ABSTRACT. Name: Suzanne M. Riehl Department: Mathematical Sciences. Spectral Functions Associated with Sturm-Liouville and Dirac Equations

Transcription

1 ABSTRACT Name: Suzanne M. Riehl Department: Mathematical Sciences Title: Spectral Functions Associated with Sturm-Liouville and Dirac Equations Major: Mathematical Sciences Degree: Doctor of Philosophy Approved by: Date: Dissertation Director NORTHERN ILLINOIS UNIVERSITY

2 ABSTRACT Two questions relating to the spectral functions associated with limit point differential equations are addressed. The equations are the second order Sturm-Liouville equation and the first order two-dimensional system known as the Dirac equation. For each equation, conditions on the coefficient functions are given so that the spectral function derivatives may be given eplicitly in the form of a series in terms of the given functions. Also, for each equation, formulae relating the spectral functions for different values of the initial condition are displayed.

3 NORTHERN ILLINOIS UNIVERSITY SPECTRAL FUNCTIONS ASSOCIATED WITH STURM-LIOUVILLE AND DIRAC EQUATIONS A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICAL SCIENCES BY SUZANNE M. RIEHL c 2001 Suzanne M. Riehl DEKALB, ILLINOIS MAY 2001

4 Certification: In accordance with departmental and Graduate School policies, this dissertation is accepted in partial fulfillment of degree requirements. Dissertation Director Date

5 ACKNOWLEDGMENTS I would like to etend my thanks to the faculty, staff, and graduate students at the Department of Mathematical Sciences, Northern Illinois University, for challenging, supporting, and encouraging me throughout my studies. In particular, I greatly appreciate the guidance and humour of my advisor, Dr. Bernard Harris. I have learned much from him and have thoroughly enjoyed the process. I also gratefully acknowledge the help of my committee: Dr. Linda Sons, Dr. Qingkai Kong, and, from the Dublin Institute of Technology, Dr. Daphne Gilbert. Their comments and questions pulled me to a higher level of mathematical maturity. As I continue in mathematics, I aim to be as effective in communicating the beauty and power of mathematics as all those who have influenced me.

6 TABLE OF CONTENTS Page Chapter 1. INTRODUCTION Overview Necessary Background Information The Operator Viewpoint SPECTRAL FUNCTIONS ASSOCIATED WITH STURM- LIOUVILLE EQUATIONS WITH POTENTIALS OF WIGNER-VON NEUMANN TYPE Introduction The Results The Riccati Equation Proof of Theorem An Eample: The Wigner-von Neumann potential A REFINEMENT OF THE STURM-LIOUVILLE CONNECTION FORMULAE Introduction Proofs of the Connection Formulae Eamples

7 v Chapter Page 4. CONNECTION FORMULAE FOR SPECTRAL FUNCTIONS ASSOCIATED WITH SINGULAR DIRAC EQUATIONS Introduction The Connection Formulae Proofs Eamples THE FORM OF THE SPECTRAL FUNCTIONS ASSOCIATED WITH DIRAC EQUATIONS Introduction The Representation of ρ α(µ) The Riccati Equation Proofs Eamination of Conditions for which Theorem 5.1 Holds Eamples A REFINEMENT OF THE DIRAC EQUATION CONNECTION FORMULA Introduction Proof of Theorem Eamples REFERENCES

8 CHAPTER 1 INTRODUCTION 1.1 Overview In this work, the spectral functions associated with two differential equations are studied. The first is the Sturm-Liouville equation (py ) + qy = λwy (1.1.1) on [0, ) together with the initial condition y(0) cos α + p(0)y (0) sin α = 0 (1.1.2) where α [0, π). The functions p, q, and w are assumed to be real valued on [0, ) with w( ) > 0 and p( ) > 0 and each of 1/p, q, w L 1 loc[0, ). Some results will be given under more restrictive conditions. The comple spectral parameter is λ := µ + iɛ, µ, ɛ R. The second equation is the Dirac equation of the form y = p λ + c + v 1 (λ c + v 2 ) p on [0, ) together with the initial condition y, y = y 1 y 2 (1.1.3) y 1 (0) cos α + y 2 (0) sin α = 0 (1.1.4)

9 2 where α [0, π). Also, c 0 is a constant, λ = µ+iɛ is the comple spectral parameter, and v 1, v 2, and p are real valued members of L 1 [0, ). Additional restrictions on the coefficient functions are imposed for some results. These equations are related in the sense that equation (1.1.1) with p w 1 is a one-dimensional Schrödinger equation and equation (1.1.3) can be viewed physically as the radial component of the Dirac partial differential equation in relativistic quantum mechanics [10]. A spectral function ρ α (µ) is associated with each equation and initial condition. This function is defined for µ R where µ = Re{λ}. The spectral function is nondecreasing and can be thought of, roughly, as a probability distribution function. The symmetric derivative ρ α(µ) of the spectral function is defined by ρ ρ α (µ + ε) ρ α (µ ε) α(µ) = lim ε 0 + 2ε Two questions regarding these spectral functions are addressed. It is known that with appropriate conditions on the coefficient functions, the spectral functions associated with (1.1.1) can be given eplicitly in the form of a series. Harris [11] considers (1.1.1) with p w 1 and proves the following. Theorem 1.1 A sequence of functions is defined as follows. v 1 (, µ) = ep(2iµ 1/2 (t )) q(t) dt ( ) t n v n+1 (, µ) = ep 2iµ 1/2 (t ) + v k (s, µ) ds v n (t, µ) 2 dt k=1 v(, µ) = iµ 1/2 + v n (, µ) n=1 S(, µ) = Re{v(, µ)}

10 3 T (, µ) = Im{v(, µ)} If there eist functions a() and η(µ) and a µ 0 > 0 such that ep (2iµ 1/2 t)q(t) dt a()η(µ) for µ µ 0 and 0 < where a() is decreasing, a( ) L 1 [0, ), and η(µ) 0 as µ, then ρ α(µ) = π 1 µ 1/2 T (0, µ) 2 ep ( 2 0 S(t, µ) dt) (S(0, µ) 2 + T (0, µ) 2 ) sin 2 α + S(0, µ) sin 2α + cos 2 α (1.1.5) for α [0, π). It is also shown that as µ, ρ 0(µ) µ 1/2 /π and ρ α(µ) µ 1/2 /π, for all α (0, π). This is consistent with the asymptotic behavior of the spectral functions in [16]. Using a different sequence of functions {v n }, Gilbert and Harris [8] show the spectral derivative can be written ρ α(µ) = 1 π T (0, µ) (S(0, µ) 2 + T (0, µ) 2 ) sin 2 α + S(0, µ) sin 2α + cos 2 α (1.1.6) where S(, µ) and T (, µ) are as defined above. A key feature is that the S and T functions are defined in terms of the potential q and can be identified with the real and imaginary parts, respectively, of m 0 (µ), the Titchmarsh-Weyl function for α = 0. Theorem 1.1 holds, for eample, if q is either decreasing on [0, ) and a member of L 1 [0, ) or if (1 + t)q(t) L 1 [0, ). In Chapter 2, this theorem is etended to include a different class of functions q. Specifically, the spectral function is proved to have the form (1.1.6) for q which are conditionally integrable in a sense that includes the Wigner-von Neumann potentials.

