and takes w (x) and (x) to e principal and nonprincipal solutions respectively. These additional restrictions are not, however, necessary for (4.8). 29 Compare (4.8) with K.Jörgens [5, p. 9.0, Corollary to Theorem 4]. 26

27 so that the oundary conditions in (8) are identical to those in (3.7) with =. In this case the Weyl Spaces in () are indentical with those in (3.6), and the associated m () - functions of (3.39) can e written as, m ; () = a; cos + a; 2 sin a; cos +, 2 [0; ), (3) a; 2 sin where a;, a; 2 and a;, a; 2 are the Rellich initial numers for a; and a; at c with respect to u (x), v (x). If 6= 0, or if = 0 and 0 (x)! 0 (x) 6 u (x) v (x), we have in any case the following corollary of Theorem 4: Corollary 4. (i) The oundary functionals R c () : = (S ()) (c) cos + (S ()) 2 (c) sin and P c (f) : = f cos + f 2 sin are equivalent if and only if = Arc cot! (S! ) 2 (c) cos S (c) sin 2 (S! ) (c) cos + S (c) sin (4) (ii) Moreover, (4) de nes a one-to-one correspondence etween the parameters and under which U ; 2 = U ; 4 and m ; a; () = cos + a; 2 sin a; cos + a; 2 sin. (5) Proof. (i) Because of (2) the oundary condition (8) may e written as, W c cos! + sin, f = 0, which is of the type () since (x) and! (x) are real-valued. Thus (i) follows immediately from Theorem 4 with w c (x) = cos! (x) + sin (x). To see that (4) de nes a one-to-one correspondence etween 2 [0; ) and 2 [0; ) it su ces to show (since (S! ) (c) and S (c) are real-valued) that A (z) := (S! ) 2 (c) z S (c) 2 (S! ) (c) z + S (c) 27

28 de nes an invertile linear fractional transformation. The invertiility follows from (9) ecause of (3.5). (ii) Because of the equivalence of the oundary functionals in (i) we have, P c () = KR c () for some constant K 6= 0 when and are related as in (4). Hence Pc ( a; ) = KR c ( a; ) Pc a; KRc = R c ( a; ) a; Rc, a; which proves (5). 6 The Weyl Spaces of Hermann Weyl Boundary conditions of the type originally given y H.Weyl in his famous paper [37] are as follows: 30 Let u (x) e a solution of (2.8) for Im 6= 0 satisfying Then W c (u ; u ) = 0. () (f) := W c (u ; f) for f 2 D (L ) (2) is a oundary functional for at c and the corresponding Weyl Spaces are, U ;u 5 := ff 2 D (L ) j R a (f) = 0, W c (u ; f) = 0g. (3) Remark 5. For = i a direct proof of the self-adjointness of operators having domains of the type (3) was given y F.Rellich in his lecture notes [23, p.39]. This proof was later improved y A.Schneider in [28] and was included in the ook of G.Hellwig [, p.238, Theorem ]. The proof can e modi ed to otain the self-adjointness of spaces of the type (3) for any, Im 6= 0. In this section we show that each oundary functional of the type (2) is equivalent to one of the oundary functionals in (3.7) for some value of 2 [0; ). It will e noted that this does not follow directly from Theorem 4 ecause the function u (x) in () is complex-valued. We require the following two lemmas: Lemma In [37], H.Weyl considers only the case = i. 28

29 (i) For each, Im 6= 0, T c; (z) := (S a;) (c) z + (S a; ) 2 (c) S a; (c) z + S a; 2 (c) (4) is an invertile linear fractional transformation which maps the real z-axis onto the -limit circle, C c;. (ii) If z = have: T c; (w), then for all with = Im 6= 0 and all w 2 C we = = = (S a; ) (c) + w S (c) 2 Im z a; ~W c S a; + w a; ; S a; + w a; Z c a Z c a 2i a; (x) n w 2 dx % c; 2 a; (x) + w a; (x) 2 dx + Im w rc; o 2 where % c; and r c; are the center of radius of the limit circle, C c;, cf. equation (24). (5) Proof. (i) The fact that Tc; is invertile follows from (3.6) ecause of (3.5). The image of the real z-axis under Tc; can e computed (Ahlfors [2, p.79], for example) and is found to e the circle with center = radius = fw c S a; ; S a;, and fw c S a; ; S a; W f. c S a; ; S a; This circle is the - limit circle, Cc;, ecause of (3.5) and (2.24). (ii) The rst part of (5) follows y computation of Im z. The rest follows y letting! c in (2.23) and using (3.5). Lemma 5.2 For xed 2 [0; ) and xed, Im 6= 0, f ; :! m ; () (6) de nes a one-to-one mapping from [0; ) onto the -limit circle, C c;. 29

