Dirac and magnetic Schrödinger operators on fractals

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ScienceDirect Journal of Functional Analysis 265 (2013) 2830–2854 www.elsevier.com/locate/jfa

Dirac and magnetic Schrödinger operators on fractals Michael Hinz a,1 , Alexander Teplyaev b,∗,2 a Department of Mathematics, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany b Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA

Received 11 October 2012; accepted 23 July 2013 Available online 20 August 2013 Communicated by L. Gross

Abstract In this paper we define (local) Dirac operators and magnetic Schrödinger Hamiltonians on fractals and prove their (essential) self-adjointness. To do so we use the concept of 1-forms and derivations associated with Dirichlet forms as introduced by Cipriani and Sauvageot, and further studied by the authors jointly with Röckner, Ionescu and Rogers. For simplicity our definitions and results are formulated for the Sierpinski gasket with its standard self-similar energy form. We point out how they may be generalized to other spaces, such as the classical Sierpinski carpet. © 2013 Elsevier Inc. All rights reserved. Keywords: Fractal; Laplacian; Dirichlet; Magnetic

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . 2. Vector analysis on the Sierpinski gasket . 3. Dirac operators . . . . . . . . . . . . . . . . . 4. Magnetic Schrödinger operators . . . . . . 5. Other fractals . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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* Corresponding author.

E-mail addresses: [email protected] (M. Hinz), [email protected] (A. Teplyaev). 1 Research supported in part by NSF grant DMS-0505622 and by the Alexander von Humboldt Foundation (Feodor

Lynen Research Fellowship Program). 2 Research supported in part by NSF grant DMS-0505622. 0022-1236/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jfa.2013.07.021

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1. Introduction The aim of the present paper is to introduce natural (local) Dirac and magnetic Schrödinger operators on fractal spaces and to prove that they are (essentially) self-adjoint. To make the paper more accessible, and to approach the most interesting examples, we formulate our definitions and results for the classical Sierpinski gasket. We are making this choice mainly because there is an abundance of literature about Sierpinski gasket, and in particular the numerical analysis and finite elements method is well developed (see [34,39,55,88,98], the references therein and newer preprints of Strichartz that concern the energy measure analysis on the Sierpinski gasket). Thus our theoretical results can be used to develop efficient numerical techniques for differential operators on fractals that involve magnetic potentials and gauge fields. We also explain how to modify them for more general fractals and other spaces. This is particularly straightforward to do for species with a resistance form, in the sense of Kigami [59–61] (see also [11,48,103]), such as the classical two-dimensional Sierpinski carpet, but many results are valid for much more general spaces. In particular, extending our results for spaces that are not locally compact will be subject of future work. Our analysis uses the concept of 1-forms in the content of the Dirichlet form theory, and is based on recent results on 1-forms and vector fields [24,21,22,54] and [52], respectively. Our space of 1-forms is a Hilbert space, which allows to identify 1-forms and vector fields, and to introduce other notions of vector analysis, as recently done in [52] (which generalizes earlier approaches to vector analysis on fractals, see [57,60,71,97,103]). This is a part of a comprehensive program to study vector equations on general non-smooth spaces which carry a diffusion process or, equivalently, a local regular Dirichlet form. The study of the Laplacian on fractal graphs was originated in physics literature (see [5,12, 30,38,83,84]), and for a selection of recent mathematical physics results see [2–4,34,39,43,56] and the references therein. Among the problems where fractal spaces seem to appear naturally we would like to mention, in particular, the spaces of fractional dimension appearing in quantum gravity ([6,74,87] and the references therein). Besides that, our motivation is coming from the theory of quantum graphs ([32,33,37,45,64–69,81] and the references therein); from the spectral theory on fractals ([8,28,44,62,63,72,77,78,80,89,90] and the references therein); form some questions of non-commutative analysis ([18,19,24,21,22,54] and the references therein) and the theory of spectral zeta functions [31,73,95,102]; and from the localization problems ([1,79,82, 91,100] and the references therein). Recall that roughly speaking, the Dirac operator is defined as the square root of the Laplace operator. (Note, however, that classical Dirac operator for diffusions is a local operator, which excludes the possibility of using the spectral theorem to define it.) Depending on context and purpose it appears in various formulations with possibly different complexity and sign conventions. On the real line D = −id/dx may for instance be regarded as the Dirac operator. Given a Riemannian manifold M, its tangent bundle ΛT ∗ M can be turned into a Clifford module, and the associated Dirac operator is defined as D = d + d ∗ , where d is the exterior derivative and d ∗ is its adjoint, cf. [14]. For a spin 1/2-particle in the plane the Dirac operator is given by D = −iσx ∂/∂x1 − iσy ∂/∂x2 , where σx and σy are the respective Pauli spin matrices, [35]. More generally, it may be defined for spinor bundles over spin manifolds, see [14,35] or [40] for background and details. Dirac operators on discrete graphs have for instance been considered in [27, Section 4] with a strong emphasis on connections to non-commutative geometry. The paper [86] follows a similar spirit and considers related spectral triples and Connes metrics. More recently Dirac operators on

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discrete graphs and related index theorems have been studied in [81]. In this reference they act on a tensor product of the form H0 ⊕ H1 , where H0 and H1 are Hilbert spaces of functions on the vertices and edges, respectively. Roughly speaking, the discrete difference operator d : H0 → H1 plays the role of the exterior derivative. Denoting its adjoint by d ∗ : H1 → H0 , the associated Dirac operator is then defined on H0 ⊕ H1 by  D=

0 d

d∗ 0

 ,

(1)

and as a consequence D 2 yields the matrix Laplacian acting on H0 ⊕ H1 . In a somewhat similar fashion [81] also investigates Dirac operators and index theorems on quantum graphs (often referred to as metric graphs or quantum wires, [64,65,67,68]), now within the context of suitable Sobolev spaces. A preceding article dealing with index theorems on quantum graphs is [37], and a related much earlier reference for Dirac operators is [17]. Different quantization schemes are reviewed in [45]. There is an extensive literature on magnetic Schrödinger operators. In the simplest cases (such as for bounded and sufficiently integrable potentials A and V ) the essential self-adjointness of magnetic Schrödinger Hamiltonians (−i∇ − A)2 + V