11 4 In Chapter 5, with suitable conditions on the coefficient functions, a representation for spectral functions associated with the Dirac equation (1.1.3) is established. This representation has the same dependence on m 0 (µ) as (1.1.6) where the real and imaginary parts of m 0 (µ) are given in terms of the coefficient functions of the Dirac equation. The spectral functions for the Sturm-Liouville and Dirac equations, however, have distinctly different asymptotic behaviors. The other central question addressed is the relationships among spectral functions for distinct initial conditions in a fied equation. Gilbert and Harris answer this question for (1.1.1) in [7]. In Chapter 3, their result is refined for special cases of the equation. In Chapters 4 and 6, connection formulae for the Dirac equation are established. There are surprising similarities and differences in the connection formulae for the two equations. 1.2 Necessary Background Information It is convenient to denote particular solutions of the Sturm-Liouville and Dirac equations as follows. We let ϕ α (, λ) and θ α (, λ) be solutions of (1.1.1) satisfying θ α (0, λ) = cos α ϕ α (0, λ) = sin α p(0)θ α(0, λ) = sin α p(0)ϕ α(0, λ) = cos α (1.2.1) or let them be solutions of (1.1.3) satisfying θ α (0, λ) = cos α sin α ϕ α (0, λ) = sin α cos α (1.2.2) where α [0, π). ϕ α (, λ) is a regular solution of the initial value problem as it satisfies the initial condition (1.1.2) or (1.1.4). Since θ α and ϕ α are linearly independent

12 5 (their Wronskian is nonzero), every solution of the Sturm-Liouville or Dirac equation can be written as a linear combination of them. It is well known that (1.1.1) and (1.1.3) fall into eactly one of two types [3, 12, 17]. If, for any λ C, every solution of the differential equation is a member of L 2 [0, ), then the equation is called limit circle at infinity. Otherwise, the equation is called limit point at infinity. This classification is independent of λ. Here we are concerned with limit point equations eclusively. In this case it is further known that for each nonreal λ, there is eactly one solution, up to constant multiples, belonging to L 2 [0, ). This solution is denoted ψ α (, λ). The Titchmarsh-Weyl function, m α (λ), is that function such that θ α (, λ) + m α (λ)ϕ α (, λ) = ψ α (, λ) L 2 [0, ). (1.2.3) The function m α (λ) is a Herglotz (or Nevanlinna-Pick) function, that is, it is analytic in the upper half-plane and maps the upper half-plane to itself. m α (λ) is also analytic in the lower half-plane with negative imaginary part. In general, these functions may not be analytic continuations of each other. Any pole of m α (λ) is simple and on the real ais. m α (λ) can be represented in the form of a Cauchy-Stieltjes integral involving the spectral function ρ α (µ). Additionally, m α (λ) and m β (λ) are related by the connection formula m β (λ) = m α(λ) cos(β α) sin(β α) m α (λ) sin(β α) + cos(β α). (1.2.4) We refer to [3, 6, 9, 12, 17, 18, 20]. Conversely, subject to normalization, the spectral function ρ α (µ) is uniquely determined by m α (λ). Since the spectral function is nondecreasing, it may be written

13 6 as the sum of nondecreasing functions ρ α = ρ (ac) α + ρ (sc) α + ρ (p) α where ρ (ac) α is absolutely continuous, ρ (sc) α is singularly continuous, and ρ (p) α is a step function. This decomposition is unique up to the addition of constants. The spectrum of a problem is defined as points of nonconstancy of the spectral function. More precisely, it is the complement of the set of points in a neighborhood of which ρ α is constant [6, 17]. It is necessarily a closed set. The absolutely continuous spectrum consists of the nonconstancy points of ρ (ac) α ; the singularly continuous and point spectra are defined similarly. Formulae connecting ρ α (λ) and m α (z) are 1 ν ρ α (ν) ρ α (µ) = lim Im{m ɛ 0 + α ( + iɛ)} d (1.2.5) π µ for all µ, ν R which are points of continuity of ρ α (λ) and m α (z) = 1 λ z dρ α(λ) + cot α (1.2.6) which holds for all α (0, π). This results of this work concern the absolutely continuous spectra associated with equations (1.1.1) and (1.1.3). The following version of (1.2.5) referred to as the Titchmarsh-Kodaira formula supplies the connection between m α (λ) and ρ α (µ) : ρ α(µ) = 1 π lim ɛ 0 + Im{m α(µ + iɛ)} (1.2.7) where the limit eists. See [3, 12, 17].

14 7 Gilbert and Harris [7] prove the following results for Sturm-Liouville limit point equations (1.1.1). Theorem 1.2 For almost all µ R, if there is an α [0, π) such that 0 < ρ α(µ) <, then we have the following. (i) ρ β(µ) eists, with 0 < ρ β(µ) < for all β [0, π). (ii) For all α 1, α 2 (0, π)\{π/2}, { } 2ρ 1 1 0(µ) ρ α 1 (µ) sin 2α 1 ρ α 2 (µ) sin 2α 2 = ρ 0(µ) ρ π/2 (µ) {tan α 1 tan α 2 } + cot α 1 cot α 2. (1.2.8) Theorem 1.3 For almost all µ R, if there is an α [0, π) such that 0 < ρ α(µ) <, and if α, α + π/2 and β are distinct members of [0, π), then ρ α(µ) ρ α+π/2 (µ) (πρ α(µ)) 2 = 1 sin 2 (2β 2α) ρ 2 α(µ) ρ β (µ) ρ α(µ) ρ α+π/2 (µ) sin2 (β α) cos 2 (β α). (1.2.9) Eastham [4] rewrites (1.2.8) for arbitrary α, β, γ, δ [0, π): sin(β γ) sin(γ δ) sin(δ β) ρ α(µ) sin(γ δ) sin(δ α) sin(α γ) ρ β (µ) + sin(δ α) sin(α β) sin(β δ) ρ γ(µ) sin(α β) sin(β γ) sin(γ α) ρ δ (µ) = 0. These theorems display the relations among the spectral derivatives. For this general setting, knowledge of ρ α(µ) for three distinct α is sufficient to uniquely determine any fourth spectral derivative. Also, (1.2.9) relates the spectral derivatives

15 8 for just three distinct initial conditions. In Chapter 3, this result is written for three arbitrary initial conditions and, for special cases of (1.1.1), a third spectral derivative is uniquely determined. We fi notation. For the comple number z = re iθ, Arg{z} represents the argument of z where θ [0, 2π) and z is taken as re iθ/ The Operator Viewpoint Spectral functions can also be approached from the point of view of the general spectral theory of linear selfadjoint operators in a Hilbert space with scalar product (f, g). Consider a set D T of elements of a Hilbert space H. If to each f D T there is associated some element T f H, then T is called an operator in H with domain D T. T is a linear operator if T satisfies T (αf + βg) = αt f + βt g for any α, β C and any f, g D T. An operator T is closed if from the eistence of the limits lim f n = f, n n lim T f n = g, with f n D T, it follows that f D T and g = T f. Now there are pairs of elements g, g for which (T f, g) = (f, g ) holds for every f D T. If D T is dense in H, then the element g is uniquely determined by the element g. T, the adjoint of T, is that operator with g = T g. An operator T is selfadjoint if T = T, that is, D T = D T and T f = T f for all f D T. A few of the many properties of selfadjoint operators are as follows. First, if a selfadjoint operator has an inverse, then this inverse is selfadjoint. The eigenvalues of a selfadjoint operator are real. Also, two eigenvectors f 1, f 2 of a selfadjoint operator corresponding to distinct eigenvalues λ 1, λ 2 are orthogonal. Let T be a closed linear operator defined on D T, a dense subset of H. If T λi

16 9 is invertible (that is, the operator (T λi) 1 eists, is defined everywhere in H, and is bounded), then λ is said to be in the resolvent set of T and is known as a regular point of the operator T. Otherwise, λ is in the spectrum of T, denoted σ(t ). For T a selfadjoint operator, σ(t ) R. The spectrum can be decomposed into the point spectrum, the singularly continuous spectrum, and the absolutely continuous spectrum. For the Sturm-Liouville case, we consider the differential epression L defined by Ly = 1 w ( (py ) + qy). Then equation (1.1.1) is equivalent to Ly = λy. The relevant Hilbert space is L 2 [0, ; w) = { f 0 } f() 2 w() d < with scalar product (f(), g()) = 0 f()g()w() d. The operator H α is defined by H α y = Ly for y D where D is the set of functions y satisfying i. y, py L 1 loc[0, ) ii. y, H α y L 2 [0, ; w) iii y(0) cos α + y (0) sin α = 0 for some fied α in [0, π).

