On a spectral theorem of Weyl

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On a spectral theorem of Weyl Nigel Higson ∗, Qijun Tan Department of Mathematics, Penn State University, University Park, PA 16802, United States of America

Received 27 August 2019; accepted 27 August 2019

Abstract We give a new proof of a theorem of Weyl on the continuous part of the spectrum of Sturm– Liouville operators on the half-line with asymptotically constant coefficients. Earlier arguments, due to Weyl and Kodaira, depended on particular features of Green’s functions for linear ordinary differential operators. We use a concept of asymptotic containment of C ∗ -algebra representations that has geometric origins. We apply the concept elsewhere to the Plancherel formula for spherical functions on reductive groups. c 2020 Elsevier GmbH. All rights reserved. ⃝ MSC 2010: primary 34L05; secondary 46L45 Keywords: spectral theory; Sturm–Liouville theory; Plancherel theorem

1. Introduction The purpose of this paper is to present a new approach to an old theorem of Hermann Weyl on the spectral theory of self-adjoint Sturm–Liouville operators on a half-line. Our aim is to introduce methods that are more geometric and more amenable to generalization than the originals. We show elsewhere [13] that the same arguments apply to HarishChandra’s Plancherel formula for spherical functions (in fact Harish-Chandra was very much inspired by Weyl’s work; compare [3, p. 38] and [1]). Sturm–Liouville theory is of course concerned with the eigenvalues and eigenfunctions of linear differential operators d d D=− · p(x) · + q(x), (1.1) dx dx ∗ Corresponding author.

E-mail addresses: [email protected] (N. Higson), [email protected] (Q. Tan). https://doi.org/10.1016/j.exmath.2020.02.001 c 2020 Elsevier GmbH. All rights reserved. 0723-0869/⃝

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initially on a closed interval [a, b]. Assume for simplicity that p(x) and q(x) are smooth, real-valued functions on [a, b], with p(x) positive everywhere. In examining the nonzero solutions of the eigenvalue problem D fλ = λ fλ,

(1.2)

it is appropriate to impose suitable self-adjoint boundary conditions. For the sake of this introduction let us choose the simplest of these, namely f λ (a) = 0 = f λ (b).

(1.3)

The elements of Sturm–Liouville theory can then be summarized as follows: Theorem 1.4. The eigenvalues λ for the above problem are real numbers, and each has multiplicity one. The set of all eigenvalues is a discrete subset of R, bounded below, and if h is any smooth function on [a, b], then ∑ ⟨ f λ , h⟩ f λ (x) h(x) = ⟨ fλ, fλ⟩ λ for x ∈ (a, b). In an influential paper from early in his career, Weyl developed an analogous theory for Sturm–Liouville operators on a half-line rather than a bounded interval [14]. Weyl’s paper addresses many issues, but our concern here is his treatment of the continuous spectrum of certain Sturm–Liouville operators, and especially his version, for the continuous spectrum, of the expansion theorem above. Assume that the coefficient functions p(x) and q(x) in (1.1) are defined on [0, ∞) and converge sufficiently rapidly to the constants 1 and 0, respectively, as x tends to infinity. For the purpose of this introduction, let us assume more than Weyl, namely that p(x) ≡ 1

and q(x) ≡ 0

if x ≫ 0

(1.5)

(this assumption is too strong to be interesting in applications, but it allows us to quickly introduce Weyl’s result). For each λ ∈ C there is a one-dimensional space of eigenfunctions Fλ for D that satisfy the boundary condition Fλ (0) = 0.

(1.6)

If we focus on the case where λ > 0, and if we choose, as we may, Fλ to be nonzero and real-valued, then our assumptions on the coefficient functions p(x) and q(x) imply that √

Fλ (x) = c(λ)ei

λx

√

+ c(λ)e−i

λx

(1.7)

for some nonzero c(λ) ∈ C and all x ≫ 0. Weyl’s result for the continuous spectrum, expressed in L 2 -terms, is as follows (there is also a pointwise result that is analogous to Theorem 1.4, which may be derived from the L 2 -result): Theorem 1.8. then ⟨ f, g⟩ =

If h and g are smooth and compactly supported functions on (0, ∞), ∑ ⟨g, Fλ ⟩⟨Fλ , h⟩ λ<0

⟨Fλ , Fλ ⟩

+

1 4π

∫ 0

∞

⟨g, Fλ ⟩⟨Fλ , h⟩ dλ √ , |c(λ)|2 λ

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The sum is over the square-integrable eigenfunctions associated to negative eigenvalues that satisfy the boundary condition (1.6), and there are finitely many of these. The integral is absolutely convergent. All the inner products in the formula have the standard L 2 -form. We shall approach Weyl’s theorem by comparing the Sturm–Liouville operator D to the simpler operator d2 . dx2 But we shall regard D0 as an operator on the full line (−∞, ∞), rather than the half-line, and the translation-invariance of D0 on the full line will be crucial. To explain why, it is helpful to make the following general definition: D0 = −

Definition 1.9.

Let A be a C ∗ -algebra and let

π : A −→ B(H )

and

π0 : A −→ B(H0 )

be nondegenerate Hilbert space representations of A. We shall say that π0 is asymptotically contained in π if (i) There is a one-parameter group of unitary operators Ut : H0 →H0 that commute with the operators in π0 [A]. (ii) There is a bounded operator W : H0 →H such that for every a ∈ A, and for every u, v ∈ H0 , [⟨ ⟩ ⟨ ⟩ ] lim u, π0 (a)v H − W Ut u, π(a)W Ut v H = 0. (1.10) 0

t→+∞

Note that asymptotic containment of representations implies weak containment of representations [6, Definition 3.4.5]. For Weyl’s theorem, we take A=C0 (R), and we define π and π0 to be the functional calculus representations π : ϕ ↦−→ ϕ(D)

and

π0 : ϕ ↦−→ ϕ(D0 ),

on H =L 2 (0, ∞) and H0 =L 2 (−∞, ∞) respectively. We define {Ut } to be the standard one-parameter unitary group of translations on L 2 (−∞, ∞) and take W : L 2 (−∞, ∞) −→ L 2 (0, ∞) to be the obvious restriction operator. The asymptotic containment of π0 in π follows from the condition (1.5) on the coefficients of D; see Lemma 3.3. Returning to the general case, we shall assume that the C ∗ -algebra A is separable and commutative, as it is in the examples of concern to us, and that the Hilbert spaces H and H0 are separable, too. Then the representations π and π0 may be decomposed into direct integrals ∫ ⊕ ∫ ⊕ H= Hλ dµ(λ) and H0 = H0,λ dµ0 (λ) (1.11) over the spectrum of A, so that the action of a∈A on each space in either integral is through the character a ↦→ λ(a). We shall assume that the spaces Hλ and H0,λ are finite-dimensional, as again is the case in the examples of concern to us.

