Spectral Theory in Hilbert Space

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62 10. BOUNDARY CONDITIONS locally absolutely continuous derivative and satisfies −u ′′ + qu = f. Conversely, if u, f ∈ L 2 (I) and this equation is satisfied, then u ∈ D(T ∗ c ) and T ∗ c u = f. Proof. Let u1 be a solution of −u ′′ 1 + qu1 = f. Assume u0 is in the domain of Tc and put f0 = Tcu0. Integrating by parts twice we get (10.5) I u0f = I u0(−u ′′ 1 + qu1) = I (−u ′′ 0 + qu0)u1 = I f0u1. So, if f is orthogonal to the domain of Tc, then u1 is orthogonal to all compactly supported elements f0 ∈ L2 (I) for which there is a solution u0 of −u ′′ 0 + qu0 = f0 with compact support. By Corollary 10.5 it follows that u1 solves −v ′′ + qv = 0 so that f = 0. Thus Tc is densely defined. The calculation (10.5) also proves the converse part of the lemma. Furthermore, if u is in the domain of T ∗ c with T ∗ c u = f we obtain 0 = 〈u0, f〉 − 〈f0, u〉 = 〈f0, u1 − u〉. Just as before it follows that u1 − u solves the equation −v ′′ + qv = 0. It follows that u solves the equation −u ′′ + qu = f so that Tc ⊂ T ∗ c . The proof is complete. Being symmetric and densely defined Tc is closeable, and we define the minimal operator T0 as the closure of Tc and denote the domain of T0 (the minimal domain) by D0. Similarly, the maximal operator T1 is T1 := T ∗ c with domain D1 ⊃ D0. Thus the maximal domain D1 consists of all differentiable functions u ∈ L 2 (I) such that u ′ is locally absolutely continuous function for which T1u = −u ′′ + qu ∈ L 2 (I). We can now apply the theory of Chapter 9. The deficiency indices of T0 are accordingly the number of solutions of −u ′′ + qu = iu and −u ′′ + qu = −iu respectively which are linearly independent and in L 2 (I). Since there are only 2 linearly independent solutions for each of these equations the deficiency indices can be no larger than 2. For the equation (10.3) the deficiency indices are always equal, since if u solves −u ′′ + qu = λu, then u solves the equation with λ replaced by λ, and linear independence is preserved when conjugating functions. Thus, for our equation there are only three possibilities: The deficiency indices may both be 2, both may be 1, or both may be 0. We shall see later that all three cases can occur, depending on the choice of q and I. We will now take a closer look at how selfadjoint realizations are determined as restrictions of the maximal operator. Suppose u1 and u2 ∈ D1. Then the boundary form (cf. Chapter 9) is (10.6) 〈(u1, T1u1), U(u2, T1u2)〉 = i (u1T1u2 − T1u1u2) = i I (−u1u ′′ 2 + u ′′ 1u2) = −i I I [u1, u2] ′ = −i lim [u1, u2] , K→I K
10. BOUNDARY CONDITIONS 63 the limit being taken over compact subintervals K of I. We must restrict T1 so that this vanishes. In some sense this means that the restriction of T1 to a selfadjoint operator T is obtained by boundary conditions since the limit clearly only depends on the values of u1 and u2 in arbitrarily small neighborhoods of the endpoints of I. This is of course the motivation for the terms boundary operator and boundary form. The simplest case is when an endpoint is an element of I. This means that the endpoint is a finite number, and that q is integrable near the endpoint. Such an endpoint is called regular; otherwise the endpoint is singular. If both endpoints are regular, we say that we are dealing with a regular problem. We have a singular problem if at least one of the endpoints is infinite, or if q /∈ L 1 (I). Consider now a regular problem. It is clear that the deficiency indices are both 2 in the regular case, since all solutions of −u ′′ +qu = iu are continuous on the compact interval I and thus in L 2 (I). We shall investigate which boundary conditions yield selfadjoint restrictions of T1. The boundary form depends only on the boundary values (u(a), u ′ (a), u(b), u ′ (b)), and the possible boundary values constitute a linear subspace of C 4 . On the other hand, the boundary form is positive definite on Di and negative definite on D−i, both of which are 2-dimensional spaces. The boundary values for the deficiency spaces therefore span two two-dimensional spaces which do not overlap. It follows that as u ranges through D1 the boundary values range through all of C 4 . The boundary conditions need to restrict the 4-dimensional space Di⊕D−i to the 2-dimensional space D of Theorem 9.2, so two independent linear conditions are needed. This means that there are 2 × 2 matrices A and B such that the boundary conditions are given by AU(a)+ u BU(b) = 0, where U = −u ′ . Linear independence of the conditions means that the 2 × 4 matrix (A, B) must have linearly independent rows. Consider first the case when A is invertible. Then the condition is of the form U(a) = SU(b), where S = −A−1 0 1 B. If J = ( −1 0 ) the boundary form is −i{(U2(a)) ∗JU1(a) − (U2(b)) ∗JU1(b)}, so symmetry requires this to vanish. Inserting U(a) = SU(b) the condition becomes (U2(b)) ∗ (S∗JS − J)U1(b) = 0 where U1(b) and U2(b) are arbitrary 2 × 1 matrices. Thus it follows that the condition U(a) = SU(b) gives a selfadjoint restriction of T1 precisely if S satisfies S∗JS = J. Such a matrix S is called symplectic. Important special cases are when S is plus or minus the unit matrix. These cases are called periodic and antiperiodic boundary conditions respectively. Another valid choice is S = J. Since det J = 1 = 0 it is clear that any symplectic matrix S satisfies | det S| = 1 (see also Exercise 10.1). In particular, it is invertible. It is clear that the inverse
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Spectral Theory in Hilbert Space Le
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Preface The aim of these notes is t
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iv CONTENTS Exercises for Chapter 1
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2 0. INTRODUCTION cos( √ λ t), i
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4 0. INTRODUCTION • Can the equat
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6 1. LINEAR SPACES of u so one may
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8 1. LINEAR SPACES Exercises for Ch
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10 2. SPACES WITH SCALAR PRODUCT on
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Page 22 and 23: 16 3. HILBERT SPACE Given a normed
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Page 26 and 27: 20 3. HILBERT SPACE ˆvn = limj→
Page 29 and 30: CHAPTER 4 Operators A bounded linea
Page 31 and 32: 4. OPERATORS 25 Proposition 4.2. A
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Page 37 and 38: CHAPTER 5 Resolvents We now conside
Page 39: EXERCISES FOR CHAPTER 5 33 Analytic
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Page 44 and 45: 38 6. NEVANLINNA FUNCTIONS Proof. B
Page 46 and 47: 40 7. THE SPECTRAL THEOREM function
Page 48 and 49: 42 7. THE SPECTRAL THEOREM as The r
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Page 57 and 58: CHAPTER 9 Extension theory We will
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Page 63: EXERCISES FOR CHAPTER 9 57 Exercise
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Page 89 and 90: CHAPTER 12 Inverse spectral theory
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112 15. SINGULAR PROBLEMS in the pr
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114 15. SINGULAR PROBLEMS Since vK
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APPENDIX A Functional analysis In t
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A. FUNCTIONAL ANALYSIS 119 other wo
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122 B. STIELTJES INTEGRALS (4) C (5
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124 B. STIELTJES INTEGRALS Definiti
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126 B. STIELTJES INTEGRALS We obtai
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APPENDIX C Linear first order syste
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C. LINEAR FIRST ORDER SYSTEMS 131 s
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