(2)

on Euclidean space can be deduced from classical perturbation theorems in Hilbert spaces, cf. [85, Section X.2]. More sophisticated pointwise methods can be found in [85, Sections X.3 and X.4]. Essential self-adjointness results for operators on manifolds may be found in the comprehensive paper [93], see also [41] for singular potentials. Discrete magnetic Schrödinger operators on lattices and graphs have for instance been discussed in [9,12,26,29,46,92,99]. The paper [92] introduces discrete magnetic Laplacians on the two-dimensional integer lattice, proves that they have no point spectrum and compares them to almost Mathieu operators and to onedimensional quasi-periodic operators. This is closely connected to the (one-dimensional) ten martini problem, solved in [7]. Periodic magnetic Schrödinger operators on the two-dimensional integer lattice are treated in [99] and their spectra, typically of band or Cantor type, are studied. In [46] magnetic Schrödinger operators on graphs are considered. Under some conditions the analyticity of the bottom of their spectra is verified and relations to corresponding operators on a quotient graph (by a suitable automorphism group) are discussed. The paper [29] also investigates discrete magnetic Laplacians and Schrödinger operators on graphs, compares their spectra and heat kernels to the original graph Laplacians, defines related Novikov–Shubin invariants and establishes a long term decay result for the heat kernel trace. In [26] the essential self-adjointness of a discrete version of (2) is shown, based on a previous result [25] for operators with zero magnetic potential. Ref. [26] also discusses gauge invariance in terms of holonomy maps. First steps towards magnetic Schrödinger operators on fractals had been taken in [9] and [12] by studying them on infinite Sierpinski lattices. Some decimation techniques for the spectrum and related numerical experiments can be found in [9]. The paper [12] sets up a renormalization group equation for the magnetic Laplacian and discusses relations to superconductivity. Another branch of literature concerns quantum graphs, see [64,65,67,68]. The paper [66] introduces magnetic Laplacians on metric graphs and, based on results in [65], provides a matrix criterion for the boundary conditions to characterize self-adjointness. In [16] a metric graph point of view is used to provide a comprehensive study of the two-dimensional periodic square graph lattice with mag-

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netic fields. The paper [32] shows that any self-adjoint vertex coupling on a metric graph can be approximated by a sequence of magnetic Schrödinger operators on a network of shrinking tubular neighborhoods. For prototype examples of fractal sets carrying a diffusion it is much less clear how to define Dirac operators and magnetic Laplacians. Note that Laplace operators had been studied in several papers and books (see for instance [10,57–59,76,96] and the references therein), but definitions and results concerning analogs of first order differential operators (gradients) were sparse [57, 70,97,101], and hardly flexible enough to fit a sufficient functional analytic context. For the Sierpinski gasket Dirac operators and spectral triples are investigated in [23,19,20], see also [42]. The spectral triple in [19] encodes the geodesic distance and the natural normalized Hausdorff measure on the Sierpinski gasket. In [23] a family of spectral triples is considered. For some parameters the natural normalized Hausdorff measure is recovered as a trace, and the associated Connes distance is shown to be Lipschitz equivalent to some root of the Euclidean distance. Also the standard energy form on the Sierpinski gasket is obtained using a trace function in [23], its abscissa of convergence is different from the one observed for the trace giving the Hausdorff measure. Here we start with the standard energy form, use the Kusuoka measure as reference measure and follow the approach of [21] and [22], where differential 1-forms and derivations based on Dirichlet forms had been introduced. In these papers a Hilbert space H of 1-forms is constructed as, roughly speaking, the completion of the tensor product F ⊗ F of the space F of energy finite functions, a concept that leads to an L2 -theory, see for instance [50] for further explanations. This approach has been studied further in [24,54] and also in [52,51], where related notions of vector analysis are discussed. In this context derivations, coderivations, Dirac and magnetic Schrödinger operators can be defined. More precisely, our Theorems 3.1 and 4.1 below state that under some conditions, analogs of (1) and (2) define essentially self-adjoint operators on suitable Hilbert spaces of functions and vector fields on fractals, respectively. Related spectral triples and metrics are discussed in [53]. For the Sierpinski gasket (or another compact space carrying a local regular resistance form) the associated Connes metric then coincides with the intrinsic metric of the Dirichlet form induced by the resistance form, see [53]. In the next section we review the approach of [21,22] to 1-forms based on energy for the specific example of the Sierpinski gasket K with its standard self-similar energy form (E, F). We recall the definitions of the gradient and divergence operators from [52] and the energy Laplacian for functions. In several places we provide auxiliary formulations in terms of harmonic coordinates. In Section 3 we define a related Dirac operator D that acts on the tensor product L2,C (K, ν) ⊗ HC of the spaces of complex-valued square integrable functions (with respect to the Kusuoka measure ν) and complex-valued vector fields on K. Theorem 3.1 proves it is self-adjoint. In Section 4 we first provide a priori estimates necessary to introduce a bilinear form E a,V associated with a magnetic Schrödinger Hamiltonian H a,V on K. Then we establish a result on its essential self-adjointness on L2,C (K, ν), Theorem 4.1, which merely follows from our definitions, preliminary estimates and a simple KLMN theorem. Finally we prove sort of a gauge invariance result, Theorem 4.2. Section 5 contains some instructions how to generalize the presented results to arbitrary finitely ramified fractals carrying a regular resistance form. In this paper we generally intend to provide a basic setup to study Dirac operators and magnetic fields on fractals. We do not discuss questions regarding the spectrum, refined pointwise statements or approximations. These topics will be addressed elsewhere. To simplify notation, sequences or families indexed by the naturals will be written with index set suppressed, e.g. (an )n stands for (an )n∈N . Similarly, limn an abbreviates limn→∞ an .