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Now let us assume temporarily that the asymptotic containment relation (1.10) is replaced by the exact containment relation ⟨ ⟩ ⟨ ⟩ u, π0 (a)v H = W u, π(a)W v H (1.12) 0

for all u, v ∈ H0 and all a ∈ A, so that the operator W is necessarily an isometric inclusion of π0 as a subrepresentation of π . It follows that the operator W decomposes into a field of operators Wλ : H0,λ −→ Hλ .

(1.13)

and that each Wλ∗ Wλ is a multiple of the identity operator on H0,λ . Of course that multiple is Trace(Wλ∗ Wλ )/dim(H0,λ ). Using the family {Wλ }, the direct integral decomposition of π in (1.11) gives rise to an alternative direct integral decomposition of π0 . Comparing the two, we find that the measure µ0 in (1.11) is absolutely continuous with respect to µ, and that indeed dµ0 Trace(Wλ∗ Wλ ) (λ) = dµ dim(H0,λ )

(1.14)

for µ-almost all λ in the support of the representation π0 . The main observation of this paper, which, aside from some functional-analytic details, is very simple, is that even when π0 is only asymptotically contained in π , we can still derive a version of the formula (1.14) in essentially the same way, if we assume the existence of operators Wλ : H0,λ →Hλ that asymptotically decompose W in the sense that lim Wλ (Ut v)0,λ − (W Ut v)λ = 0.

t→+∞

(1.15)

See Section 2 for a precise account of the assumptions we make. As for (1.15), it is easiest to understand its meaning by examining the adjoint operators Cλ = Wλ∗ : Hλ −→ H0,λ . In the context of the Sturm–Liouville problem, the spaces Hλ and H0,λ may be understood as λ-eigenspaces for the operators D and D0 , respectively, and the asymptotic formula (1.15) simply asserts that each eigenfunction of D is mapped by Cλ to an eigenfunction of D0 to which it is asymptotic in the sense of (1.7). This proves the existence of the operators Cλ in this context, and also computes the trace in (1.14) in terms of |c(λ)|2 . Weyl’s formula in Theorem 1.8 is an immediate consequence.1 Other interesting examples of asymptotic containment of representations come from representation theory. In brief, if G is a real reductive group with maximal compact subgroup K , and if P = Mp Ap Np is a minimal parabolic subgroup, then the representation of C ∗ (G) on L 2 (G/Mp Np ) is asymptotically contained in the representation of C ∗ (G) on L 2 (G/K ). The case of S L(2, R) is illustrated in Fig. 1. The figure should make it clear that the asymptotic containment in this example has a very geometric origin. To fit this example within the framework of this paper we take the C ∗ -algebra A in our asymptotic containment to be the commutative C ∗ -subalgebra of C ∗ (G) that To be accurate, the Radon–Nikodym derivative formula only determines µ on the positive part of the spectrum because the necessary assumptions on Wλ only hold there. A separate argument is required for the nonpositive spectrum; see Section 4. 1

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Fig. 1. The homogeneous spaces G/K and G/Mp Np for the reductive group G = S L(2, R), realized as coadjoint orbits.

is generated by the K -bi-invariant compactly supported smooth functions on G. It is naturally represented on the Hilbert spaces H and H0 of K -fixed vectors within L 2 (G/K ) and L 2 (G/Mp Np ), respectively. The K -invariant functions on G/K and G/Mp Np can be + identified with functions on A+ p and Ap respectively, where Ap is a dominant chamber in Ap . A suitable operator W may be defined using restriction of functions from Ap to + A+ p (followed by a translation away from the walls of Ap to make W bounded). The minimal parabolic is defined by a one-parameter subgroup of A, and right-translation on G/Mp Np by this one-parameter group gives the necessary one-parameter unitary group on H0 . The counterpart of Weyl’s theorem in this example is Harish-Chandra’s Plancherel theorem for spherical functions. See [13]. In fact the present paper grew out of a project in noncommutative geometry involving the Plancherel formula [4,5]. The reader is referred to [1] for an interesting and thorough discussion of the relation between Weyl’s theorem and harmonic analysis on symmetric spaces. After this paper was written the authors were lucky to enjoy a very stimulating conversation with Joseph Bernstein, who pointed out that he had obtained very similar results in unpublished work from the 1980s (see the final remarks in [2, Sec. 0.2] for hints of this). Some of the spectral-theoretic methods from [12], which studies harmonic analysis on p-adic spherical varieties, are also very closely related to the methods of this paper. See especially Section 8 of [12]. Here is a brief outline of the present paper. We shall discuss asymptotic containment of representations in Section 2. The main result is 2.14. We shall apply our method to the positive, continuous spectrum of Sturm–Liouville operators in Section 3, and we shall briefly address the nonpositive spectrum in Section 4. We shall consider not only the operators discussed in this introduction, but also a nontrivial example related to the representation theory of S L(2, R). In an Appendix we quickly review the Weyl–Kodaira approach to Theorem 1.8 for the sake of comparison.

2. Asymptotic containment of representations In this section we shall describe our approach to Weyl’s theorem. We shall formulate the method in fairly general terms, applicable to examples beyond Weyl’s theorem, although we shall not strive for the upmost generality in the assumptions that we make.

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Let A be a separable, commutative C ∗ -algebra with Gelfand spectrum Λ, so that of course A ∼ = C0 (Λ) for some locally compact space Λ. We shall view elements of A as continuous functions on Λ without further comment. Let us suppose that we are given two non-degenerate representations of A on separable Hilbert spaces: π : A −→ B(H )

and

π0 : A −→ B(H0 ).

We shall assume that π0 is asymptotically contained in π , as in Definition 1.9, with unitary group {Ut } on H0 and operator W : H0 → H as described in the definition. Our analysis of the relation between π and π0 will be based on following formula, which is an immediate consequence of (1.10). Lemma 2.1.

If a ∈ A and if g, h ∈ H0 , then ∫ 1 T ⟨g, π0 (a)h⟩ H0 = lim ⟨W Ut g, π(a)W Ut h⟩ H dt. □ T →+∞ T 0 We shall make the following assumptions concerning the representation π0 ; in the case of Sturm–Liouville operator they will be easy to verify using Fourier analysis. Assumption 2.2.