17 10 If L is regular at 0 and limit point at infinity, then the operator H α is selfadjoint for each α in [0, π). The spectral theorem applies to selfadjoint operators. This implies that there eists an associated non-decreasing spectral function from which the spectral properties of the operator can be determined. A selfadjoint operator for the Dirac equation may be defined mutatis mutandis. We refer to [3, 6, 9, 12, 17, 21] and the references listed therein.

18 CHAPTER 2 SPECTRAL FUNCTIONS ASSOCIATED WITH STURM-LIOUVILLE EQUATIONS WITH POTENTIALS OF WIGNER-VON NEUMANN TYPE 2.1 Introduction Let ρ α (µ) denote the spectral function associated with the equation y + (λ q)y = 0 (2.1.1) on [0, ) with the initial condition y(0) cos α + y (0) sin α = 0 (2.1.2) where α [0, π) and q is real-valued. We suppose q is such that (2.1.1) is in the limit point case at infinity so that ρ α (µ) is unique, up to the addition of a constant function. As is usual, we normalize by taking ρ α (0) = 0. If q L 1 [0, ), the spectral function is known to be absolutely continuous for µ > 0, [20]. Under quite weak etra conditions on q, the spectral function is calculated in [8] and [11] in the form of a series. If q is conditionally integrable but not L 1 [0, ), the results of [8] and [11] do not apply. For certain of these q, however, Behncke [1, 2] shows that (2.1.1) is limit point at infinity and has absolutely continuous spectrum on the positive real line away from the set of resonance points. An eample of such a q is furnished by

19 12 the Wigner-von Neumann potential q(t) = A sin(2ct). (2.1.3) t + 1 The purpose of the present chapter is to etend the results of [11] to cover potentials of this type and thus to give a series representation for the spectral derivative ρ α(µ) for µ Λ 0, where Λ 0 is computable. In 2.5, we present ρ α(µ) in the case where q is given by (2.1.3). We note that asymptotic representations for spectral functions associated with (2.1.1), (2.1.2) are developed for the case when q L 1 [0, ) but is small at infinity, in [5] and the earlier papers cited therein. These representations require q L 1 [0, ) and do not appear to be applicable to the case when q is given by (2.1.3). In this work, the oscillation of the conditionally convergent q is utilized in order to represent the spectral functions. 2.2 The Results We write λ =: µ + iɛ and λ 1/2 =: γ + iδ where µ, ɛ, γ, δ R and the principal branch of the square root is used. We take µ, γ > 0 and δ 0 which implies that ɛ 0. We set v 1 (, λ) := v 2 (, λ) := e 2iλ1/2 (t ) q(t) dt (2.2.1) e 2iλ1/2 (t ) v 1 (t, λ) 2 dt (2.2.2) and for n 3, v n (, λ) := ( e 2iλ1/2 (t ) v n 1 (t, λ) 2 + 2v n 1 (t, λ) n 2 m=1 v m (t, λ) ) dt. (2.2.3)

20 13 We further write S(t, λ) + it (t, λ) := iλ 1/2 + v n (t, λ) (2.2.4) n=1 where S and T are real-valued. The conditions imposed on q throughout this chapter are referred to as conditions (Q) and are as follows. We suppose that there eist functions a, b, η 1, η 2 so that (i) v 1 (, λ) a()η 1 (λ), where η 1 (λ) 0 as λ and a() decreases to 0 as. (ii) v 2 (, λ) b()η 2 (λ), where η 2 (λ) 0 as λ and b() is a decreasing L 1 [0, ) function. (iii) There eists K > 0 so that a(t)b(t) dt K b() for 0. 4 (iv) v 1 (t, λ) dt < for 0 <. 0 Theorem 2.1 If q satisfies conditions (Q) and if Λ 0 R is so large that 9η 2 (λ) b(t) dt + Kη 1 (λ) 1 0 for all λ = µ + iɛ with µ Λ 0, then for all µ Λ 0 ρ 0(µ) = 1 T (0, µ) π and, for α (0, π), ρ α(µ) = 1 π T (0, µ) (S(0, µ) 2 + T (0, µ) 2 ) sin 2 α + S(0, µ) sin 2α + cos 2 α. We note here that the proof will show 0 < ρ α(µ) < for all α [0, π) and µ Λ 0 and that S(µ) 0 and T (µ) µ 1/2 as µ. The theorem is proved in 2.4 after the groundwork is laid in 2.3. An immediate consequence is as follows.

21 14 Corollary 2.2 Λ 0 is an upper bound for the set of resonances. Proof. A resonance point is an eigenvalue embedded in the continuous spectrum and the spectral derivative does not eist as finite number at such a point. But since 0 < ρ α(µ) < for all α [0, π) and µ Λ 0, it follows that Λ 0 is an upper bound for the set of resonances. Q.E.D. The following representation of v 2 (, λ) is useful in applications of the theorem and will be proved in 2.4. It is also possible to represent each v n (, λ) in an analogous manner. v 2 (, λ) v 1(, λ) e 2iλ1/2 (t ) v 2iλ 1/2 iλ 1/2 1 (t, λ)q(t) dt, 0. (2.2.5) 2.3 The Riccati Equation Equation (2.1.1) is assumed to be in the limit point case at infinity which implies that for Im{λ} > 0, there is a L 2 [0, ) solution ψ(, λ), unique up to constant multiples. We denote by θ α (, λ) and ϕ α (, λ) the solutions of (2.1.1) satisfying the initial conditions θ α (0, λ) = cos α ϕ α (0, λ) = sin α θ α(0, λ) = sin α ϕ α(0, λ) = cos α. (2.3.1) These solutions, together with ψ(, λ), enable us to define the Titchmarsh-Weyl m-function by ψ(, λ) = θ α (, λ) + m α (λ)ϕ α (, λ). (2.3.2) It is known, [3, 12, 20], that m α (λ) is well defined, nonzero, and analytic for Im{λ} > 0. Further, if Im{λ} > 0, ψ(, λ) 0 for any [0, ); this follows from the limit

22 15 point construction (see, for eample, [3, chapter 9]). The ratio ψ (, λ)/ψ(, λ) is unique and by equating representations of ψ (0, λ)/ψ(0, λ) for distinct values of the initial value parameter, we obtain the connection formula m α (λ) = m β(λ) cos(α β) sin(α β) m β (λ) sin(α β) + cos(α β). (2.3.3) The spectral derivative, ρ α(µ), may be obtained from the Titchmarsh-Kodaira formula (1.2.7) wherever lim ɛ 0 m α (µ + iɛ) eists. denote Let y(, λ) denote a solution of (2.1.1) which does not vanish for [0, ) and v(, λ) := y (, λ) y(, λ). It may readily be shown that v(, λ) satisfies the Riccati equation v = λ + q v 2 (2.3.4) for [0, ). We seek a solution of (2.3.4) of the form v(, λ) = iλ 1/2 + v n (, λ) (2.3.5) n=1 where lim v(, λ) = iλ 1/2. Following the analysis of [8] we substitute (2.3.5) into (2.3.4) and obtain n 1 v (, λ) = (v n + 2iλ 1/2 v n ) = q v1 2 (v 2 n + 2v n v m ). (2.3.6) n=1 n=2 The sum may be decomposed as m=1 v 1 + 2iλ 1/2 v 1 = q (2.3.7) v 2 + 2iλ 1/2 v 2 = v 2 1 (2.3.8)