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2. Vector analysis on the Sierpinski gasket This section recalls a few items of the concept of 1-forms and vector fields based on Dirichlet forms as studied in [24,21,22,54] and [52], respectively. For simplicity we formulate definitions and results for the Sierpinski gasket, some comments on finitely ramified fractals are provided in Section 5. For investigations of other physical models on the Sierpinski gasket see for instance [34,98]. Let {p1 , p2 , p3 } be the vertex set of an equilateral triangle in R2 . The Sierpinski gasket K is the unique nonempty compact subset of R2 satisfying the self-similarity relation K=

3 

ϕi K,

i=1

where ϕi (x) = x/2 + pi /2. For our purposes the embedding in R2 is inessential, only the associated post-critically finite self-similar structure (K, {1, 2, 3}, {ϕ1 , ϕ2 , ϕ3 }) in the sense of [58] matters. Let (E, F) denote the standard self-similar energy form on K, obtained as the increasing limit E(u) := lim En (u) n

of a sequence of rescaled discrete energies  n   2 5 u(x) − u(y) En (u) = 3 x∼ y n

on approximating graphs. For precise definitions and background we refer to [13,22,57,58]. The form (E, F) is a regular resistance form in the sense of [61]. In particular, E 1/2 is a Hilbert norm on the space F/∼ obtained from F by factoring out constants, and there is some constant c > 0 such that

f ∞  cE(f )1/2

(3)

for all f ∈ F/∼. The space F is a dense subalgebra of C(K), and in particular E(f g)1/2  E(f )1/2 g L∞ (K,ν) + f L∞ (K,ν) E(g)1/2 ,

f, g ∈ F.

(4)

For any f ∈ F we can define a unique (nonatomic) Borel energy measure νf on K, see for instance [101], and polarization yields mutual energy measures νf,g for f, g ∈ F . The space of nonconstant harmonic functions on K is two dimensional. Let {h1 , h2 } be a complete energy orthonormal system for it. The Kusuoka energy measure ν is defined by ν := νh1 + νh2 , and this definition does not depend on the choice of the complete energy orthonormal system {h1 , h2 }. By construction all energy measures νf,g are absolutely continuous with respect to ν and

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have integrable densities Γ (f, g). In particular, we can find Borel versions Zij of the functions Γ (hi , hj ) and a Borel set K0 ⊂ K such that for any x ∈ K0 , the real (2 × 2)-matrix   Zx := Zij (x) ij =1,2 is symmetric, non-negative definite and of rank one, and we have ν(K \ K0 ) = 0. For x ∈ K \ K0 we may define Zx to be the zero matrix. The matrix valued mapping x → Zx is Borel measurable. See for instance [48,70,103]. Note that every Zx , x ∈ K, acts as a projection in R2 , and for fixed x the space (R2 / ker Zx , ·, Zx ·R2 ) is isometrically isomorphic to the image space (Zx (R2 ), ·, Zx ·R2 ). In addition, we may assume K0 is such that h(x) = 0 for all x ∈ K0 . According to [58, Theorem 2.4.1] the regular resistance form (E, F) defines a local regular Dirichlet form on L2 (K, ν). The scalar product on L2 (K, ν) will be denoted by ·,·L2 (K,ν) . The measures νf , f ∈ F , coincide with the energy measures in the sense of [36], and the operation (f, g) → Γ (f, g) taking two members f, g ∈ F into the density Γ (f, g) coincides with the carré du champ in the sense of [15]. Remark 2.1. Here we consider the L2 -space L2 (K, ν) with respect to the Kusuoka measure. Note that the energy measures νf,g are singular with respect to the naturally associated renormalized self-similar Hausdorff measure on K, see [13,47,49]. The choice of the Kusuoka measure simplifies many technicalities. However, it also carries a geometric meaning, for instance, it allows to discuss vector fields of bounded length, cf. formula (25), or related Connes metrics, cf. [53]. The spectrum of the Laplacian ν associated with (E, F) on L2 (K, ν) (see below) will be subject of further studies. Setting   h(x) := h1 (x), h2 (x)

and

y := h(x)

we obtain a homeomorphism h from K onto its image h(K) in R2 , and the latter may be viewed with coordinates y. The collection of functions of the form f = F ◦ h with F ∈ C 1 (R2 ) is dense in F , and for any such function the Kusuoka–Kigami formula  2 E(f ) = Z∇F (y) R2 dν (5) K

holds, where ∇F is the usual gradient of F in R2 . More generally, by the chain rule [36, Theorem 3.2.2] the energy measure νf of such f is given by |Z∇F (y)|R2 dν. By L2,C (X, ν) we denote the natural complexification L2 (X, ν) + iL2 (X, ν) of L2 (X, ν). The closed form E on L2 (X, ν) can be complexified by setting E(f, g) := E(f1 , g1 ) − iE(f1 , g2 ) + iE(f2 , g1 ) + E(g1 , g2 )

(6)

for any f = f1 + if2 and g = g1 + ig2 from FC := F + iF . This yields a positive definite quadratic form E on L2,C (X, ν). That is, E is conjugate symmetric, linear in the first argument, and E(f )  0 for any f ∈ FC . We will use a similar terminology for the mappings considered in what follows. The form E is densely defined and closed. Similarly, and in a way consistent with (6), also the energy measure νf,g and their densities Γ (f, g) can be complexified.

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Consider FC ⊗ FC endowed with the symmetric bilinear form 

a ⊗ b, c ⊗ dH = bdΓ (a, c) dν,

(7)

K

a ⊗ b, c ⊗ d ∈ FC ⊗ FC . It is positive definite and generates a seminorm. Let HC denote the Hilbert space obtained by factoring out zero seminorm elements and completing. Following [21,22] we refer to it as the space of 1-forms on K. For simplicity we will not distinguish between an element a ⊗ b and its equivalence class in HC . To rewrite several items in coordinates we also define the space  

SC := span f ⊗ g: f = F ◦ h, g = G ◦ h with F, G ∈ CC1 R2 . Theorem 2.1. The space SC is dense in HC . Proof. Note first that the collection S C of elements f ⊗ g with f = F ◦ h, F ∈ CC1 (R2 ) and g ∈ FC is dense in HC : By the definition of HC it suffices to approximate finite linear com (n) binations i ai ⊗ bi ∈ FC ⊗ FC by elements of S C . For fixed i let (fi )n be a sequence E-converging to ai . Then  2  (n)     (n)    dν, a ⊗ b − f ⊗ b = b i b j Γ a i − fi i i i i  i

H

i

ij K

which converges to zero by the boundedness of the functions bi . On the other hand every element

(n) i ai ⊗ bi of SC can be approximated by elements of SC : For fixed i let (gi )n E -converge to bi . The estimate (3) implies uniform convergence, and therefore also  2     (n)  (n) 2   b i − gi ai ⊗ bi − a i ⊗ gi  = Γ (ai , aj ) dν  i

goes to zero.

