ON THE GEOMETRY OF SEMICLASSICAL LIMITS ON DIRICHLET SPACES BATU GU¨NEYSU 7 1 0 2 n Abstract. ThispaperisacontributiontosemiclassicalanalysisforabstractSchrödinger a typeoperatorsonlocallycompactspaces: LetX beametrizableseperablelocallycompact J space, let µ be a Radon measure on X with a full support. Let (t,x,y) p(t,x,y) 8 7→ be a strictly positive pointwise consistent µ-heat kernel, and assume that the generator 1 H 0 of the corresponding self-adjoint contraction semigroup in L2(X,µ) induces a ] repgu≥lar Dirichlet form. Then, given a function Ψ : (0,1) (0, ) such that the limit G limt→0+p(t,x,x)Ψ(t) exists for all x X, we prove that fo→r ever∞y potential w : X R ∈ → D one has . lim Ψ(t)tr e−tHp+w = e−w(x) lim p(t,x,x)Ψ(t)dµ(x) < h t→0+ Z t→0+ ∞ t (cid:0) (cid:1) a forthe Schrödingertype operatorHp+w,providedw satisfiesverymildconditionsat , ∞ m that are essentially only made to guarantee that the sum of quadratic forms H +w/t is p [ self-adjoint and bounded from below for small t, and to guarantee that 1 e−w(x) lim p(t,x,x)Ψ(t)dµ(x)< . v Z t→0+ ∞ 8 Theproofisprobabilisticandreliesonaprincipleofnotfeelingtheboundaryforp(t,x,x). 9 Inparticular,thisresultimpliesanewsemiclassicallimitresultforpartitionfunctionsvalid 9 4 onarbitraryconnectedgeodesicallycompleteRiemannianmanifolds,andonealsorecovers 0 a previously established semiclassical limit result for possibly locally infinite connected . weighted graphs. 1 0 7 1. Introduction 1 : v To motivate all following results, let us take a look at the case when (X,g) is either the Xi Euclidean Rm or a closed Riemannian m-manifold, noting that for the sake of exposition we will not care about self-adjointness issues and other technicalitities for the moment. r a Given a function f : R R, classical results by Helffer/Robert [11] (see also [1] for closed → X’s) show that for a sufficiently bounded smooth potential w : X R, one has → Z (g;w;f;~) QM (1) lim = 1, ~→0+ Zcl(g;w;f;~) where Z (g;w;f;~) := tr f ~2∆ +w QM g (cid:0) (cid:0) (cid:1)(cid:1) denotes the “quantum partition function” and 1 Z (g;w;f;~) := f p 2 +w(q) dp dq, cl (2π~)m Z | |g∗ ∧ T∗X (cid:0) (cid:1) 1 2 B. GU¨NEYSU denotes its classical analogue. Note that the integral in Z (g;w;f;~) is a globally well- cl defined integral of a differential form in Ω2m(T∗X), as dp dq is (up to a constant) just ∧ the coordinate representation of the m-th power of the symplectic form on T∗X. The proofs of these results use an asymptotic expansion of ~mZ (g;w;f;~) as ~ QM → 0+, and which naturally require strong global growth conditions on the potential w, in addition to smoothness. As their formulation is built upon cotangent bundles and forms, semiclassical limit results such as (1) for general f’s clearly cannot be formulated in other settings than manifold-like spaces. On the other hand, the most important choice of f for quantum physics1 and geometry is f(λ) := e−λ, where then the numerator in (1) becomes the usual quantum partition function from statistical physics and the denominator its classical analogue. Now (1) reflects the fact every quantum data having a classical analogue should by approximated by the latter as ~ 0+. Moreover, in the exponential case the momentum integration → (thatis,thep-integration)inZ (g;w;f;~)factorsandbecomesaGaussianintegral, leading cl straightforwardly to the equivalence of (1) to (2) lim(2πt)m/2tr e−t(∆g+w/t) = e−wdµ , t→0+ Z g (cid:0) (cid:1) X with dµ the Riemannian volume measure. This simple observation is used as a tool in a g proofgiven in[20], but it isactually much morethanjust a tool: the abovereformulationof (1)doesnotrefertomanifoldsatall, aslangasoneconsiders ∆ abstractlyasthegenerator g of a strongly continuous self-adjoint contraction semigroup on an L2-space. Looking for an abstract formulation of (2), it clearly remains to clarify the meaning of the regularizing factor (2πt)m/2. Of course it is tempting to believe that this factor stems from the on- diagonal singularity of the Euclidean heat kernel. Another aspect of results like (2) is completely hidden in the Riemannian setting, namely the choice of measure on the RHS of (2). While in the Riemannian case it is the measure of the underlying Hilbert space, something else happens on weighted graphs: It has been shown recently in [5], that for the canonically given self-adjoint Laplacian ∆ in ℓ2(X,µ) on a possibly locally infinite b,µ connected weighted graph (X,b,µ) (cf. Example 2.8) one has (3) lim tr e−t∆b,µ+w/t = e−w(x). t→0+ (cid:0) (cid:1) X x∈X What is seemingly