23 v n + 2iλ 1/2 v n = (v 2n 1 + 2v n 1 n 2 ) v m m=1 16, n 3. (2.3.9) These equations are satisfied by the {v n (, λ)} of (2.2.1) (2.2.3). The term-by-term differentiation and rearrangement of v(, λ) are justified by the absolute uniform convergence of (2.3.5) which is proved in the following lemma. Lemma 2.3 If q satisfies the conditions (Q) for 0 < and Im{λ} 0 then there eists Λ 0 R such that for all λ = µ + iɛ with µ Λ 0 and n 2 v n (, λ) η 2(λ)b() 2 n 2 for 0 <. Proof. We use induction on n. The result is clear for n = 2. Suppose it were true for n = 2,..., k; then v k+1 (, λ) η 2(λ) 2 k 1 ( η 2(λ)b() 2 k 1 v k v k v v k k 1 m=2 v m ( η2 (λ) 2 b(t) 2 + 2η 2(λ)η 1 (λ)a(t)b(t) + 2η 2(λ) 2 b(t) 2 2 2k 4 2 k 2 2 k 2 ) dt m=2 [ ( 4η 1 (λ) a(t)b(t) dt ) ] η 2 k 3 2 (λ) b(t) 2 dt [ ] Kη 1 (λ) + 9η 2 (λ) b(t) dt. 0 ) 1 dt 2 m 2 The result now follows if Λ 0 is so large that the bracketed term is less than or equal to 1 for all λ with µ Λ 0. Q.E.D. Lemma 2.3 implies that v n (, λ) is uniformly absolutely convergent and hence n=1 that v(, λ) = iλ 1/2 + v n (, λ) is continuous in and λ for Re{λ} Λ 0, Im{λ} n=1

24 17 0, and [0, ). Also, from (2.3.9) for n 3, v n = 2iλ 1/2 v n v 2 n 1 2v n 1 v m n=1 so v n 2 λ1/2 η 2 (λ)b() + η 2(λ) 2 b() 2 + 4η 2(λ) 2 b() 2 + 2η 1(λ)η 2 (λ)a()b() 2 n 2 2 2n 6 2 n 3 2 n 3 [ ] η 2(λ)b() 2 n 3 λ 1/2 + η 2(λ)b() 2 n 3 + 4η 2 (λ)b() + 2η 1 (λ)a(). For each λ, the epression in brackets is bounded. A similar calculation shows v 1, v 2 are bounded for each λ. Hence v(, λ) is differentiable and therefore indeed represents a solution to (2.3.4) for [0, ), Im{λ} 0 and Re{λ} Λ 0 where Λ 0 is computed as in Lemma Proof of Theorem 2.1 We show first that v(, λ), the solution of (2.3.5) with lim v(, λ) = iλ 1/2, is equal to ψ (, λ) ψ(, λ) where ψ(, λ) is the L2 [0, ) solution of (2.1.1) for Im{λ} > 0. It is clear that a solution of (2.1.1) which does not vanish for [0, ) is represented by { } (constant) ep v(t, λ) dt. 0 To prove that this solution is in L 2 [0, ), we show that for Im{λ} > 0 there eists X with { } Re v(t, λ) dt < δ 0 2 for X

25 18 where, in the notation of 2.2, δ = Im{λ 1/2 }. We note from (2.3.5) and Lemma 2.3 that { } Re v(t, λ) dt 0 { } { } = δ + Re v 1 (t, λ) dt + Re v n (t, λ) dt 0 0 n=2 δ + v 1 (t, λ) dt + v n (t, λ) dt 0 n=2 0 δ + v 1 (t, λ) dt + 2η 2 (λ) b(t) dt. 0 0 If X is so large that sup [X, ) 0 v 1 (t, λ) dt + 2η 2 (λ) 0 b(t) dt < δ 2, (2.4.1) then the result follows. Such an X must eist by conditions (Q) since the left-hand side of (2.4.1) is bounded while, for any δ > 0, the right hand side goes to infinity with. Since the L 2 [0, ) solution is unique (up to constant multiples), we conclude { } ψ(, λ) = (constant) ep v(t, λ) dt, and hence that v(, λ) = ψ (, λ)/ψ(, λ). For α = 0, (2.3.1) and (2.3.2) show that 0 v(0, λ) = ψ (0, λ) ψ(0, λ) = m 0(λ) for Im(λ) > 0, µ Λ 0. The solution v(, λ), and v(0, λ) in particular, is defined for Im(λ) = 0, µ Λ 0. Thus lim ɛ 0 + m 0(µ + iɛ) eists and the Titchmarsh-Kodaira formula (1.2.7) and (2.2.4) yield Equation (2.3.3) with β = 0 now gives ρ 0(µ) = 1 π lim Im{m 0(µ + iɛ)} = 1 T (0, µ). ɛ 0 + π ρ α(µ) = 1 π T (0, µ) (S(0, µ) 2 + T (0, µ) 2 ) sin 2 α + S(0, µ) sin(2α) + cos 2 α

26 19 and the theorem is proved. Q.E.D. To verify (2.2.5), the alternate representation of v 2 (, λ), we use integration by parts together with (2.2.2) and (2.3.8) to see that v 2 (, λ) = v 1(t, λ) 2 e 2iλ1/2 (t ) 2iλ 1/2 1 e 2iλ1/2 (t ) v iλ 1/2 1 (t, λ)v 1(t, λ) dt = v 1(, λ) 2 1 e 2iλ1/2 (t ) v 2iλ 1/2 iλ 1/2 1 (t, λ)(q(t) 2iλ 1/2 v 1 (t, λ)) dt = v 1(, λ) 2 + 2v 2iλ 1/2 2 (, λ) 1 e 2iλ1/2 (t ) v iλ 1/2 1 (t, λ)q(t) dt. The result now follows. A similar technique may be used to ehibit corresponding formulae for the other {v n (, λ)}. 2.5 An Eample: The Wigner-von Neumann potential We show now that q(t) = A sin(2ct), c > 0, A R (2.5.1) t + 1 satisfies the hypotheses of Theorem 2.1. The net lemma is useful in integrations involving this q. Lemma 2.4 If β > 0, k R where k and λ 1/2 are not both 0, and Im{λ} 0, then e 2iλ1/2 (t ) (t + 1) β e ikt dt = ( + 1) β e ik i(2λ 1/2 + k) + E(, λ, β, k) where E(, λ, β, k) 2β( + 1) β 1 (2 λ 1/2 k ) 2.