We shall suppose that we are given:

(i) An open subset Λ0 ⊆ Λ and a locally trivial continuous field of finite-dimensional Hilbert spaces {H0,λ }λ∈Λ0 over Λ0 (or in other words a Hermitian vector bundle). (ii) A dense subspace H0 ⊆ H0 and a linear map h ↦→ {h 0,λ } from H0 into the continuous sections of {H0,λ } such that H0,λ = { h 0,λ : h ∈ H0 } for every λ ∈ Λ0 . (iii) A Borel measure µ0 on Λ0 such that ⟨h 0,λ , g0,λ ⟩ is a µ0 -integrable function of λ, for every h, g ∈ H0 , and such that ∫ ⟨ ⟩ ⟨ ⟩ h 0,λ , g0,λ H a(λ) dµ0 (λ) h, π0 (a)g H = 0

Λ0

0,λ

for every h, g ∈ H0 and every a ∈ A. Assumption 2.3. We shall assume that the action of the one-parameter unitary group {Ut } on the Hilbert space H0 maps the subspace H0 into itself, and that the continuous field {H0,λ }λ∈Λ0 carries a continuous, unitary action {Ut,λ } of R such that (Ut h)0,λ = Ut,λ h 0,λ for every h ∈ H0 and every λ ∈ Λ0 . Next, we shall make assumptions on the representation π that are similar to those in Assumption 2.2, except that we shall in addition assume that π has multiplicity one: the fibers in the field of Hilbert spaces that decomposes π have dimension one. This assumption is not altogether necessary (finite-dimensionality would suffice), but it simplifies the statements of the results that follow, along with their proofs, and it is satisfied in the situations of interest to us. Here are the details.

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Assumption 2.4.

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We shall suppose that there are given:

(i) A locally trivial continuous field of one-dimensional Hilbert spaces {Hλ }λ∈Λ over Λ (that is, a Hermitian line bundle over Λ). (ii) A dense subspace H ⊆ H such that if h ∈ H0 then W Ut h ∈ H for all t ≫ 0. (iii) A family of linear maps ελ : h ↦→ h λ from H into Hλ so that {h λ } is a continuous section, and a Borel measure on Λ such that ⟨h λ , gλ ⟩ is a µ-integrable function of λ, for every h, g ∈ H, and such that ∫ ⟨ ⟩ ⟨ ⟩ h, π(a)g H = h λ , gλ H a(λ) dµ(λ) λ

Λ

for every h, g ∈ H and every a ∈ A. Finally, we shall make the following assumption concerning the asymptotic relation between the fields {Hλ } and {H0,λ }. As we noted in the introduction, and as we shall see clearly in the next section, in the Sturm–Liouville context this means that the operator Cλ below maps each λ-eigenfunction for D to a λ-eigenfunction for D0 to which it is asymptotic. Assumption 2.5. linear maps

We shall assume that there is given a continuous family of injective (λ ∈ Λ0 )

Cλ : Hλ −→ H0,λ

with the property that if h belongs to H0 , and if {vλ } is a continuous section of {Hλ }, and K is a compact subset of Λ0 , then ⏐⟨ ⟩ ⟨ ⟩ ⏐⏐ ⏐ lim sup ⏐ Cλ vλ , (Ut h)0,λ H − vλ , (W Ut h)λ H ⏐ = 0. λ

0,λ

t→+∞ λ∈K

Using the four assumptions listed above we shall prove: Theorem 2.6. The measure µ0 is absolutely continuous with respect to µ on the open set Λ0 , with Radon–Nikodym derivative dµ0 Trace(Cλ∗ Cλ ) (λ) = . dµ dim(H0,λ ) Here is the proof in outline. We are assuming that ∫ ⟨ ⟩ ⟨h, π0 (ϕ)h⟩ H0 = h 0,λ , h 0,λ H ϕ(λ) dµ0 (λ). Λ0

0,λ

(2.7)

We shall obtain from our remaining assumptions a new integral formula, namely ∫ ⟨ [ ] ⟩ ⟨h, π0 (ϕ)h⟩ H0 = h 0,λ , Av Cλ Cλ∗ h 0,λ H ϕ(λ)dµ(λ), (2.8) Λ0

0,λ

where the operator Av[Cλ Cλ∗ ] : H0,λ → H0,λ is defined by the averaging formula ∫ [ ] 1 T Av Cλ Cλ∗ = lim U−t,λ Cλ Cλ∗ Ut,λ dt T →∞ T 0

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(since we are dealing here with operators on the finite-dimensional space H0,λ , the limit certainly exists). At this point, we can appeal to the following uniqueness result for spectral decompositions: Lemma 2.9. Let {Tλ } be a measurable field of positive operators on {H0,λ }λ∈Λ0 . Suppose that ∫ ∫ ⟨ ⟩ ⟨ ⟩ h 0,λ , Tλ h 0,λ H ϕ(λ) dµ(λ) = h 0,λ , h 0,λ H ϕ(λ) dµ0 (λ) Λ0

0,λ

0,λ

Λ0

for every h ∈ H0 and every continuous and compactly supported function ϕ. Then Tλ is a scalar multiple of the identity for µ-almost all λ ∈ Λ0 . In addition µ0 is absolutely continuous with respect to µ on Λ0 and dµ0 (λ) · I H0,λ Tλ = dµ µ-almost everywhere on Λ0 . Proof. For each point λ0 ∈ Λ0 there exists h ∈ H0 for which the section h 0,λ is nonzero at λ0 . It follows immediately from the uniqueness part of the Riesz representation theorem that µ0 is absolutely continuous with respect to µ near λ0 with Radon–Nikodym derivative ⟨ ⟩ h 0,λ , Tλ h 0,λ H dµ0 ⟩ 0,λ . (λ) = ⟨ dµ h 0,λ , h 0,λ H 0,λ

Since the derivative is independent of {h 0,λ } this implies that dµ0 Tλ = (λ) · I H0,λ , dµ almost everywhere, as required. □ Returning to the proof of Theorem 2.6, Lemma 2.9 tells us that the operator Av[Cλ Cλ∗ ] is a scalar multiple of the identity for µ-almost-all λ ∈ Λ0 . The computation ( [ ]) Trace Av Cλ Cλ∗ = Trace(Cλ Cλ∗ ) = Trace(Cλ∗ Cλ ), determines the multiple, and the theorem follows. So it remains to establish the integral formula (2.8): Lemma 2.10. If h ∈ H0 and if ϕ is a continuous and compactly supported function on Λ0 , then ∫ ⟨ [ ] ⟩ ⟨h, π0 (ϕ)h⟩ H0 = h 0,λ , Av Cλ Cλ∗ h 0,λ H ϕ(λ)dµ(λ). Λ0

0,λ

Proof. It suffices to prove this formula for all functions ϕ that are supported on compact sets K ⊆ Λ0 over which the field {Hλ } is trivializable, and so we shall assume that here. In addition we shall use the notation Wt = W Ut : H0 −→ H.