H

i

ij K

2

Recall that we use the coordinate notation y = y(x) = (h1 (x), h2 (x)). Theorem 2.2. There are a measurable field of Hilbert spaces {HC,x }x∈X and surjective linear ⊕ maps ω → ωx from HC onto HC,x such that the direct integral K HC,x ν(dx) is isometrically isomorphic to HC and in particular, 

ω 2H = ωx 2H,x ν(dx), ω ∈ HC . K

For ν-a.e. x ∈ X the fiber HC,x is isomorphic to C2 / ker Zx , and for any f = F ◦ h and g = G ◦ h with F, G ∈ CC1 (R2 ) we have  2

f ⊗ g 2 = ZG(y)∇F (y) 2 dν. (8) H

C

K

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Identity (8) obviously extends (5). Proof. For fixed x ∈ K0 define a linear map ϕx : SC → C2 / ker Zx by ϕx (g ⊗ f ) := Zx G(y)∇F (y). Since h(x) = 0 and linear functions belong to CC1 (R2 ), the linear map ϕx is surjective. Consequently there exists an isomorphism Φx from SC / ker ϕx onto C2 / ker Zx , given by   Φx (f ⊗ g)x = Zx G(y)∇F (y),

(9)

where (f ⊗ g)x = κx (f ⊗ g) denotes the equivalence class mod ker ϕx of f ⊗ g and κx the canonical epimorphism. From (9) we obtain (f ⊗ g)x = g(x)(f ⊗ 1)x

(10)

for any f, g ∈ FC and x ∈ K0 . We write HC,x := SC / ker ϕx and endow this space with the norm   (f ⊗ g)x  Zx G(y)∇F (y) 2 . := H,x C

(11)

Then Φx becomes an isometric isomorphism between Hilbert spaces. For x ∈ K \ K0 we set HC,x = {0} and κx := 0. Due to the Borel measurability of x → Zx the function x → Zx G(y)∇F (y) is Borel measurable and therefore also x → (f ⊗ g)x for all f ⊗ g ∈ SC . The countable set of elements (f ⊗ g)x with f = F ◦ h and g = G ◦ h such that F , G are polynomials with rational coefficients is dense in each fiber HC,x , respectively. This implies that {HC,x }x∈X is a measurable field of Hilbert spaces. For every x ∈ X the fiber HC,x is finite dimensional and therefore κx extends uniquely to a surjective bounded linear map κx : HC → HC,x . For f ⊗ g ∈ SC as above we have 

f ⊗ g 2H =

 |g|2 Γ (f ) dν =

K

G(y) 2 Zx ∇F (y) 2 ν(dx) =



C

K

  (f ⊗ g)x 2 ν(dx). H,x

K

Using bilinearity and the denseness of SC , this extends to an isometric embedding ⊗ κ : HC →

HC,x ν(dx) K

of HC into

⊗ K

HC,x ν(dx). In fact κ is onto: Assume that ω ∈ 

0 = ω, f ⊗ gH = K



 ωx , (f ⊗ g)x H,x ν(dx) =

 K

⊕ K

HC,x ν(dx) is such that

  g(x) ωx , (f ⊗ 1)x H,x ν(dx)

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for all f ⊗ g ∈ SC . Note that we have used (10). Then necessarily ωx , (f ⊗ 1)x H,x = 0 for ν-a.a. x. Therefore ωx must be zero on HC,x for such x and integrating, we have ω = 0 in ⊗ K HC,x ν(dx). 2 The definitions c(a ⊗ b) := (ca) ⊗ b − c ⊗ (ab)

and

(a ⊗ b)d := a ⊗ (bd)

(12)

for a, b, c ∈ FC ⊗ FC and d ∈ L∞,C (K, ν) extend continuously to uniformly bounded actions on HC ,

cω H  c L∞ (X,ν) ω H

and ωc H  c L∞ (X,ν) ω H ,

ω ∈ HC .

(13)

For ω ∈ HC and c ∈ FC we have the equality cω = ωc in HC . See [21,22,54]. By νω,η (A) := ω1A , ηH

(14)

for ω, η ∈ HC and Borel set A ⊂ K, we define weighted energy measures for 1-forms. Note that for any f ∈ FC we have νf ⊗1 = νf . Note also that every νω,η is absolutely continuous with respect to ν, and x → ωx , ηx H,x is a version of its density. To stress the similarity to the energy density we also use the notation ΓH,x (ωx , ηx ) := ωx , ηx H,x .

(15)

A derivation operator ∂ : FC → HC can be defined by setting ∂f := f ⊗ 1. It satisfies the Leibniz rule, ∂(fg) = f ∂g + g∂f,

f, g ∈ FC .

(16)

The linear operator ∂ is bounded, more precisely,

∂f 2H = E(f ),

f ∈ FC .

(17)

By the closedness of (E, FC ) in L2,C (K, ν) the derivation ∂ may be viewed as an unbounded closed operator from L2 (K, ν) into H. In coordinates y = y(x), the operator ∂ agrees with the usual gradient operator ∇ in R2 . This can be phrased as a corollary of Theorem 2.2 in terms of the isomorphisms Φx from (9). It may be viewed as a ‘pointwise formula’ for the derivation ∂. Corollary 2.1. For any f = F ◦ h and g = G ◦ h with F ∈ CC1 (R2 ) and G bounded Borel measurable on R2 we have   Φx (g∂f )x = Zx G(y)∇F (y) for ν-a.e. x ∈ K, and in particular,   Φx (∂f )x = Zx ∇F (y).