27 20 Proof. This follows readily after integration by parts. Q.E.D. Note also that For q as in (2.5.1), v 1 (, λ) is written v 1 (, λ) = A 2i E(, λ, β, k) = E(, λ, β, k) (2.5.2) We apply Lemma 2.4 and (2.5.2) to (2.5.3) which gives e 2iλ1/2 (t ) (t + 1) 1 (e 2ict e 2ict ) dt. (2.5.3) ( ) A( + 1) 1 e 2ic v 1 (, λ) = 4 λ 1/2 + c e 2ic λ 1/2 c A 2i( E(, λ, 1, 2c) E(, λ, 1, 2c) ). (2.5.4) Thus, v 1 (, λ) A ( + 1) 1 4 ( 2 ) λ 1/2 c + A 2 ( + 1) 2 ( λ 1/2 c) 2 A ( + 1) 1 λ 1/2 c and we take a() := A ( + 1) 1, η 1 (λ) := ( λ 1/2 c) 1. Condition (i) of (Q) is satisfied. Integration by parts also shows that v 1 (t, λ) dt < for 0 < 0 and so condition (iv) of (Q) is satisfied. To verify that v 2 (, λ) satisfies condition (ii) of (Q) we use (2.5.4) in (2.2.5). The first term of (2.2.5) is now v 1 (, λ) 2 2iλ 1/2 = A2 ( + 1) 2 32iλ 1/2 + A( + 1) 1 E 1 (, λ) 4iλ 1/2 ( e 4ic (λ 1/2 + c) + e 4ic 2 (λ 1/2 c) 2 ) 2 λ c 2 ( ) e 2ic λ 1/2 + c e 2ic + E 1(, λ) 2 λ 1/2 c 2iλ 1/2 (2.5.5)

28 21 where E 1 (, λ) := A 2i( E(, λ, 1, 2c) E(, λ, 1, 2c) ) thus A ( + 1) 2 E 1 (, λ) 2( λ 1/2 c). 2 The second term of (2.2.5) is 1 e 2iλ1/2 (t ) v iλ 1/2 1 (t, λ)q(t) dt = A 2 8λ 1/2 e 2iλ1/2 (t ) (t + 1) 2 ( e 4ict λ 1/2 + c + ) e 4ict λ 1/2 c 1 λ 1/2 + c 1 dt λ 1/2 c + A e 2iλ1/2 (t ) (t + 1) ( 1 e 2ict e 2ict) E 2λ 1/2 1 (t, λ) dt. Some calculation then shows that v 2 (, λ) A 2 ( + 1) 2 2 λ 1/2 ( λ 1/2 2c) + 7 A 2 ( + 1) 3 8 λ 1/2 ( λ 1/2 c)( λ 1/2 2c) 2 2 A 2 ( + 1) 2 λ 1/2 ( λ 1/2 2c) 2 (2.5.6) We have supposed, in addition to the considerations of Theorem 2.1, that λ 1/2 c > 1 and λ 1/2 > 2c. It follows from (2.5.6) that v 2 (, λ) b()η 2 (λ) where b() := 2 A 2 ( + 1) 2 and η 2 (λ) := 1 λ 1/2 ( λ 1/2 2c) 2. We note that condition (iii) of (Q) is met since a(t)b(t) dt = 2 A 3 (t + 1) 3 dt = A 3 ( + 1) 2 = A 2 b(). Finally Λ 0 can be taken to be the maimum of (18A A + c) 2 and (2c + 1) 2. This estimate is rather crude. For eample, if A = c = 1, then Λ 0 = However, using

29 22 the values of A and c in the calculations above, Λ Note that in this case, λ = 1 is a resonance point and ρ α(1) does not eist. (The integral for v 1 (, λ) is divergent for λ 1/2 = c = 1.) But we have shown that the hypotheses of Theorem 2.1 are fulfilled by the Wigner-von Neumann potential (2.5.1) for µ > 23 and hence the representation of the spectral function derivative follows.

30 CHAPTER 3 A REFINEMENT OF THE STURM-LIOUVILLE CONNECTION FORMULAE 3.1 Introduction In this chapter we consider connection formulae for the spectral functions ρ α (µ) for µ R associated with the Sturm-Liouville equation (py ) + qy = λwy (3.1.1) on [0, ) together with the boundary condition y(0) cos α + p(0)y (0) sin α = 0 (3.1.2) where α [0, π). The spectral parameter is written λ = µ + iɛ, with µ, ɛ R. Two variations of the problem are considered. (I) Equation (3.1.1) is in the limit-point case at infinity and the coefficient functions p, q and w satisfy i. p, q, w are real valued on [0, ), ii. w() > 0 and p() > 0 for [0, ), iii. 1/p, q, w L 1 loc[0, ). (II) The real valued functions p, q and w in (3.1.1) are such that in addition to the requirements of (I),

31 24 i. there is a Λ 0 R such that for all µ Λ 0, the spectral derivative ρ α(µ) is continuous and 0 < ρ α(µ) < for α [0, π) ii. there eist real valued functions S(µ), T (µ) such that for µ Λ 0, ρ α(µ) = 1 π T (µ) (S(µ) 2 + T (µ) 2 ) sin 2 α + S(µ) sin 2α + cos 2 α (3.1.3) with S(µ) 0 and T (µ) µ 1/2 as µ. Condition (II) is satisfied for special cases of the Sturm-Liouville equation with p w 1 and, for eample, either of the following conditions on q. a. q L 1 [0, ) and there eists a Λ 0 R and functions a() and η(µ) such that ep(2iµ 1/2 t)q(t) dt a()η(µ) for µ Λ 0 and 0 < where a( ) is a decreasing L 1 [0, ) function and η(µ) 0 as µ. See [7, 9, 11]. b. Conditions (Q) of 2.2. Condition (I) places minimal restrictions on (3.1.1) and the results of Gilbert and Harris [7] and Eastham [4] apply. It is known that the spectral derivatives for three initial conditions α, α + π/2, and β are related and that given the spectral derivatives for two initial conditions, the third derivative must satisfy a quadratic equation. Here, this result is given in greater generality and a corollary is proved. The primary result of this chapter, Theorem 3.3, states that when condition (II) is satisfied, a third derivative is uniquely determined in most cases. Theorem 3.1 For the problem (I) and almost all µ R, if there is an α [0, π) such that 0 < ρ α(µ) < and if α, β, and γ are distinct members of [0, π), then

32 25 ρ β(µ), ρ γ(µ) eist with 0 < ρ β(µ) <, 0 < ρ γ(µ) < and [ sin 2 (β α) ρ γ(µ) sin2 (β γ) ρ α(µ) ] 2 sin2 (γ α) ρ β (µ) = ( ) 4 sin 2 (β γ) sin 2 1 (γ α) ρ α(µ)ρ β (µ) π2 sin 2 (β α). (3.1.4) Thus, sin2 (β α) ρ γ(µ) must be one of the two choices sin 2 (β γ) ρ α(µ) + sin2 (γ α) 1 ρ β (µ) ± 2 sin (β γ) sin (γ α) ρ α(µ)ρ β (µ) π2 sin 2 (β α). (3.1.5) Equation (3.1.4) is a more general form of (1.2.9), the connection formula for three initial conditions given by Gilbert and Harris in [7]. A corollary is as follows. Corollary 3.2 For the problem (I), if ρ α(µ), ρ β(µ) eist as finite positive numbers for distinct α, β [0, π), then ρ α(µ)ρ β(µ) 1 π 2 sin 2 (β α) (3.1.6) For the problem (II), this result is true for all µ Λ 0. The cases of the following theorem arise depending on whether the spectral derivative for the initial condition 0 is known. There are three cases: α, β are both nonzero, one of α, β is zero and the other is in (0, π/2), and one is zero and the other is in (π/2, π). Since α, β play interchangeable roles, we assume α < β. Theorem 3.3 For the problem (II) and µ Λ 0,

33 26 (i) if α, β (0, π) and γ [0, π) or if α = 0, β (0, π/2) and γ (0, π) then sin 2 (β α) ρ γ(µ) = sin2 (β γ) ρ α(µ) + sin2 (γ α) ρ β (µ) sin (β γ) sin (γ α) ρ α(µ)ρ β (µ) π2 sin 2 (β α). (3.1.7) (ii) if α = 0, β (π/2, π) and γ (0, π) then sin 2 (β α) ρ γ(µ) = sin2 (β γ) ρ α(µ) + sin2 (γ α) ρ β (µ) 1 2 sin (β γ) sin (γ α) ρ α(µ)ρ β (µ) π2 sin 2 (β α). (3.1.8) We use the principal branch of the square root function. Theorem 3.3 omits the case where α = 0 and β = π/2. In this situation, additional information on the sign of S(µ) is required before ρ γ(µ) can be determined uniquely, as will be made clear in the proof. The results are proved in 3.2 and eamples are given in Proofs of the Connection Formulae Proof of Theorem 3.1. For almost all µ R for which ρ α(µ) eists with 0 < ρ α(µ) <, m α (µ + iɛ) converges to a finite non-real limit as ɛ 0 + and satisfies (1.2.7), where m α (λ) is the Titchmarsh-Weyl function for λ in the upper half-plane. For such µ, let X α (µ) = lim ɛ 0 + Re{m α(µ + iɛ)}. Then, m α (µ) = lim ɛ 0 + m α(µ + iɛ) = X α (µ) + iπρ α(µ) (3.2.1) where the imaginary part of (3.2.1) follows from (1.2.7). Since lim ɛ 0 + m α (µ) eists as a finite non-real limit if and only if the same is true of lim ɛ 0 + m β (µ) (see [7],