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According to Lemma 2.1, ⟨h, π0 (ϕ)h⟩ H0 = lim

T →+∞

1 T

∫ 0

T⟨

⟩ Wt h, π(ϕ)Wt h H dt.

(2.11)

Now use Assumption 2.4 to write the integrand on the right hand side of (2.11) as ∫ ⟨ ⟩ ⟨ ⟩ Wt h, π(ϕ)Wt h H = (Wt h)λ , (Wt h)λ H ϕ(λ)dµ(λ). λ

Λ0

Since the space Hλ are one-dimensional, we can write ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ (Wt h)λ , (Wt h)λ H = (Wt h)λ , vλ H · vλ , (Wt h)λ H , λ

λ

λ

where {vλ } is a continuous section of {Hλ } with ∥vλ ∥ Hλ = 1 for all λ ∈ K . So ⟨h, π0 (ϕ)h⟩ H0 ∫ ∫ ⟨ ⟩ ⟨ ⟩ 1 T (Wt h)λ , vλ H · vλ , (Wt h)λ H ϕ(λ)dµ(λ)dt. = lim λ λ T →+∞ T 0 Λ0 Consider now the difference ⟨ ⟩ ⟨ ⟩ (Ut h)0,λ , Cλ vλ H · Cλ vλ , (Ut h)0,λ H 0,λ 0,λ ⟨ ⟩ ⟨ ⟩ − (Wt h)λ , vλ H · vλ , (Wt h)λ H , λ

(2.12)

(2.13)

λ

which we can write as [⟨ ⟨ ⟩ ⟩ ⟨ ⟩ ] (Ut h)0,λ , Cλ vλ H Cλ vλ , (Ut h)0,λ H − vλ , (Wt h)λ H λ 0,λ 0,λ [⟨ ⟩ ⟨ ⟩ ]⟨ ⟩ + (Ut h)0,λ , Cλ vλ H − (Wt h)λ , vλ H vλ , (Wt h)λ H . λ

0,λ

λ

The terms in the square brackets converge to 0, as t → +∞, uniformly in λ ∈ K . In addition, since ⏐⟨ ⟩ ⏐ ⏐⟨ ⟩ ⏐ ⏐ (Ut h)0,λ , Cλ vλ ⏐ = ⏐ Ut,λ h 0,λ , Cλ vλ ⏐ ≤ ∥h λ ∥ · ∥Cλ vλ ∥, H H 0,λ

0,λ

we see that ⟨(Ut h)0,λ , Cλ vλ ⟩ H0,λ is uniformly bounded in t and λ ∈ K . It follows from this and Assumption 2.5 that ⟨(Wt h)λ , vλ ⟩ Hλ is uniformly bounded too. So the expression (2.13) converges to zero as t → ∞, uniformly in λ ∈ K . Observe next that ⟨ ⟩ ⟨ ⟩ (Ut h)0,λ , Cλ vλ H · Cλ vλ , (Ut h)0,λ H 0,λ 0,λ ⟨ ⟩ ⟨ ⟩ = Ut,λ h 0,λ , Cλ vλ H · Cλ vλ , Ut,λ h 0,λ H 0,λ 0,λ ⟨ ⟩ ∗ = h 0,λ , U−t,λ Cλ Cλ Ut,λ h 0,λ H . 0,λ

It follows from our analysis of (2.13) that the inner integral in (2.12) is asymptotic to the integral ∫ ⟨ ⟩ h 0,λ , U−t,λ Cλ Cλ∗ Ut,λ h 0,λ H ϕ(λ) dµ(λ) Λ0

0,λ

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(that is, the difference converges to zero as t → +∞). As a result, ⟨h, π0 (ϕ)h⟩ H0 ∫ ∫ ) ⟩ 1 T( ⟨ = lim h 0,λ , U−t,λ Cλ Cλ∗ Ut,λ h 0,λ H ϕ(λ) dµ(λ) dt, 0,λ T →+∞ T 0 Λ0 It now follows from Fubini’s theorem, that ⟨h, π0 (ϕ)h⟩ H0 ∫ ( ∫ T ) ⟨ ⟩ 1 = lim h 0,λ , U−t,λ Cλ Cλ∗ Ut,λ h 0,λ H dt ϕ(λ)dµ(λ). 0,λ T →+∞ Λ T 0 0 The integral in the parentheses is uniformly bounded in T . Therefore we can interchange the limit as T → +∞ and the integral over Λ0 to obtain (2.8), as required. □ Theorem 2.6 gives a formula for the measure µ0 in terms of the measure µ. But since our goal is to obtain information about the measure µ, we should invert this formula: Theorem 2.14. The measure µ is absolutely continuous with respect to the measure µ0 on Λ0 , and the Radon–Nikodym derivative of µ with respect to µ0 on Λ0 is dµ dim(H0,λ ) (λ) = . □ dµ0 Trace(Cλ∗ Cλ )

3. Sturm–Liouville operators In this section we shall apply the approach of Section 2 to Sturm–Liouville operators on the half-line. So let d d · p(x) · + q(x), D=− dx dx where the coefficient functions p(x) and q(x) are smooth and real-valued on (0, ∞), and where p(x) is everywhere positive. We shall assume that D is a self-adjoint operator on some domain that includes Cc∞ (0, ∞). We shall study the following examples (which may be generalized considerably). Example 3.1. If p and q are in fact smooth on [0, ∞) and eventually constant, with p positive, as in Section 1, then D is essentially self-adjoint on the domain of smooth, compactly supported functions on [0, ∞) that vanish at 0. Example 3.2. If G=S L(2, R) and K =S O(2), then the symmetric space G/K may be identified with the hyperbolic plane (with G acting as isometries on the plane). The Laplace–Beltrami operator ∆ is essentially self-adjoint on the space of smooth and compactly supported functions on G/K . On K -invariant functions it acts as d d ∆ = − 2 − coth(r ) , dr dr where r is the radial coordinate in the polar coordinate system associated to the action of K . Now identify the K -fixed part of L 2 (G/K ) with L 2 (0, ∞) using the radial coordinate

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11

1

and multiplying by sinh(r ) 2 (the latter comes from the formula dArea = sinh(r )dr dθ). We obtain an essentially-self adjoint operator d ∆ = D + 41 = − 2 − 41 csch2 (x) + 14 dx on L 2 (0, ∞) (we have subtracted the term 1/4 from ∆ with Lemma 3.3 in mind). Associated to the unbounded self-adjoint operator D on the Hilbert space H = L 2 (0, ∞) is the functional calculus representation π : C0 (R) −→ B(H ) π : ϕ ↦−→ ϕ(D). We shall compare π to the functional calculus representation π0 : C0 (R) −→ B(H0 ) π0 : ϕ ↦−→ ϕ(D0 ), where D0 = −d 2 /d x 2 and H0 = L 2 (−∞, ∞). Here we view −d 2 /d x 2 is an essentially self-adjoint operator on the domain of smooth, compactly supported functions, and we take D0 to be its self-adjoint extension. Define Ut : H0 → H0 to be the translation operator (Ut h)(x) = h(x − t). Obviously each ϕ(D0 ) commutes with each Ut . Denote by W : H0 −→ H the orthogonal projection (which restricts functions on (−∞, ∞) to functions on (0, ∞), of course). The following computation checks that π0 is asymptotically contained in π , assuming that the coefficients of D converge to constant values. Lemma 3.3.