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Because HC is anti-isomorphic to itself, we regard its elements also as vector fields and ∂ as a generalization of the classical gradient operator. Let FC∗ denote the dual of FC with the norm



u F ∗ = sup u(f ) : f ∈ FC , E1 (f )  1 , where E1 (f ) := E(f ) + f 2L2 (K,ν) . The space FC∗ may be thought of as a ‘space of distributions’. The symbol ·,· will denote the dual pairing between FC and FC∗ . Note that FC ⊂ FC∗ and f, g = f, gL2 (K,ν) for f, g ∈ FC . Given a vector field of the form g∂f with f, g ∈ BC , its divergence ∂ ∗ (g∂f ) can be defined similarly as in [52] by 

∂ ∗ (g∂f )(ϕ) := −

gΓ (f, ϕ) dν, K

seen as an element in FC∗ . This extends continuously to a bounded linear operator ∂ ∗ from HC into FC∗ such that ∂ ∗ v(ϕ) = − v, ∂ϕH ,

v ∈ HC , ϕ ∈ F C ,

see [52, Lemma 3.1] for a proof. Note that here the divergence ∂ ∗ v is defined in a distributional sense. By restricting the space of test functions further this definition can be modified to fit into the context of distributions on p.c.f. fractals as studied in [88]. In coordinates we have the following expression. Theorem 2.3. Let f = F ◦ h, g = G ◦ h and u = U ◦ h with F, U ∈ CC1 (R2 ) and G bounded Borel measurable on R2 . Then  ∗ ∂ (g∂f )(u) = divZx (G∇F )(U )(y) dν, K

where for ν-a.e. x ∈ K, divZx (G∇F )(·)(y) := −



Zij (x)(G∇F )j (y)

ij

∂(·) (y). ∂yi

(18)

(18) may be seen as a bounded linear functional on CC1 (R2 ). In a sense it remotely reminds of the divergence in Riemannian coordinates. Proof. We have  − g∂f, ∂uH = −



 G(y)∇F (y), Zx ∇U (y) C2 ν(dx)

K

=−

 

K

ij

Zij (x)G(y)

∂F (y) ∂U (y)ν(dx). ∂yj ∂yi

2

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Let ν denote the energy Laplacian on K, that is, the infinitesimal generator of (E, F) in L2 (K, ν). Its complexification is the non-positive definite self-adjoint operator ν with domain dom ν associated with the closed form (E, FC ) on L2,C (K, ν) by E(f, g) = − f, ν gL2 (K,ν) for f ∈ FC and g ∈ dom ν . For general g ∈ FC we can set (ν g)(f ) := −E(f, g),

f ∈ FC ,

(19)

to define ν g as an element of FC∗ . Then (19) may be written as the dual pairing f, ν g. Since −∂ ∗ ∂g(f ) = E(f, g), we observe the FC∗ -identity ν g = ∂ ∗ ∂g

(20)

for any g ∈ FC . In coordinates we have the following. Theorem 2.4. For any f = F ◦ h with F ∈ CC2 (R2 ) we have f ∈ dom ν and ν f (x) = Tr(Zx Hess F )(y) for ν-a.e. x ∈ K. Moreover, given arbitrary u = U ◦ h with U ∈ CC1 (R2 ), the identity 

 divZx (∇F )(U )(y) dν =

K

ν f u dν K

holds. Proof. The first statement is a special case of [103, Theorem 8]. For the second note that 

 divZx (∇F )(U )(y)ν(dx) = −

K



 ∇F (y), Zx ∇U (y) C2 ν(dx)

K

= −E(f, u)  = ν f u dν.

2

K

Remark 2.2. In the classical Euclidean case we have div(G∇F ) = GF + ∇F ∇G for G ∈ C 1 (R2 ) and F ∈ C 2 (R2 ), where ∇,  and div denote the Euclidean gradient, Laplacian and divergence, respectively. In our case we have ∂ ∗ (g∂f ) = gν f + Γ (f, g) for f, g ∈ F , by [52, Lemma 3.2], seen as an equality in FC∗ . This may again be written in coordinates. Given f = F ◦ h and g = G ◦ h with F ∈ C 2 (R2 ) and G ∈ C 2 (R2 ) we have

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 divZx (G∇F )(U )(y) dν K







 ∇F (y), Zx ∇G(y) C2 U (y) dν

G(y) Tr(Zx Hess F )(y)U (y) dν +

= K

K

for any U ∈ C 1 (R2 ), as may be seen from the previous theorems. The divergence ∂ ∗ may also be seen in an L2 -sense as an unbounded linear operator from HC into L2,C (K, ν). As usual an element v ∈ HC is said to be a member of dom ∂ ∗ if there is some v ∗ ∈ L2,C (K, ν) such that v ∗ , uL2 (K,ν) = − v, ∂uH for all u ∈ FC . In this case we set ∂ ∗ v := v ∗ . The operator −∂ ∗ is the adjoint (codifferential) of ∂, and since ∂ is a densely defined and closed operator, dom ∂ ∗ is dense in HC and ∂ ∗ is closed. Note that for the previous distributional definition we obtain   ∂ ∗ v(ϕ) = ∂ ∗ v, ϕ L

2 (K,ν)

.

We end this section by a remark that allows to retrieve some more explicit information about the domain dom ∂ ∗ of the divergence operator ∂ ∗ in L2 -sense. Remark 2.3. As the quadratic form (E, FC ) is closed, the operator ∂ is closed. From (3) it follows that (E, FC ) has a spectral gap. Therefore the range Im ∂ of ∂ is a closed subspace of HC and the space HC decomposes orthogonally into Im ∂ and its complement (Im ∂)⊥ . By orthogonality we observe (Im ∂)⊥ = ker ∂ ∗ . A comprehensive Hodge and deRham theory for the Sierpinski gasket has been developed in [24]. A more general Hodge theory valid for finitely ramified fractals is provided in [54]. Strongly local Dirichlet forms on compact metric spaces are considered in [50]. There the space ker ∂ ∗ has been identified as the space of harmonic forms (or vector fields). Let dom ν denote the domain of ν in L2,C (K, ν). We obtain the following explicit description of dom ∂ ∗ . Corollary 2.2. We have 

dom ∂ ∗ = v ∈ HC : v = ∂f + w: f ∈ dom ν , w ∈ ker ∂ ∗ . For any v = ∂f + w with f ∈ dom ν and w ∈ ker ∂ ∗ we have ∂ ∗ v = ν f, and for any g ∈ dom ν formula (20) holds in L2 (K, ν). Note that the right hand side of this identity is explicitly seen to be dense in H: The denseness of the range of Green’s operator in L2 (K, ν) can be used to see that dom ν is dense in FC . Therefore the image ∂(dom ν ) of dom ν under the derivation ∂ is dense in Im ∂. Proof. According to Remark 2.3 any v ∈ dom ∂ ∗ ⊂ H admits a unique orthogonal decomposition v = ∂f + w with f ∈ FC and w ∈ ker ∂ ∗ . Since

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ν f = ∂ ∗ ∂f = ∂ ∗ v is in L2 (K, ν) we obtain f ∈ dom ν . Conversely, any vector field v = ∂f + w ∈ H with f ∈ dom ν and w ∈ ker ∂ ∗ is a member of dom ∂ ∗ since for any u ∈ FC we have − v, ∂uH = ∂f, ∂uH = E(f, u) = − ν f, uL2 (K,ν) . The last statements of the corollary are obvious consequence.