34 27 Lemma 1.1), it follows from (1.2.7) that for µ satisfying (3.2.1), ρ β(µ) also eists with 0 < ρ β(µ) < for all β [0, π). Hence by [7, Equation (5.6)] we have ρ β(µ) ρ α(µ) 1 = lim ɛ 0 + m α (µ + iɛ) sin (β α) + cos (β α) 2 = 1 X α (µ) sin(β α) + cos(β α) + iπρ α(µ) sin(β α) 2 So ρ α(µ) ρ β (µ) = (X α(µ) sin(β α) + cos(β α)) 2 + π 2 ρ α(µ) 2 sin 2 (β α) (3.2.2) and 0 = X α (µ) 2 sin 2 (β α) + X α (µ) sin (2β 2α) + cos 2 (β α) + π 2 ρ α(µ) 2 sin 2 (β α) ρ α(µ) ρ β (µ) The last equation is quadratic in X α (µ), so X α (µ) is one of cos (β α) ± ρ α (µ) ρ β (µ) π2 ρ α(µ) 2 sin 2 (β α). (3.2.3) sin(β α) Equation (3.2.2) is also valid when β is replaced by γ, hence, using (3.2.3) for X α (µ), the two choices for ρ α(µ) ρ γ(µ) are cos(β α) ± sin(γ α) ρ α(µ) ρ β (µ) π2 ρ α(µ) 2 sin 2 (β α) sin(β α) + cos(γ α) 2 + π 2 ρ α(µ) 2 sin 2 (γ α).

35 28 This is equivalent to ρ α(µ) ρ γ(µ) = ρ sin(β γ) ± sin(γ α) α (µ) ρ β (µ) π2 sin 2 2 (β α) sin(β α) + π 2 ρ α(µ) 2 sin 2 (γ α) and the formulae (3.1.4) (3.1.5) follow upon rearranging. Q.E.D. Proof of Corollary 3.2. By Theorem 1.2, if 0 < ρ α(µ) < and 0 < ρ β(µ) < for distinct α, β [0, π), then 0 < ρ γ(µ) < for all γ [0, π). Hence, (3.1.5) must give real numbers. That is, 1 ρ α(µ)ρ β (µ) π2 sin 2 (β α) 0 (3.2.4) from which the result follows. Note that for the problem (II), the inequality holds for all µ Λ 0 since the spectral derivatives are assumed finite and positive for these µ and all α [0, π). Q.E.D. Proof of Theorem 3.3. In the case of (II), we can write ρ α(µ), ρ β(µ) in terms of S(µ), T (µ) using (3.1.3). The radicand in (3.1.5) is a perfect square: 1 ρ αρ β π 2 sin 2 (β α) = π T (S 2 + T 2 ) sin α sin β + S sin(α + β) + cos α cos β. (3.2.5) We will see that correctly selecting the branch of the square root depends on whether the quantity within absolute value in (3.2.5) is negative or nonnegative. Temporarily,

36 29 let us suppose it is nonnegative. Then the positive branch of (3.1.5) can be written π T { (S 2 + T 2 )(sin α sin(β γ) + sin β sin(γ α)) 2 + S(sin 2 (β γ) sin 2α + sin 2 (γ α) sin 2β + 2 sin(β γ) sin(γ α) sin(α + β)) + (cos α sin(β γ) + cos β sin(γ α)) 2}. Trigonometric identities are used to simplify this epression. The coefficient of (S 2 + T 2 ) becomes (sin α sin(β γ) + sin β sin(γ α)) 2 = (sin α(sin β cos γ cos β sin γ) + sin β(sin γ cos α cos γ sin α)) 2 = (sin γ(sin β cos α cos α cos β)) 2 = sin 2 γ sin 2 (β α). The constant term is simplified in a similar way and the coefficient of S becomes sin 2 (β γ) sin 2α + sin 2 (γ α) sin 2β + 2 sin(β γ) sin(γ α) sin(α + β) = 1 sin(β γ) [cos(2α β + γ) cos(γ + β)] 2 1 sin(γ α) [cos(2β + γ α) cos(γ + α)] 2 = sin(β γ) sin(α + γ) sin(β α) + sin(γ α) sin(β + γ) sin(β α) [ 1 = sin(β α) 2 cos(β 2γ α) 1 ] 2 cos(β + 2γ α) = sin 2 (β α) sin 2γ

37 30 That is, the positive branch of (3.1.5) simplifies to π T { (S 2 + T 2 ) sin 2 (β α) sin 2 γ + S sin 2 (β α) sin 2γ + sin 2 (β α) cos 2 γ } which is identically sin2 (β α) by (3.1.3). Therefore, when the quantity within ρ γ(µ) absolute value in (3.2.5) is nonnegative, the positive branch yields the correct value for the spectral derivative. In a similar way, if the epression within absolute value is negative, then the negative branch of (3.1.5) will simplify as above and is the correct choice. We now use the fact that ρ γ(µ) is continuous for µ Λ 0. This implies that the same branch of the square root must be used for each µ Λ 0, once α, β are fied. That is, the sign of the quantity within absolute value in (3.2.5) can not change for µ Λ 0. It is clear that for α, β 0 and large µ, this quantity is positive since S(µ) 0 and T (µ). Thus the positive branch gives the correct value. On the other hand, if α = 0, the quantity within absolute value tends to cos β and so the positive or negative branch of (3.1.5) is chosen depending on the quadrant in which β lies. This establishes formulae (3.1.7) and (3.1.8). Note if α = 0, and β = π/2, then the epression in (3.2.5) is π S. This tends to 0, but it is not clear whether T the approach is from above or below. Hence information on the sign of S is required before the branch can be selected. Q.E.D. 3.3 Eamples For p w 1 and q = 0 in (3.1.1), ρ α(µ) may be computed directly and ρ α(µ) = 1 π µ µ sin 2, α + cos 2 α µ > 0. (3.3.1)

38 31 Here S(µ) 0 and T (µ) = µ. To illustrate Theorem 3.3, we set α = π/3 and β = π/2. Then ρ α(µ) = 1 4 µ π 3µ + 1 and ρ β(µ) = 1 1. Since α and β are both π µ nonzero, the positive branch of the square root function must be used. For γ = 0, (3.1.7) gives 1/4 ρ 0(µ) = π = [ 6µ µ + (2)(1) π 4 µ. ( ) ] 3 3µ 2 2 µ Hence, ρ 0(µ) =, which agrees with (3.3.1). The value obtained from (3.1.8), π using the negative branch of the square root function, would indicate (incorrectly) that ρ 0(µ) is 1 µ. This rejected value does not have the correct asymptotic π 12µ + 1 behavior as µ. If γ = 5π/6, then (3.1.7) gives [ 1/4 25µ/4 + 3/4 ρ 5π/6 (µ) = π 4 + (2) µ µ/4 + 3/4 = π 4. µ ( ) ] 3 3µ (1) 2 2 Hence, ρ 5π/6 (µ) = 1 4 µ π µ + 3. The rejected value is 1 4 µ and although it has the π 49µ + 3 same asymptotic behavior as the correct value, it can not be written in the form of (3.1.3). Thus to uniquely determine a spectral derivative for a third initial condition, it is not sufficient to know only the asymptotic behavior of ρ γ(µ) rather, knowledge of the asymptotic behavior of the component functions S and T in (3.1.3) is required. To illustrate Theorem 3.3(ii), let α = 0, β = 5π/6, and γ = π/3. Then (3.1.8)