Assume that the coefficients of D satisfy

lim p(x) = 1,

x→∞

lim p ′ (x) = 0

x→∞

and

lim q(x) = 0.

x→∞

If ϕ ∈ C0 (R), and if g, h ∈ L 2 (−∞, ∞), then ] [⟨ ⟩ ⟨ ⟩ lim W Ut g, ϕ(D)W Ut h L 2 (0,∞) − g, ϕ(D0 )h L 2 (−∞,∞) = 0. t→+∞

Proof. We shall prove that   lim ϕ(D)W Ut h − W Ut ϕ(D0 )h  L 2 (0,∞) = 0 t→+∞

(3.4)

for every h ∈ L 2 (−∞, ∞), which will suffice. The set of all ϕ ∈ C0 (R) satisfying (3.4) is a norm-closed subalgebra of C0 (R), and it therefore suffices to show that the resolvent functions ϕ(λ) = (λ ± i)−1 belong to it. Moreover it suffices to check (3.4) for each of these two functions ϕ and for a dense set of functions h in L 2 (−∞, ∞). Let ϕ(x) = (x ± i)−1 . We shall calculate the limit (3.4) when h = (D0 ± i I ) f

and

f ∈ Cc∞ (−∞, ∞).

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If f ∈ Cc∞ (−∞, ∞), and if t ≫ 0, then W Ut f is a smooth and compactly supported function on (0, ∞), and we compute that ϕ(D)W Ut h − W Ut ϕ(D0 )h = (D ± i I )−1 W Ut (D0 ± i I ) f − W Ut f = (D ± i I )−1 (D − D0 )W Ut f, where, in the last line, we are regarding D0 as a differential operator acting on the smooth and compactly supported functions on (0, ∞). Our assumptions on the coefficients of D imply that lim ∥(D − D0 )W Ut f ∥ = 0,

t→+∞

and so (3.4) is proved for ϕ(x) = (x ± i)−1 , as required. □ Assumptions 2.2 and 2.3 about the representation π0 from the previous section are easily obtained from the Fourier transform ∫ ∞ ˆ h(ξ ) = h(x)e−iξ x d x, −∞

as follows. To begin, let Λ0 = (0, ∞), and for λ ∈ Λ0 define√ H0,λ to be√ the twodimensional vector space of functions on the line spanned by ei λx and e−i λx . Equip H0,λ with the inner product that makes these two functions an orthonormal basis. The family {H0,λ }λ>0 obviously forms a continuous field of Hilbert spaces over Λ0 with constant and finite fiber dimension. Now let H0 be space of smooth and compactly supported functions in H0 . The Fourier transform associates to each h ∈ H0 a continuous section {h 0,λ } of the continuous field, namely √ √ √ √ h(− λ)e−i λx . h 0,λ = ˆ h( λ)ei λx + ˆ Moreover it follows from Plancherel’s formula that ∫ ⟨h 0,λ , g0,λ ⟩ H0,λ ϕ(λ) dµ0 (λ), ⟨h, ϕ(D0 )g⟩ L 2 (−∞,∞) = Λ0

where 1 dλ √ . 4π λ So Assumption 2.2 is satisfied. The unitary actions dµ0 (λ) =

√

Ut,λ : aei

λx

+ be−i

√ λx

√

↦−→ e−i

λt

√

aei

λx

(3.5) √

+ ei

λt

√

be−i

λx

on the fibers H0,λ decompose the translation action on L 2 (−∞, ∞), as in Assumption 2.3. Let us turn now to the representation π of C0 (R). General theory guarantees that π has a measurable direct integral decomposition ∫ ⊕ L 2 (0, ∞) ∼ Hλ dµ(λ). (3.6) = R

This means that there exists: (i) A Borel-measurable field of Hilbert spaces, {Hλ }λ∈R , as in [7, Part II, Chapter 1]. (ii) A Borel measure µ on R.

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(iii) A unitary isomorphism from L 2 (0, ∞) to the Hilbert space of square-integrable sections of the measurable field, h ↦→ {h λ }λ∈R , under which the representation π corresponds to the representation of C0 (R) on square-integrable sections by pointwise multiplication. Thus if g ∈ H and if ϕ ∈ C0 (R), then (π (ϕ)g)λ = ϕ(λ)gλ for µ-almost every λ ∈ R. See [7, Part II, Chapter 6, Theorem 2]. We need to upgrade this measurable decomposition to a continuous decomposition, as required by Assumption 2.4. We do not know the full extent to which this is possible, but the Gelfand–Kostyuchenko method, which we shall now review, handles the examples of concern to us. (See [2, Section 1] for a concise account of the Gelfand–Kostyuchenko method, as well as applications that are closely related to those in this paper.) The inclusion of the topological vector space Cc∞ (0, ∞) into L 2 (0, ∞) factors through a Hilbert–Schmidt operator. That is, there is a commuting diagram Cc∞ (0, ∞) continuous

inclusion

↘

→ L 2 (0, ∞) ↗

(3.7)

Hilbert-Schmidt

K

where K is a Hilbert space. This has the following consequence: Lemma 3.8 (See for Example [11, Chapter VII, Section 1]). For all λ ∈ R there exist continuous linear operators ελ : Cc∞ (0, ∞) −→ Hλ

(3.9)

such that if h ∈ Cc∞ (0, ∞), and if {h λ }λ∈R is the associated square-integrable section of {Hλ }λ∈R , then h λ = ελ (h) for µ-almost every λ ∈ R. □ Since Cc∞ (0, ∞) is dense in the Hilbert space L 2 (0, ∞), the maps ελ have dense range for µ-almost every λ. The adjoint operators ελ∗ : Hλ∗ −→ Cc∞ (0, ∞)∗