2

Remark 2.4. (i) Assume f = F ◦ h. If F ∈ CC2 (R2 ) then ∂f ∈ dom ∂ ∗ by Theorem 2.4. For general F ∈ CC1 (R2 ), however, we will not have f ∈ dom ν and therefore also not ∂f ∈ dom ∂ ∗ . (ii) Even if f = F ◦ h with F ∈ CC2 (R2 ) a simple vector field g∂f can generally not be expected to be an element of dom ∂ ∗ . Let PIm ∂ denote the orthogonal projection in HC onto Im ∂. It has been introduced and studied in [22], see also [54]. Given some measurable (or energy finite) g, the operator f → PIm ∂ (g∂f ) is a complicated non-local first order pseudodifferential operator. See [55] for some first results concerning pseudo-differential operators on fractals. 3. Dirac operators The definitions of the derivation ∂ and the divergence ∂ ∗ allow to define related Dirac operators. Consider the Hilbert space L2,C (K, ν) ⊕ HC and write 

 (f, ω), (g, η) ⊕ := f, gL2 (K,ν) + ω, ηH

for its scalar product. We define the Dirac operator associated with (E, F) to be the unbounded linear operator D on L2,C (K, ν) ⊕ HC with domain dom D := FC ⊕ dom ∂ ∗ , given by   D(f, ω) := −i∂ ∗ ω, −i∂f ,

(f, ω) ∈ dom D.

In matrix notation we have  D=

0 −i∂

−i∂ ∗ 0

 .

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Remark 3.1. Here the signs and imaginary factors are chosen in a way that fits with the following section. Of course other, possibly more physical choices, can be considered by obvious modifications. The next theorem was obtained in abstract form in [21], but for the convenience of the reader we sketch a proof. Note that the results in Section 2 show that D is a pointwise defined local operator. Theorem 3.1. The operator (D, dom D) is self-adjoint operator on L2,C (K, ν) ⊕ HC . Proof. Obviously dom D is dense in L2,C (K, ν) ⊕ HC . Recall that by definition

dom D ∗ = (g, η) ∈ L2,C (X, m) ⊕ HC : there exists some (g, η)∗ ∈ L2,C (X, m) ⊕ HC      such that (f, ω), (g, η)∗ ⊕ = D(f, ω), (g, η) ⊕ for all (f, ω) ∈ dom D . Note that (−i∂)∗ = −i∂ ∗ . For arbitrary (f, ω) and (g, η) from dom D we have 

    −i∂ ∗ ω, −i∂f , (g, η) ⊕ = −i∂ ∗ ω, g L

+ −i∂f, ηH  = ω, −i∂gH + f, −i∂ ∗ η L (X,m) 2    = (ω, f ), −i∂ ∗ η, −i∂g ⊕ . 2 (X,m)



Consequently dom D ⊂ dom D ∗ and D ∗ ω = Dω for all ω ∈ dom D. Now the self-adjointness follows from the closedness of ∂ and ∂ ∗ . 2 Our next aim is to justify our nomenclature by showing that in an appropriate sense, D 2 := D ◦ D is the Laplacian. The Kusuoka Laplacian ν acts on functions, and we need to discuss a second Laplacian which acts on 1-forms. From [50] we recall the following. Let 

dom ν,1 := ω ∈ dom ∂ ∗ : ∂ ∗ ω ∈ FC and for ω ∈ dom ν,1 , define ν,1 ω := ∂∂ ∗ ω.

(21)

To ν,1 we refer as the form Laplacian associated with (E, F). The following result had been shown in [50, Theorem 6.1]. Theorem 3.2. Definition (21) yields a self-adjoint operator (ν,1 , dom ν,1 ) on HC . Now set dom D 2 := dom ν ⊕ dom ν,1 , clearly a dense subspace of L2,C (K, ν) ⊕ HC . Then the following is immediate. Lemma 3.1. For any (f, ω) ∈ dom ν ⊕ dom ν,1 we have D 2 (f, ω) = (ν f, ν,1 ω).

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In matrix notation this is

 D = 2

ν 0

0 ν,1

 .

4. Magnetic Schrödinger operators The notions defined in Section 2 also allow to define a suitable generalization of the quantum mechanical Hamiltonian H A,V = (−i∇ − A)2 + V

(22)

for a particle moving in Euclidean space Rn subject to a real (magnetic) vector potential A and a real-valued (electric) potential V . The main result of this section is Theorem 4.1 below, which tells that there exists a self-adjoint operator on L2,C (K, ν) that generalizes (22). Another result, Theorem 4.2, is a gauge invariance result and tells that, roughly speaking, we may restrict attention to divergence free vector potentials. We collect some preliminaries. Let H = {v ∈ HC : v = v} be the space of real vector fields. An element a ∈ H may be seen as a bounded linear mapping a : FC → HC defined by f → f a. It admits the estimate f a H  f L∞ (K,ν) a H . For a ∈ H, define a bounded linear mapping a ∗ : HC → FC∗ by v → a ∗ v, where   ∗  a v (ϕ) := ϕΓH (a, v) dν = aϕ, vH , ϕ ∈ FC . K

The boundedness follows from

   = sup ϕΓ (a, v) dν : ϕ ∈ F with E (ϕ)  1 1 C H F∗

 ∗  a v 

K

together with  ϕΓH (a, v) dν  ϕ L (K,ν) a, vH  cE(ϕ)1/2 a H v H , ∞ K

where we have used (3) and the Cauchy–Schwarz inequality in HC . In terms of duality we observe a ∗ v(ϕ) = a ∗ v, ϕ. Given f ∈ FC we have in particular a ∗ af := a ∗ (af ) ∈ FC∗ with   ∗  a af (ϕ) = ϕf ΓH (a) dν. K

For u ∈ FC∗ and f ∈ FC define the product f u ∈ FC∗ by (f u)(ϕ) := u, ϕf ,

ϕ ∈ FC ,

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for convenience we also use the notation uf := f u. Recall that for a ∈ H we have ∂ ∗ a ∈ FC and therefore 

      ∂ ∗ a f (ϕ) = ∂ ∗ a, ϕf = − a, ∂(ϕf ) H .