39 32 gives Hence, ρ π/3 (µ) = 1 π [ 1/4 3µ/4 + 25/4 ρ π/3 (µ) = π 4 (2)(1) µ 4 µ 3µ + 1. = π 3µ/4 + 1/4 4. µ ( 3 2 ) ] 3 4µ Finally, consider the inequality in Corollary 3.2. By (3.1.3), ρ π/2 (µ) = T (µ) π(s(µ) 2 + T (µ) 2 ) and ρ 0(µ) = T (µ). Hence, for α = 0, β = π/2, the bound of π (3.1.6) is attained when S(µ) = 0. This occurs, for eample, when p w 1, q = 0. Note that in the case of (II) for α, β 0, the left side of (3.1.6) tends to 0 as µ while the right side is at least 1/π 2. The inequality is more significant when α = 0 for then the limit of the left side is not 0.

40 CHAPTER 4 CONNECTION FORMULAE FOR SPECTRAL FUNCTIONS ASSOCIATED WITH SINGULAR DIRAC EQUATIONS 4.1 Introduction In this chapter, we consider connection formulae for spectral functions ρ α(µ), µ R, associated with the Dirac equation given by y = p λ + c + v 1 (λ c + v 2 ) p on [0, ) together with the initial condition y, y = y 1 y 2 (4.1.1) y 1 (0) cos α + y 2 (0) sin α = 0 (4.1.2) where α [0, π). Also, c 0 is a constant, λ = µ + iɛ is the comple spectral parameter, and v 1, v 2, and p are real valued members of L 1 [0, ). To each parameter α [0, ), one may define θ α and ϕ α as solutions of (4.1.1) that satisfy for all λ θ α (0, λ) = cos α sin α ϕ α (0, λ) = Then the Titchmarsh-Weyl m α (λ) function is defined by sin α cos α. (4.1.3) ψ α (, λ) = θ α (, λ) + m α (λ)ϕ α (, λ) L 2 [0, ). (4.1.4)

41 34 We assume (4.1.1) is in the limit point case at infinity, which implies that m α (λ) is well defined and unique for Im{λ} > 0. Also, in the limit point case, the L 2 [0, ) solution of (4.1.1) is unique up to constant multiples; thus it follows that ψ α (, λ) and ψ β (, λ) are linearly dependent. Following Hille [12], the Wronskian of ψ α (0, λ) and ψ β (0, λ) is 0, that is, cos α m α (λ) sin α sin α + m α (λ) cos α cos β m β (λ) sin β sin β + m β (λ) cos β = 0. (4.1.5) This gives the m connection formula m β (λ) = m α(λ) cos(β α) sin(β α) m α (λ) sin(β α) + cos(β α). (4.1.6) The spectral functions may then be defined in terms of these m α (λ) functions through the Titchmarsh-Kodaira formula (1.2.7). The purpose of this chapter is to show how the spectral derivatives ρ α(µ) of (4.1.1) for distinct initial conditions are related. The assumptions are minimal: the equation must be in the limit point case at infinity and µ must be such that 0 < ρ α(µ) < for all α [0, ). Hinton and Shaw [14] prove these assumptions are met when, for eample, p, v 1, and v 2 are integrable and µ > c. It is notable that the connection formulae for (4.1.1) are superficially similar to those for the Sturm-Liouville equation (3.1.1) considered in Chapter 3 and by Gilbert and Harris in [7]. It will be seen in Chapter 5 that under additional assumptions on p, v 1, and v 2, the spectral derivatives of (4.1.1) all tend to 1 π as µ. Recall that in the Sturm-Liouville case, with appropriate restrictions on p, q, and w, ρ α(µ) π 1 µ ±1/2, depending on whether α = 0 or α (0, π). Despite the ρ α(µ) having distinctly different behaviors in the two cases, their connection formulae have the same structure.

42 35 The results are given in 4.2 and proved in 4.3. Eamples are given in The Connection Formulae Theorem 4.1 For almost all µ, if there is an α [0, π) such that ρ α(µ) eists with 0 < ρ α(µ) <, then (i) ρ β(µ) eists with 0 < ρ β(µ) < for all β [0, π). (ii) For distinct α, β, γ, δ [0, π), the spectral derivatives associated with (4.1.1) satisfy 0 = sin(β γ) sin(γ δ) sin(δ β) ρ α(µ) sin(γ δ) sin(δ α) sin(α γ) ρ β (µ) + sin(δ α) sin(α β) sin(β δ) ρ γ(µ) sin(α β) sin(β γ) sin(γ α) ρ δ (µ). (4.2.1) As in the Sturm-Liouville equation case, if ρ α(µ), ρ β(µ) eist and satisfy 0 < ρ α(µ) <, 0 < ρ β(µ) < for distinct α, β [0, π), then ρ γ(µ) eists and satisfies 0 < ρ γ(µ) <, for all γ [0, π). See [7, Remark 5.2]. Theorem 4.2 If there are distinct α, β [0, π) such that ρ α(µ), ρ β(µ) eist with 0 < ρ α(µ) <, 0 < ρ β(µ) <, then, for any γ [0, π), the spectral derivatives associated with (4.1.1) satisfy [ sin 2 (β α) ρ γ(µ) sin2 (β γ) ρ α(µ) ] 2 sin2 (γ α) ρ β (µ) = ( ) 4 sin 2 (β γ) sin 2 1 (γ α) ρ α(µ)ρ β (µ) π2 sin 2 (β α). (4.2.2)

43 36 Thus, sin2 (β α) ρ γ(µ) must be one of the two choices sin 2 (β γ) ρ α(µ) + sin2 (γ α) 1 ρ β (µ) ± 2 sin(β γ) sin(γ α) ρ α(µ)ρ β (µ) π2 sin 2 (β α). (4.2.3) Corollary 4.3 If 0 < ρ α(µ) <, 0 < ρ β(µ) < for distinct α, β [0, π), then ρ α(µ)ρ β(µ) 1 π 2 sin 2 (β α). (4.2.4) Theorem 4.1 is the analog of Theorem 2.1 in [7] in the form given by Eastham [4]. Similarly, Theorem 4.2 is comparable to Theorem 2.2 in [7] where there need be no special relationship among the three initial conditions. The analog to the corollary is Corollary 3.2 of Chapter 3. In [4], Eastham obtains several relationships among the spectral derivatives associated with Sturm-Liouville equation for special values of α, β, γ, δ. Analogous corollaries are valid for the Dirac equation (4.1.1) and are listed here. Corollary 4.4 Suppose ρ α(µ) eists and satisfies 0 < ρ α(µ) < for all α [0, π). Then does not depend on α. 1 ρ α(µ) 1 ρ α+π/2 (µ) Corollary 4.5 Suppose ρ α(µ) eists and satisfies 0 < ρ α(µ) < for all α [0, π). Then for any fied η, does not depend on α, (α 0, π/2). ( ) 1 ρ η+α(µ) 1 csc 2α ρ η α(µ)