(3.10)

are therefore injective for µ-almost every λ. That is, for almost every λ∈R the map ελ∗ is defined and embeds Hλ∗ into the space of distributions on R. Keeping in mind the Hilbert space isomorphism Hλ∗ ∼ = Hλ , it follows from Lemma 3.8 and the definitions that if h ∈ Cc∞ (0, ∞), and if {vλ }λ∈R is a measurable section of {Hλ }λ∈R , then ∫ ∞ ⟨vλ , h λ ⟩ Hλ = ⟨vλ , ελ (h)⟩ Hλ = ελ∗ (vλ ) · h, 0

for µ-almost every λ ∈ R, where the right-hand integral is the pairing between distributions and test functions. Using this and the property (iii) above, we find that if Vλ = ελ∗ (vλ ), then ∫ ∞ ∫ ∞ DVλ · h = λVλ · h 0

0

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for µ-almost every λ ∈ R (the operator D is applied to Vλ in the sense of distributions) and since Cc∞ (0, ∞) is separable it follows that DVλ = λVλ for µ-almost every λ. Thus for almost every λ, the morphism ελ∗ embeds Hλ into the space of λ-eigendistributions for D on (0, ∞). The latter is 2-dimensional and consists of smooth functions on (0, ∞). Let us study the implications of all this for the operators in Example 3.1. Lemma 3.11. Let D be as in Example 3.1. For µ-almost every λ ∈ R the operator ελ∗ maps Hλ isomorphically to the one-dimensional space of (smooth) solutions of the differential equation DG λ = λG λ that satisfy the boundary condition G λ (0) = 0. Proof. We can repeat the Gelfand–Kostyuchenko method above using the space of functions in Cc∞ [0, ∞) that vanish at 0 in place of Cc∞ (0, ∞). If f and g belong to this space, then for almost every λ we can write ⟨ελ (Dh), gλ ⟩ Hλ − ⟨ελ (h), (Dg)λ ⟩ Hλ = ⟨ελ (Dh), gλ ⟩ Hλ − ⟨ελ (h), λgλ ⟩ Hλ ∫ ∞ ∫ ∞ (Dh)(x)G λ (x) d x − h(x)λG λ (x) d x = ∫0 ∞ ∫0 ∞ = (Dh)(x)G λ (x) d x − h(x)(DG λ )(x) d x, 0

(3.12)

0

ελ∗ (gλ ).

Assume now that in addition h ′ (0) = 1. Calculating the difference where G λ = of integrals using the fundamental theorem of calculus we find that ∫ ∞ ∫ ∞ (Dh)(x)G λ (x) d x − h(x)(DG λ )(x) d x = p(0)G λ (0). 0

0

The top expression in (3.12) is an integrable function of λ, and therefore so is G λ (0). If ϕ is any continuous and compactly supported function on R, then by (iii) above the integral of the left-hand side of (3.12), times ϕ(λ), is equal to zero, and so ∫ ∞ G λ (0) ϕ(λ) dµ(λ) = 0, 0

It follows that G λ (0) = 0 for almost every λ. The lemma follows from this because the elements gλ span Hλ , for almost all λ. □ Now form the family of one-dimensional eigenfunction spaces { Fλ : [0, ∞)→C : D Fλ = λFλ , Fλ (0) = 0 }.

(3.13)

These assemble to form the fibers of a smooth vector bundle using the usual topology of convergence of smooth functions. Equip each with the norm ∥Fλ ∥ = |Fλ′ (0)| to obtain a continuous field of one-dimensional Hilbert spaces over R for which λ ↦→ Fλ is a continuous section if λ ↦→ Fλ′ (0) is continuous. Lemma 3.11 shows that for almost every λ the morphism ελ∗ is a vector space isomorphism from the Hilbert space fiber Hλ in the direct integral decomposition (3.6)

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to the fiber (3.13). The morphism is not necessarily isometric, but we can remedy this possible shortcoming by changing the inner products on the Hλ , and the measure µ, using ⟨ , ⟩ Hλ := ∥vλ ∥−2 Hλ · ⟨ , ⟩ Hλ

and

dµ(λ) := ∥vλ ∥2Hλ · dµ(λ)

where vλ is chosen so that if Fλ = ελ∗ (vλ ) then Fλ′ (0) = 1. With these changes, we obtain a new direct integral decomposition of the form (3.6) (the map h ↦→ {h λ } from L 2 (0, ∞) to square-integrable sections is not changed), and now the morphisms ελ∗ are unitary isomorphisms, for almost every λ. Now take H = Cc∞ (0, ∞), and if h ∈ H, then according to the definitions, if Fλ′ (0) = 1, then ∫ ∞ ∗ ελ (h λ ) = Fλ (x)h(x) d x · Fλ . 0

for almost every λ. The right hand side is a continuous section of the field {Hλ } since the function Fλ depends continuously (in fact analytically) on λ. This verifies Assumption 2.4 for the Sturm–Liouville operators from Section 1. As for the Laplace–Beltrami operator from Example 3.2, we can repeat the Gelfand– Kostyuchenko analysis, as in Lemma 3.8 and the discussion following the lemma, using the space of smooth, compactly supported, K -invariant functions on G/K in place of Cc∞ (0, ∞), and obtain, for almost every λ, embeddings of Hλ into the K -invariant λ-eigenfunctions of D. But the latter space is actually one-dimensional already (the λ-eigenfunctions are distinguished from one another by their values at eK ∈ G/K ) and the family of all such eigenspaces spaces carries the structure of a continuous field of one-dimensional Hilbert spaces, since there are explicit formulas for the eigenfunctions that vary smoothly with λ. See [8, Chap. 2, Thms 1.1 & 1.2]. The argument above then handles Assumption 2.4 in this case. Finally, we need to verify Assumption 2.5. For this purpose we shall assume a bit more about the coefficients of D, namely that ∫ ∞ ∫ ∞ |1 − p(x)−1 | d x < ∞ and |q(x)| d x < ∞. (3.14) x0

x0

Certainly these conditions hold in our examples. Proposition 3.15. Let λ > 0 and let Fλ be the λ-eigenfunction of D with F ′ (0) = 1. If (3.14) holds, then there is a unique nonzero λ-eigenfunction F0,λ of D0 such that ⏐ ⏐ lim ⏐ Fλ (x) − F0,λ (x)⏐ = 0. x→∞

The convergence is uniform over compact sets of eigenvalues λ in (0, ∞).