Now let a ∈ H and V ∈ L∞ (K, ν). The preceding considerations ensure that      E a,V (f, g) := −ν + 2ia ∗ ∂ + a ∗ a + i ∂ ∗ a + V f, g , f, g ∈ FC , provides a well-defined form E a,V on FC × FC that is linear in the first and conjugate linear in the second argument. The right hand side in this definition is interpreted as a dual pairing. Proposition 4.1. Let a ∈ H and V ∈ L∞ (K, ν). (i) For any f, g ∈ FC we have   E a,V (f, g) = (−i∂ − a)f, (−i∂ − a)g H + f V , g.

(23)

(ii) The form E a,V is a quadratic form on FC . Before proving Proposition 4.1, we verify a product rule. Lemma 4.1. For a ∈ H and f ∈ FC we have   ∂ ∗ (f a) = ∂ ∗ a f + a ∗ ∂f, seen as an equality in FC∗ . Proof. Using the Leibniz rule (16) we see that 

      ∂ ∗ a f (ϕ) = − a, ∂(f ϕ) H = − a, ϕ∂f H − a, f ∂ϕH = − a ∗ ∂f (ϕ) + ∂ ∗ (f a)(ϕ)

for any ϕ ∈ FC .

2

We prove Proposition 4.1. Proof. The right hand side of (23) rewrites E(f, g) + M(f, g) + f V , g, where M(f, g) := i ∂f, agH − i af, ∂gH + f a, gaH . For the first summand on the right hand side we observe

(24)

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i∂f, agH = ∂f, −iagH = −iag, ∂f H and similarly for the second. Consequently M(g, f ) = M(f, g), and therefore the expression (24) is conjugate symmetric. We show that (24) equals E a,V (f, g). Note first that 

af, ∂gH = f dΓH (a, ∂g) = a ∗ ∂g(f ). X

By Lemma 4.1 this equals        ∂ ∗ (ga)(f ) − g ∂ ∗ a (f ) = − ag, ∂f H − ∂ ∗ a, gf H = − ∂f, agH − ∂ ∗ a f, g H . Next, note that 

∂f, agH =

  gΓH (∂f, a) dν = a ∗ (∂f )(g) = a ∗ (∂f ), g .

K

Consequently        M(f, g) = 2i a ∗ ∂f, g + i ∂ ∗ a f, g + a ∗ af, g , and taking into account (20), identity (23) in (i) follows. Statement (ii) is a straightforward consequence. 2 Recall formula (15). To

 H∞ := ω ∈ H: v = v and ΓH,· (v) ∈ L∞ (K, ν)

(25)

we refer as the space of (real) vector fields of bounded length. This space is nontrivial, note that (11) implies 2 2   ΓH,x (g∂f )x = G(y) Zx ∇G(y) C2 for any f, g ∈ SC , and if G and ∇F are bounded, we have g∂f ∈ H∞ . Assume a ∈ H∞ . Then the multiplication f → f a may be seen as bounded linear operator a : L2,C (K, ν) → HC by f → f a, because 

f a H = 2

  |f |2 ΓH,· (a) dν  f 2L2 (K,ν) ΓH,· (a)L

∞ (K,ν)

.

(26)

K

From (11) it easily follows that  

S = span f ⊗ g: f = F ◦ h, g = G ◦ h with F, G ∈ C 1 R2 is a subset of H∞ . The estimate (26) allows the application of classical perturbation arguments to prove the closedness of E a,V and the essential self-adjointness of an analog of (22).

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Theorem 4.1. Let a ∈ H∞ and V ∈ L∞ (K, ν). (i) The bilinear form (E a,V , FC ) is closed. (ii) The unique self-adjoint non-negative definite operator associated with (E a,V , FC ) is given by H a,V = (−i∂ − a)∗ (−i∂ − a) + V , and the domain of ν is a domain of essential self-adjointness for H a,V . Proof. Recall that E a,V (f, g) equals (24) for any f, g ∈ FC . We show that for any ε > 0 and any f ∈ FC , M(f, f )  ε2 E(f ) + Ca,V f 2

L2 (K,ν) ,

(27)

where Ca,V > 0 is a constant bounded by V L∞ (K,ν) plus a multiple of ΓH,· (a) L∞ (K,ν) . Then the result follows by the classical KLMN theorem, cf. [85, Theorem X.17]. By the boundedness of V clearly f V , f L

2 (K,ν)

 V L

∞ (K,ν)

f 2L2 (K,ν) .

The bound (26) covers the summand f a, f aH , and applying it once more,     2 ∂f, f aH  ∂f H f a H  1 ε 2 ∂f 2 + 1 ΓH,· (a)

f L2 (K,ν) . H L∞ (K,ν) 2 ε2 Taking into account also (17), we arrive at (27).

2

Remark 4.1. Theorem 4.1 is a rather simple result and can certainly be improved. For instance, to obtain essential self-adjointness for magnetic Schrödinger operators on Euclidean spaces it is sufficient that the vector potential A is locally in L4 and div A is locally in L2 , while the scalar potential V may be taken to be locally in L2 . See [75]. What we would like to point out here is that any analog of such a hypothesis will again require the weighted energy measure νa of the 1-form a as in (14) to be absolutely continuous with respect to the reference measure ν. If in (22) a gradient ∇Λ of a real-valued function Λ is added to the vector potential A, then the operator H A+∇Λ,V is unitarily equivalent to H A,V , more precisely, H A+∇Λ,V = eiΛ H A,V e−iΛ . This property is usually referred to as gauge invariance, cf. [26,75]. We observe a similar behavior in our case. Theorem 4.2. Assume V ∈ L∞ (K, ν) and a, b ∈ H∞ . If b = a + ∂λ with some λ ∈ F , then dom H b,V = dom H a,V and H b,V = eiλ H a,V e−iλ .

