Spectral Theory in Hilbert Space

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66 10. BOUNDARY CONDITIONS uniformly to ũ, so that K |uj−u| 2 converges both to 0 and to K |ũ−u|2 . It follows that u = ũ a.e. in K. A bounded Hermitian form on a Hilbert space H is a map H × H ∋ (u, v) ↦→ B(u, v) ∈ C such that |B(u, v)| ≤ Cuv for some constant C. It is clear that the boundedness of a Hermitian form is equivalent to it being continuous as a function of its arguments. The boundary form i(〈u, T1v〉 − 〈T1u, v〉) is a bounded Hermitian form on D1, i.e., it is a Hermitian form in u, v and is bounded by u1v1, and by Lemma 10.8 the boundary form at a, i.e., u ′ (a)v(a) − u(a)v ′ (a), is also a bounded Hermitian form (bounded by 2CKu1v1 if a ∈ K). Since i(〈u, T1v〉 − 〈T1u, v〉) = −i lim x→b [u, v](x) + i[u, v](a) we see that i limx→b(u ′ (x)v(x) − u(x)v ′ (x)), the boundary form at b, is also a bounded Hermitian form on D1. Since the forms at a and b vanish if u is in the domain of Tc, i.e., if u vanishes near a and b, it follows that they also vanish if u ∈ D0. In particular, if u ∈ D0, then u(a) = u ′ (a) = 0. Now T0 is the adjoint of T1 so it follows that this is the only condition at a for an element of D1 to be in D0, since this guarantees that the form at a vanishes. Of course, u ∈ D0 also requires that the form at b vanishes. Now let Ta be the closure of the restriction of T1 to those elements of D1 which vanish near b, and let Da be the domain of Ta. Then Lemma 10.7 and the boundedness of the forms at a and b show that the boundary form at b vanishes on Da and that dim D1/D0 ≥ dim Da/D0 ≥ 2. We obtain the following theorem. Theorem 10.9. If the interval I has one regular endpoint a, then n+ = n− ≥ 1. If n+ = n− = 1, then the boundary form at the singular endpoint vanishes on D1, and any selfadjoint restriction of T1 is given by a boundary condition of the form (10.7) at a and no condition at all at b. Proof. If n+ = n− = 0 then T1 = T0 so that T1 is selfadjoint, according to Theorem 9.1. But then we can not have dim D1/D0 ≥ 2. If n+ = n− = 1, then 2 = dim D1/D0 ≥ dim Da/D0 ≥ 2 so that we must have D1 = Da. Thus the boundary form at the singular endpoint vanishes on D1, and the boundary form at a vanishes precisely if we impose a boundary condition of the form (10.7). If n+ = n− = 2 we obtain a selfadjoint restriction of T1 by imposing two appropriate boundary conditions. One of them can be a condition of the form (10.7), and then a condition at the singular endpoint also has to be imposed. There are also selfadjoint restrictions obtained by imposing coupled boundary conditions. See Exercise 10.3. Whether one obtains deficiency indices 1 or 2 when one endpoint is regular clearly only depends on conditions near the singular endpoint.
EXERCISES FOR CHAPTER 10 67 It is customary to say that a singular endpoint is in the limit point condition if the deficiency indices are 1 and in the limit circle condition if the deficiency indices are 2. The terminology derives from the methods Weyl [12] used in 1910 to construct the resolvent of a Sturm-Liouville operator. If an interval has two singular endpoints in the limit point condition it is clear that T1 is selfadjoint, since the boundary form vanishes on D1. No boundary conditions are therefore required in this case, and one often says that the operator Tc is essentially selfadjoint, since its closure T0 = T1 is selfadjoint. If one or both of the endpoints are in the limit circle condition, we have a situation similar to when the endpoint is regular, and need to impose boundary conditions of a similar type. Note, however, that at a limit circle endpoint the limits of u(x) and u ′ (x) for an element u ∈ D1 do not necessarily exist. To formulate the boundary conditions in explicit terms one may instead use the idea of Exercise 10.3. It is clearly an important problem to find explicit conditions on the interval and the coefficient q which guarantee limit point or limit circle conditions. A large number of different criterions for this are available today. We end the chapter by proving a simple criterion, known already to Weyl (with a more complicated proof), for the limit point condition at an infinite interval endpoint. Theorem 10.10. Suppose q is bounded from below near ∞. Then (10.3) is in the limit point condition at ∞. Proof. Suppose q > C on [a, ∞). Let u be the solution of −u ′′ + qu = (C + i)u with initial data u(a) = 1, u ′ (a) = 0. We have (Re(u ′ u)) ′ = Re(|u ′ | 2 + u ′′ (x)u(x)) = |u ′ | 2 + Re((q − C − i)|u| 2 = |u ′ | 2 + (q − C)|u| 2 ≥ 0. Thus Re(u ′ u) is increasing and Re(u ′ (0)u(0)) = 0 so Re(u ′ u) ≥ 0. But (|u| 2 ) ′ = 2 Re(u ′ u) so |u| 2 is increasing. It follows that |u| 2 ≥ 1 so that one can not have u ∈ L 2 (0, ∞). Thus deficiency indices are < 2. For a more general result, see Exercise 10.4. Exercises for Chapter 10 Exercise 10.1. Show that a symplectic 2 × 2 matrix S is of the form e iθ P where θ ∈ R and P is a real 2 × 2 matrix with determinant 1. Also show that the inverses and adjoints of symplectic matrices are symplectic. Exercise 10.2 (Hard!). Suppose that all solutions of −v ′′ +qv = λv are in L 2 (I) for some real or non-real λ. Show that this is then true for all complex λ.
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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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12 2. SPACES WITH SCALAR PRODUCT Pr
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14 2. SPACES WITH SCALAR PRODUCT Ex
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
Page 42 and 43: 36 6. NEVANLINNA FUNCTIONS However,
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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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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Spectral Theory in Hilbert Space

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