□

This is standard in differential equations and we will omit the proof here, but see for example Weyl’s paper [14]. Of course Proposition 3.15 is obvious for the operators from Example 3.1. In any case, using Proposition 3.15 we define injective operators Cλ : Hλ −→ H0,λ

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by Cλ : Fλ ↦→ F0,λ where Fλ and F0,λ are as in Proposition 3.15. If h is a smooth, compactly supported function on R, and if vλ = Fλ , then ∫ ∞ ⟨ ⟩ ⟨ ⟩ Cλ vλ , (Ut h)0,λ H − vλ , (Wt h)λ H = (F0,λ (x) − Fλ (x))h(x−t)d x (3.16) λ

0,λ

0

(this formula holds as long as t is large that h(x−t) is supported on the positive x-axis). Proposition 3.15 implies that if λ is confined to a compact set in (0, ∞), then the integral converges to zero, uniformly in λ, as required by Assumption 2.5. We arrive therefore the following result, which is Weyl’s theorem for the positive spectrum of D: Theorem 3.17. Let D be one of the operators from Example 3.1. Let g and h be smooth and compactly supported functions on [0, ∞). If 0<α<β, and if P[α,β] is the spectral projection for D associated to the interval [α, β], then ∫ β 1 dλ 1 ⟨g, Fλ ⟩⟨Fλ , h⟩ ⟨g, P[α,β] h⟩ = √ 2 4π α |c(λ)| λ where Fλ is the nonzero λ-eigenfunction with Fλ (0) = 0 and Fλ′ (0) = 1, and c(λ) is characterized by √ √ ) ( lim Fλ (x) − c(λ)ei λx − c(λ)e−i λx = 0 x→+∞

(the inner products are standard L 2 -inner products and the integral is absolutely convergent). Proof. We shall compute ∥P[α,β] h∥2 (the formula in the statement of the theorem will follow by polarization). First, according to the definition of a direct integral decomposition, ∫ β 2 ∥P[α,β] h∥ = ∥h λ ∥2Hλ dµ(λ). α

Now let {vλ } be the section of of the theorem. Then ∫ β ∫ β 2 ∥h λ ∥ Hλ dµ(λ) = α

α

{Hλ } for which ελ∗ (vλ ) = Fλ , with Fλ as in the statement ⏐ ⏐ ⏐⟨vλ , h λ ⟩ H ⏐2 λ

⟨vλ , vλ ⟩ Hλ

β

∫ dµ(λ) = α

|⟨Fλ , h λ ⟩ L 2 |2 dµ(λ), ⟨vλ , vλ ⟩ Hλ

and applying Theorem 2.14 we get ∫ β ∫ β |⟨Fλ , h λ ⟩ L 2 |2 2dµ0 (λ) ∥h λ ∥2Hλ dµ(λ) = ⟨vλ , vλ ⟩ Hλ Trace(Cλ∗ Cλ ) α α ∫ β |⟨Fλ , h λ ⟩ L 2 |2 =2 dµ0 (λ). α ⟨C λ vλ , C λ vλ ⟩ H0,λ It follows from our definition of Cλ that this is ∫ β |⟨Fλ , h λ ⟩ L 2 |2 dµ0 (λ), |c(λ)|2 α and the theorem follows from the explicit formula for µ0 in (3.5).

□

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There is a similar theorem for the operator in Example 3.2. The only change is that Fλ is taken to be the λ-eigenfunction of D on (0, ∞) corresponding to the K -equivariant λ eigenfunction G/K →C of the shifted Laplace–Beltrami operator with value 1 at eK .

4. Non-positive spectrum In this section we shall look at the non-positive part of the spectrum of a Sturm– Liouville operator D of the types considered in the previous section. The methods of this paper really have nothing to contribute here, and for that reason we shall be extremely brief. The value λ=0 belongs to the spectrum of D of any of the operators from Section 1 because the spectrum is closed. But for the purposes of fully determining the measure µ we need to determine whether or not 0 is an eigenvalue of the self-adjoint operator D, or in other words whether or not µ({0}) > 0. The answer is that λ=0 is not an eigenvalue. For the Sturm–Liouville differential operators from Section 1, the λ=0 eigenfunctions have the form F0 (x) = c1 + c2 x

(x ≫ 0),

and the only possibility for a square-integrable eigenfunction is c1 =c2 =0, in which case F0 is identically zero. But any eigenfunction for the self-adjoint operator D would in particular be a square-integrable eigenfunction for the differential operator D. For the Laplace–Beltrami operator from Example 3.2 one can employ a similar argument, using a version of Proposition 3.15 in place of the simple asymptotic formula for F0 given above. The negative part of the spectrum for the operators that we discussed in Section 1 needs to be handled differently, since square-integrable eigenfunctions are certainly possible in this case. But one can prove, using the same methods that go into the proof of Proposition 3.15, that: Proposition 4.1. If we assume that ∫ ∞ eαx |1 − p(x)−1 | d x < ∞ and 1

∞

∫

eαx |q(x)| d x < ∞.

1

for some α > 0, then the operator D has at most finitely many L 2 -eigenfunctions satisfying the boundary condition Fλ (0) = 0. □ One can say more using perturbation theory. The operators D from Section 1 are relatively compact perturbations of the positive operators −d/d x · p(x) · d/d x. So the negative parts of their spectra consist of at most countably many eigenvalues, accumulating only at 0. Compare [9, Chapter IV, Theorem 5.35]. But Proposition 4.1 rules out the possibility of accumulation at 0. Hence the negative spectra are finite in this case. As for the Laplace–Beltrami operator from Example 3.2, it is not difficult to show that D ≥ 0, so there is no negative spectrum at all.

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Fig. 2. The contour for the integral in (A.1).

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix. Review of Kodaira’s approach In this appendix we shall review Weyl’s approach to Theorem 1.8, as improved by Kodaira [10] (see also [15]). Our aim in doing so is to indicate how different it is from the approach taken in the body of this paper. Let D be a self-adjoint Hilbert space operator. If α<β, and if both α and β are absent from the spectrum of D, then according to the Riesz functional calculus, the spectral projection for D associated to the interval (α, β) is ∫ 1 (ν − D)−1 dν, (A.1) P(α,β) = 2πi Γ where the contour Γ is indicated in Fig. 2. The contributions to the integral in (A.1) from the vertical components of the contour Γ decrease to zero in norm as the height of the contour decreases to zero, and so (∫ β−iε ) ∫ β+iε 1 P(α,β) = lim (ν − D)−1 dν − (ν − D)−1 dν , (A.2) ε→0+ 2πi α−iε α+iε or equivalently ∫ β 1 (D−λ−iε)−1 − (D−λ+iε)−1 dλ (A.3) ε→0+ 2πi α (these are norm limits). The integrand on the right-hand side of (A.3) is uniformly bounded in ε>0 and in λ ∈ R, and by approximating a general self-adjoint operator D by operators that do not contain α or β in their spectrum, we find that: P(α,β) = lim

Lemma A.4 (Kodaira). The formula (A.3) holds for any self-adjoint operator D and any interval [α, β], as long as α and β do not belong to the point spectrum of D (the limit in (A.3) is now a strong limit of a uniformly bounded family of operators). □ The formula (A.3) is of particular value when D is a Sturm–Liouville operator because, as we shall see, the resolvent operators (D−λ±iε)−1 may be computed quite