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Remark 4.2. Recall Remark 2.3. Let P⊥ denote the orthogonal projection onto (Im ∂)⊥ . By Theorem 4.2 the operator H a,V is uniquely determined up to unitary equivalence by V and P⊥ a. In this sense we can always restrict attention to divergence free vector potentials a ∈ H∞ ∩ ker ∂ ∗ . In the classical case the condition ∂ ∗ a = 0 is referred to as Coulomb gauge condition, see for instance [94]. To prove Theorem 4.2 we first verify the following lemma. Lemma 4.2. Let a ∈ H∞ , let λ = Λ ◦ h, Λ ∈ C 1 (R2 ) and set b := a + ∂λ. Then we have   E a,0 e−iλ f = E b,0 (f ),

f ∈ FC .

Proof. Note first that under the stated hypotheses we also have b ∈ H∞ . Assume that f = F ◦ h with F ∈ CC1 (R2 ). By (9) we have       Φx −i∂ e−iλ f x = −iZx −i∇Λ(y)e−iΛ(y) F (y) + e−iΛ(y) ∇F (y)    = Φx −e−iλ ∂λ − ie−iλ ∂f x for ν-a.a. x ∈ X, and as the Φx are isomorphisms also     −i∂ e−iλ f x = −e−iλ ∂λ − ie−iλ ∂f x in the spaces HC,x . Integrating, we obtain   −i∂ e−iλ f = −e−iλ ∂λ − ie−iλ ∂f in HC and therefore eiλ (−i∂ − a)e−iλ f = (−i∂ − b)f, what implies the desired equality. For general f ∈ FC let (fn )n with fn = Fn ◦ h and Fn ∈ CC1 (R2 ) be a sequence approximating f in E . Then a,0 E (f ) − E a,0 (fn ) = E(f ) − E(fn ) + 2 ∂f, af H − ∂fn , afn H + af 2 − afn 2 . H H The first term clearly tends to zero, for the second we have the estimate 2 ∂f − ∂fn , af H + 2 ∂fn , af − afn H  2E(f − fn )1/2 af H + 2 f − fn L∞ (K,ν) a H sup E(f )1/2 , n

which tends to zero by (3). The third does not exceed     a(f − fn ) af H + afn H , H what is similarly seen to converge to zero.

2

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We prove Theorem 4.2. Proof. For λ as in the theorem we obviously have ∂λ ∈ H∞ . Let (an ) ⊂ S be a sequence approximating a in H and let (λn )n with λn = Λn ◦ h and Λn ∈ C 1 (R2 ) be a sequence approximating λ in E. Set bn := an + ∂λn . Then Lemma 4.2 implies   E an ,V e−iλn f = E bn ,V (f ),

f ∈ FC .

(28)

Now we first claim that E b,V (f ) = lim E bn ,V (f ), n

f ∈ FC .

(29)

This follows from b,V   E (f ) − E bn ,V (f )  2 ∂f, (b − bn )f + f b 2 − f bn 2 H H H because we have the upper bound 2E(f )1/2 f L∞ (K,ν) b − bn H  2E(f ) b − bn H for the first summand, and for the second,     (b − bn ) f L (K,ν) f b H + f bn H . ∞ H Combining, we arrive at (29). Next, note that we also have     E a,V e−iλ f = lim E an ,V e−iλn f , n

f ∈ FC .

(30)

To see this, note that a,V  −iλ    E e f − E an ,V e−iλn f          E a,V e−iλ f − E a,V e−iλn f + E a,V e−iλn f − E an ,V e−iλn f 2  2        2 ∂ e−iλ − e−iλn f , a H + ae−iλ f H − ae−iλn f H 2  2      + 2 ∂ e−iλn f , a − an H + ae−iλn f H − an e−iλn f H , what can be estimated by         2∂ 1 − ei(λ−λn ) e−iλ f H + 2 1 − ei(λ−λn ) ∂ e−iλ f H         +  1 − ei(λ−λn ) e−iλ f a H e−iλ f a H + e−iλn f a H    + 2∂ e−iλn f H a − an H        + e−iλn f (a − an )H e−iλn f a H + e−iλn f a H . By the chain rule, [36, Theorem 3.2.2], we have

(31)

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E 1−e

i(λ−λn )



 =

i(λ−λ ) 2 n Γ (λ − λ ) dν = E(λ − λ ), e n n

K

and consequently  lim n

iλ 2   e f Γ 1 − ei(λ−λn ) dν  lim f L (K,ν) E(λ − λn ) = 0. ∞ n

K

The function z → 1 − e−iz is continuous and bounded, and by (3) we have limn λn = λ uniformly on K. Therefore  2   lim 1 − ei(λ−λn ) Γ e−iλ f dν = 0 n

K

by bounded convergence. Combining these estimates we see that the first line in (31) converges to zero. The other terms in (31) obey similar bounds, use (3), (4) and note that   sup E e−iλn = sup n



n

−iλ 2 e n Γ (λn ) dν = sup E(λn ) < ∞. n

K

They all converge to zero, and (30) becomes evident. Clipping (28), (29) and (30) we obtain the equality of closed forms   E a,V e−iλ f = E b,V (f ),

f ∈ FC ,

which implies the coincidence of the associated self-adjoint operators and therefore Theorem 4.2. 2 5. Other fractals The results of the preceding sections easily carry over to regular resistance forms on finitely ramified fractals for which it is possible to find a complete (up to constants) energy orthonormal system h1 , . . . , hk of harmonic functions and that satisfy some further regularity conditions. In this case h(x) := (h1 (x), . . . , hk (x)), x ∈ X, defines a homeomorphism from X onto its image h(X) in Rk . Examples satisfying these hypotheses comprise quantum graphs, the residue set of the Apollonian packing, hexagaskets or post-critically infinite Sierpinski gaskets. A counterexample for which these assumptions fail to hold is the Vicsek set. See [103], in particular Section 8, and [54,59,61] for background and precise definitions. We can then define a (normed) Kusuoka energy measure as the sum ν := νh1 + · · · + νhk of the corresponding energy measures, cf. [103, Definition 3.5]. By [103, Theorem 3] the form (E, F) is a regular Dirichlet form on L2 (X, ν). Following Kusuoka [70] one can construct a matrix valued function Z on X as before, such that Zij is a version of the density of νhi ,hj with respect to ν. By [103, Theorem 6] the collection of functions F ◦ h with F ∈ CC1 (Rk ) is dense in FC and formula (5) still holds. All results of Section 2 may be rewritten for X in place of the Sierpinski gasket K. With minor modifications the constructions of Section 4 apply to any local regular Dirichlet form on a locally compact separable metric space that admits energy densities (i.e. a carré du

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