44 37 We adopt the convention of using mod π values if the parameter falls outside the interval [0, π). As in [4], the proofs of these corollaries follow quickly from (4.2.1) with α + β = γ + δ. Spectral functions are related to the Titchmarsh-Weyl m α (λ) functions and, for different initial conditions, these m functions are connected by (1.2.4). Further, for special cases of the Sturm-Liouville and Dirac equations, it is possible to relate the spectral functions to solutions of a Riccati equation. (See [11] and Chapter 2 for the Sturm-Liouville case and Chapter 5 for the Dirac case.) Since it is known that the cross ratio of four solutions of a Riccati equation is constant, [15], it is reasonable to epect that the spectral functions are also connected. The results of [7], Chapter 3, and the present chapter bear out that prediction and also show the structure of the connection formulae for the two equations is similar. The degree of similarity is surprising given that the Riccati equations and the spectral functions for the two equations are very different. Additional insight might be gained by eamining these connection formulae from a geometric view. 4.3 Proofs Proof of Theorem 4.1. (i) The proof of this is the same as the proof of the corresponding statement in the Sturm-Liouville equation case; see [7, Theorem 2.1i]. The properties of Herglotz functions and (1.2.7) imply that for almost all µ R, if ρ α(µ) eists with 0 < ρ α(µ) <, then m α (µ + iɛ) converges to a finite nonreal limit as ɛ 0 + in which case by (4.1.6), m β (µ + iɛ) also converges to a finite nonreal limit as ɛ 0 +. It then follows from (1.2.7) that ρ β(µ) eists and has the required properties, for all β [0, π). (ii) m α (µ + iɛ) converges to a finite non-real limit as ɛ 0 + for almost all µ for

45 38 which ρ α(µ) eists with 0 < ρ α(µ) < for some α [0, π). For such a µ we denote m α (µ) = lim ɛ 0 + m α(µ + iɛ) = X α (µ) + iπρ α(µ) (4.3.1) where X α is real valued and the imaginary part follows from the Titchmarsh-Kodaira formula. Then by (1.2.7) and (4.1.6) πρ β(µ) = Im { (Xα (µ) + iπρ } α(µ)) cos(β α) sin(β α) (X α (µ) + iπρ α(µ)) sin(β α) + cos(β α) = = πρ α(µ) X α (µ) sin(β α) + cos(β α) + iπρ α(µ) sin(β α) 2 (4.3.2) πρ α(µ) (X α (µ) 2 + π 2 ρ α(µ) 2 ) sin 2 (β α) + X α (µ) sin 2(β α) + cos 2 (β α) Hence ρ α(µ) ρ β (µ) 1 = ( X α (µ) 2 + π 2 ρ α(µ) 2 1 ) sin 2 (β α) + X α (µ) sin 2(β α). (4.3.3) The coefficients of sin 2 (β α) and sin 2(β α) depend on α and µ, not β. So replacing β in turn by γ and δ in (4.3.3) gives three equations in two variables which must therefore be linearly dependent. Thus, det ρ α(µ) ρ β (µ) 1 sin2 (β α) sin 2(β α) ρ α(µ) ρ γ(µ) 1 sin2 (γ α) sin 2(γ α) ρ α (µ) ρ δ (µ) 1 sin2 (δ α) sin 2(δ α) = 0. (4.3.4) Formula (4.2.1) follows upon epanding about the first column and using trigonometric identities. Specifically, note that sin 2 (c a) sin 2(d a) sin 2 (d a) sin 2(c a)

46 39 = 1 (sin(2c 2a) + sin(2a 2d) + sin(2d 2c)) 2 = 2 sin(c a) sin(d a) sin(c d) and that 2 sin(c a) sin(d a) sin(c d) + 2 sin(b a) sin(d a) sin(b d) 2 sin(c d) sin(b d) sin(c b) = 2 sin(c d) sin(b d) sin(c b) This completes the proof of Theorem 4.1. Q.E.D. Proof of Theorem 4.2. As in the proof of Theorem 4.1, ρ α(µ), ρ β(µ) and X α (µ) = Re{m α (µ)} are related by equation (4.3.2). Rearranging yields an equation quadratic in X α : 0 = X α (µ) 2 sin 2 (β α) + X α (µ) sin (2β 2α) So X α (µ) is one of cos (β α) ± + cos 2 (β α) + π 2 ρ α(µ) 2 sin 2 (β α) ρ α(µ) ρ β (µ) ρ α(µ) ρ β (µ) π2 ρ α(µ) 2 sin 2 (β α). (4.3.5) sin(β α) Equation (4.3.2) remains valid when β is replaced by γ. Replacing X α (µ) by (4.3.5) in this epression and rearranging gives formulae (4.2.2) and (4.2.3). Q.E.D. Proof of Corollary 4.3. If 0 < ρ α(µ) <, 0 < ρ β(µ) <, then by Theorem 4.2, ρ γ(µ) eists for all γ [0, π). So the radicand in (4.2.3) must be nonnegative and the result follows. Q.E.D.

47 Eamples If the coefficient functions p, v 1, and v 2 L 1 [0, ), then 0 < ρ α(µ) < for all µ > c and α [0, π). See [13, 14]. Hence the connection formulae of this chapter hold. As a particular eample, we take p v 1 v 2 0. Then the spectral derivatives can be computed directly from (1.2.7) and (4.1.4) and ρ α(µ) = µ2 c 2 π(µ + c cos 2α) µ 2 c 2 π(µ + c cos 2α) µ > c µ < c (4.4.1) for α [0, π). To illustrate Theorem 4.2, we take α = π/4, β = π/2 and compute the choices for ρ 0(µ). Since, for µ > c ρ π/4(µ) = 1 π µ2 c 2 µ and ρ π/2(µ) = 1 µ2 c 2, π µ c Theorem 4.2 yields [ 1 2ρ 0(µ) π ] 2 ( ) 3µ c µ c = π 2 2 µ2 c 2 µ + c and the choices for ρ 0(µ), µ > c are 1 µ2 c 2 π 5µ 3c and 1 µ2 c 2. π µ + c If a third derivative is known, say ρ π/3 (µ) = µ2 c 2, the ambiguity is resolved and µ c/2 ρ 0(µ) = 1 µ2 c 2 for µ > c by Theorem 4.1. The calculation for µ < c is similar. π µ + c

48 41 It will be seen in Chapter 6 that it is possible in some cases to uniquely determine a third derivative given derivatives for just two initial conditions. Finally, in this simple eample with p v 1 v 2 0, equality in Corollary 4.3, is attained when α = 0, β = π/2. In Chapter 5, ρ α(µ) for a special case of the Dirac equation will be epressed in terms of two functions S(µ) and T (µ) as was done by Harris for a special case of the Sturm-Liouville equation, [11]. In this notation, it is clear that whenever S(µ) = 0 and α = 0, β = π/2, (3.1.6) is an equality.

49 CHAPTER 5 THE FORM OF THE SPECTRAL FUNCTIONS ASSOCIATED WITH DIRAC EQUATIONS 5.1 Introduction In this chapter, we consider the spectral functions ρ α (µ) for µ R associated with a Dirac equation given by y = p λ + c + v 1 (λ c + v 2 ) p on [0, ) together with the initial condition y, y = y 1 y 2 (5.1.1) y 1 (0) cos α + y 2 (0) sin α = 0 (5.1.2) where α [0, π). Also, c 0 is a constant, λ = µ + iɛ is the comple spectral parameter, and the real valued p, v 1 and v 2 are bounded and members of L 1 [0, ). This guarantees the equation is in the limit point case at infinity [13, 14]. We show that with suitable additional restrictions on the coefficient functions p, v 1, and v 2 and for µ large enough, the spectral derivative ρ α(µ) can be written as a series. The form of this representation is similar to that of the spectral derivative of the Sturm-Liouville equation (3.1.1), yet there are striking differences in the behavior of the functions for the two equations. The main result is given in 5.2 and the groundwork is developed in 5.3. The theorem is proved in 5.4, specific conditions for which it holds are established in