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explicitly. Let us consider then d d D=− · p(x) · + q(x) dx dx where p and q are smooth, real-valued functions on (0, ∞) (we shall impose further conditions on p and q later on). Assume that D defines a self-adjoint operator on L 2 (0, ∞) on a given domain that (i) includes the smooth, compactly supported functions on (0, ∞) and (ii) is invariant under multiplication by smooth functions on (0, ∞) that are locally constant outside of a compact set. If ν ∈ / R (or more generally if ν belongs to the resolvent set of D), then there exist nonzero ν-eigenfunctions Fν and G ν that vary smoothly with ν, the first of which agrees with an element in dom(D) near 0 and the second of which agrees with an element of dom(D) near ∞. Indeed, if h is any smooth and compactly supported function on (0, ∞), then the function f = (D − ν)−1 h belongs to dom(D), and moreover D f = ν f + h. It follows that D f =ν f near 0 and near ∞. Because the set of all (D−ν)−1 h is dense in L 2 (0, ∞), we obtain, for at least some h, functions f that are nonzero near 0 and near ∞. They agree there with functions in dom(D), and they extend to nonzero ν-eigenfunctions Fν and G ν on (0, ∞), as required. Note that eigenfunctions Fν and G ν must be linearly independent, for otherwise they would belong to dom(D), which is impossible if ν ∈ / Spec(D). Consider now the integral kernel defined by { Fν (y)G ν (x) x ≥ y kν (x, y) = (A.5) Fν (x)G ν (y) x ≤ y. The associated integral operator K ν can certainly be defined on the domain of smooth, compactly supported functions h on (0, ∞), and moreover since ∫ ∞ ∫ x (K ν h)(x) = Fν (x) G ν (y)h(y) dy + G ν (x) Fν (y)h(y) dy. x

0

the range consists of smooth functions in dom(D). We compute directly that (D − ν)K ν h = Wr(Fν , G ν )h,

(A.6)

where Wr(Fν , G ν ) is the Wronskian ( ) Wr(Fν , G ν )(x) = p(x) Fν′ (x)G ν (x) − Fν (x)G ′ν (x) . As is well known, this is a constant function of x∈(0, ∞); moreover the constant value determines a nondegenerate bilinear form on the 2-dimensional space of νeigenfunctions. Since Fν and G ν are linearly independent, we can therefore normalize them so that Wr(Fν , G ν ) = 1,

(A.7)

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in which case it follows from (A.6) that (D−ν)−1 h = K ν h

(A.8)

for all smooth and compactly supported functions h on (0, ∞). In order to apply (A.8) to the limit formula (A.3) we shall make the following additional assumptions concerning the eigenfunctions G ν : (G1) For all λ > 0 the limits G+ λ = lim G λ+iε ε↘0

and

G− λ = lim G λ−iε ε↘0

1

exist in the usual C -topology (uniform convergence of the functions and their derivatives on compact sets of (0, ∞); note that, using D, this implies convergence in the C 2 -topology, and indeed in the C ∞ -topology). Moreover the convergence is uniform over compact sets of positive λ. − (G2) For all λ > 0 the functions G + λ and G λ are linearly independent. The limit functions G ± λ obey the relation − + − Wr(G + λ , G λ ) · Fλ = G λ − G λ

(A.9)

− for λ > 0. Indeed G + λ and G λ are λ-eigenfunctions for the differential operator D, and − as a result, the Wronskian Wr(G + λ , G λ ) is a constant function, so if we write − Hλ = Wr(G + λ , G λ ) · Fλ , − then the three functions Hλ , G + λ and G λ all belong to the two-dimensional space of − λ-eigenfunctions for D. To verify that Hλ = G + λ −G λ we therefore just need to observe that + − ± Wr(Hλ , G ± λ ) = Wr(G λ −G λ , G λ ),

which is a consequence of (A.7). Theorem A.10. If 0<α<β, then under the assumptions (G1) and (G2) above, the spectral projection P(α,β) for D is given by the formula ∫ ∞ (P(α,β) h)(x) = p(α,β) (x, y)h(y) dy, 0

for all smooth and compactly supported functions h on (0, ∞), where ∫ β 1 − Fλ (x)Fλ (y) Wr(G + p(α,β) (x, y) = λ , G λ ) dλ. 2πi α Proof. It follows from Lemma A.4, (A.8) that ∫ β (∫ ∞ ) ( ) 1 (P(α,β) h)(x) = lim kλ+iε (x, y)−kλ−iε (x, y) h(y) dy dλ ε↘0 2πi α 0 and from (A.5), together with assumption (G1) and (A.9), that ( ) − lim kλ+iε (x, y)−kλ−iε (x, y) = Wr(G + λ , G λ )Fλ (x)Fλ (y). ε↘0

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The convergence is uniform over compact subsets of y ∈ (0, ∞) and compact subsets of λ ∈ (0, ∞). Hence ∫ ∞ (∫ β ) 1 − Wr(G + , G )F (x)F (y) dλ h(y) dy (P(α,β) h)(x) = λ λ λ λ 2πi 0 α as required. □ At this point we finally turn to Sturm–Liouville operators with eventually constant coefficient functions, as in Section 1. If we write √ √ Fν (x) = c(ν) exp(i νx) + c(−ν) exp(−i νx) (x ≫ 0) using the usual principal branch of the square root function, equal to the positive square root on the positive axis (we shall avoid the eigenvalues ν < 0), then using (A.7) we compute that for λ > 0 and ν = λ±iε, √ i G ν (x) = (x ≫ 0). (A.11) √ exp(±i νx) 2c(∓ν) ν The sign in the exponential is needed to ensure that G ν is an L 2 -function at infinity, which is of course necessary if G ν is to agree with a function in dom(D) at infinity. It follows easily from (A.11) that (G1) and (G2) are satisfied, that √ ±i G± (x ≫ 0), √ exp(±i λx) λ (x) = 2c(∓λ) λ and that i − Wr(G + √ . λ , Gλ ) = 2|c(λ)|2 λ Therefore Theorem A.10 gives ∫ β 1 dλ 1 Fλ (x)Fλ (y) p(α,β) (x, y) = √ . 4π α |c(λ)|2 λ This is a reformulation of Weyl’s theorem, as stated in Section 1.

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Please cite this article as: N. Higson and Q. Tan, On a spectral theorem of Weyl, Expositiones Mathematicae (2020), https://doi.org/10.1016/j.exmath.2020.02.001.

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Please cite this article as: N. Higson and Q. Tan, On a spectral theorem of Weyl, Expositiones Mathematicae (2020), https://doi.org/10.1016/j.exmath.2020.02.001.