30 Proof. From the representation (3.39) we have, m ; () = T c; (cot ) for all 2 [0; ), (7) so the result is immediate from Lemma 5. (i). Remark 5.2 There are ovious analogous of Lemma 5. and 5.2 associated with the functions m ; () de ned in Remark 2.2 for < c. In this case the linear fractional transformation, T;, maps the real z-axis onto the circle C ; (de ned y (2.22)) for each, Im 6= 0. Remark 5.3 There are also analogous of Lemmas 5. and 5.2 associated with the left-handed m ()- functions of Remark 2.8. From (5) and (3.5) it is clear that w2 Cc; if and only if W c a; + w a; ; a; + w a; = 0. (8) Now x 2 [0; ), and consider the family of Weyl Spaces, U ;u 5, for di erent choices of u. Because of (8) a solution of (2.8) for Im 6= 0 satisfying () must expressile in the form, u (x) = K a; (x) + w a; (x) with w 2 C c; (9) and K 6= 0. (The possiility that u (x) could e linearly dependent with a; (x) is ruled out y (2.24).) Hence for each, Im 6= 0, the class, F c;, of nonequivalent oundary functionals type (2) associated with can e descried as follows: F c; := () j () = W c a; + Q a;; and Q 2 C c; (0) We now show that the one-to-one correspondence in (6) identi es, for each 2 [0; ), the Weyl-oundary-functional in F c; which is equivalent to the oundary functional Rc () of (3.7). More precisely we have: Theorem 5. Fix 0, Im 0 6= 0 and x 2 [0; ). Then the oundary functional, R Q c; 0 () := W c a; 0 + Q a; 0 ; with Q 2 C c; 0 () is equivalent to the oundary functional, R c () := (S ()) (c) cos + (S ()) 2 (c) sin (2) if and only if Q = m ; ( 0 ). (3) 30

31 Proof. If Q = m ; ( 0 ) then from (3.33) and (3.9) we have, R Q c; 0 () = w ; ( 0 ) W c c; 0, = w ; ( 0 ) R c (), and therefore R Q c; 0 () and R c () are linearly dependent since w ; ( 0 ) 6= 0. Now suppose Q 6= m ; ( 0 ). By assumption Q 2 C c; 0 and therefore y Lemma 5.2 there exists 0 2 [0; ) such that From (3.33) and (3.9) we again have, R Q c; 0 () = Q = m ;0 ( 0 ). w ;0 ( 0 ) W c c; 0 ; = ut 0 6= then yields a contradiction ecause of (3.2). 0 w ;0 ( 0 ) R c (), Remark 5.4 The choice of a single point Q on the circle Cc; 0 for a single 0, Im 0 6= 0, uniquely xes points P (Q) on Cc; for every other, Im 6= 0, such that the oundary functionals R Q c; 0 () and R P (Q) c; () are equivalent. The collection of all these points P (Q) is the function m ; () where the value of is determined y Q according to the equation (3). Remark 5.5 N.Levinson de nes Weyl Spaces in the limit circle case with oundary functionals of the type (), cf. [4, p.242, Theorem 4., and p.244, Equa. (4.0)]. For each oundary functional of the type () Levinson de nes an 0 m () 0 -function (depending on 0 and Q) y, 3 m () := Q ( 0) R c a a; (x) a;0 (x) + Q a;0 (x) dx + ( 0 ) R c a a; (x) a;0 (x) + Q a;0 (x) dx (4) The meromorphic character of this m () function arises easily from the fact that oth numerator and denominator are entire in. The representation (4) is however, nothing special. For if one solves equation (3.35) for m ; () in terms of m ; 0 one nds, 32 m ; () = m ; 0 0 R c a a; (x) a; 0 (x) + m ; 0 a; 0 (x) dx + 0 R c a a; (x) a; 0 (x) + m ; 0 a; 0 (x) dx (5) 3 Sign di erences arise from Levinson s choice of the parameter in the oundary condition at a, cf. [4, p.23]. 32 The representation (5.5) and the argument that the right hand side is meromorphic in was noticed y D.B.Sears in [25, p.209, section 4]. 3

32 which holds aritrarily in ; 0 so long as ; 0 are not zeros of w ; (). Putting 0 = 0 and letting e de ned y (3) it is readily seen that (5) reduces to (4). Hence the 0 m () 0 -function de ned y the formula (4) is precisely the function m ; () where Q determines as in (3). It will e noted that Levinson allows his Weyl Space to depend on the choice of 0 as well as on the choice of Q 2 Cc; 0. As we have already noted in Remark 5.4 the Weyl Space and corresponding m ()-function is xed merely y the choice of Q on Cc; 0. Thus the Weyl Space does not depend in any special way on 0 itself. Remark 5.6 The choice of in the representation (0) has no a ect on the class F c; of (nonequivalent) oundary functionals Weyl type. This follows from Theorem 5 since the functionals Rc () depend on only. Remark 5.7 There are ovious analogous of Theorem 5 and Remarks 5.4, 5.5, and 5.6 relating the oundary functionals Ra () of (2.6) to oundary functionals of Weyl type at a. Similarly there are analogous of Theorem 5 and Remarks 5.4, 5.5, and 5.6 relating the oundary functionals of (2.7) 0 for a < < c (cf. Remark 2.2) to oundary functionals of Weyl type at. From Theorem 5, Remark 5.4, and Remark 5.7 the following corollary is selfevident: Corollary 5.3 Let m ; () and m ; 2 () denote the right handed and lefthanded m ()-functions for L ; de ned in (3.3) and (3.3) 0 respectively. For xed ; 2 [0; ) we have: 33 ff 2 D (L ) jr a (f) = 0, R c (f) = 0g = ff 2 D (L ) jw a c; + m; 2 () c; ; f = 0 and W c a; 0 + m; 0 a; 0; f = 0 where and 0 are xed complex numers which are not zeros of w ; ()g = ff 2 D (L ) j W a c; + m; 2 () c; ; f = 0 and W c a; 0 + m; () a; 0; f = 0 (6) for all, Im 6= 0g. 33 For the interval [a; c); Ra (f) is de ned as in (.6), and for the interval (a; c) it is de ned as in (.6) 0, Remark