different now from the Riemannian case (2) is that on the right-hand side the integration is not with respect to the Hilbert space measure µ. On the other hand, as for the heat kernel on graphs one has lim p (t,x,x) = 1/µ(x), b,µ t→0+ 1Althoughotherchoicesoff areofcourseneededinquantumphysics: Forexample,ithasbeenpointed to the author by B. Helffer that the choice f(λ) :=Slog(1+e−S/λ), S >0, is used in solid state physics (cf. [12] for a mathematical description of the De Haas - van Halfen effect). ON THE GEOMETRY OF SEMICLASSICAL LIMITS 3 one can rewrite the exponential integral according to e−w(x) = e−w(x) lim p (t,x,x)µ(x). b,µ X X t→0+ x∈X x∈X This lead us to the following observations: Firstly, an abstract formulation of (2) on a locally compact space X with a well-behaved Radon measure µ should be built from an abstract µ-heat kernel p(t,x,y) (cf. Section 2), replacing the Laplace operator with the self-adjoint generator H of the semigroup induced by p(t,x,y). Secondly, taking once p more the Riemannian case (2) into account, where one has lim p (t,x,x)(2πt)m/2 = 1, g t→0+ the analysis should be ultimately built assuming the existence of a function Ψ : (0,1) → (0, ) such that the limit lim p(t,x,x)Ψ(t) exists for all x X. By what we have t→0+ ∞ ∈ observed so far, the natural formulation of (2) now becomes (4) lim Ψ(t)tr e−t(Hp+w/t) = e−w(x) lim p(t,x,x)Ψ(t)dµ(x). t→0+ Z t→0+ (cid:0) (cid:1) Withoutlosinganyimportantexample, wewillassumethatH stemsfromaregularDirich- p let form, and that p(t,x,y) satisfies a probabilistic principle of not feeling the boundary (cf. the new Definition 2.3). Our main result, Theorem 2.9, states that in this situation the formula (4) holds for every possibly unbounded continuous potential w : X R, under very mild global assumptions → on w that are only made to make all involved quantities well-defined at all. As we have indicated above, Theorem 2.9 is completely new even from a conceptually point of view. We believe it is a remarkable fact that it is possible to formulate such a result in a possibly nonlocal Dirichlet space setting, allowing to treat Riemannian manifolds and weighted graphs simultaniously. As a corollary to this result, we obtain a new semiclassical limit result for arbitrary complete connnected Riemannian manifolds (Corollary 2.11), which is seemingly the first result of this type for noncompact manifolds other than Rm. In addition, Corollary 2.11 seems to be even new in the Euclidean Rm, in the sense that the conditions on the potential are weaker than in all previously known results. It is important to note that Corollary 2.11 does not impose any curvature conditions on the geometry. On the other handthis result can besignificantly refined incase the Ricci curvatureis bounded from below, as is shown in Corollary 2.12, using the volume doubling machinery. Finally, we prove that Theorem 2.9 allows to recover the formula (3) for graphs (Corollary 2.13). We close this section with some remarks about the proof of Theorem 2.9: In [1, 11] the authors develop an asymptotic expansion of the underlying trace in order to prove the semiclassical limit (see also [2] for a different analytic approach). Our proof, on the other hand, is probabilistic and in the spirit of [20], which treats the Euclidean Laplace operator. In fact, the machinery developed in this paper entails that it is actually sufficient to know the leading order term of the usual t 0+ asymptotic expansion of the unperturbed → heat kernel diagonal e−tHp(x,x), in the sense that then one can use path integrals as a perturbative tool to reduce everything to the latter expansion. One moral of this story is 4 B. GU¨NEYSU that, if one is only interested in its semiclassical limit for the trace, asymptotic expansions forthetraceoftheperturbedoperatorareanoverkill. Inaddition, asindicatedabove, these expansions naturally require many strong local and global assumptions on the underlying data. We point out that the proof of Theorem 2.9 becomes significantly more complicated than the one from [20]. The main reason for this is that the on-diagonal values of the Euclidean heat kernel e−tHg(x,x) = (2πt)m/2 do not depend on x at all, and in addition one has many translation-invariance arguments that are based on Wiener measures. To compensate this lack of symmetries (which of course aren’t even present on arbitrary Riemannian manifold) in the general case, we were lead to the new Definition 2.6, and we found that a principle of not feeling the boundary plays an essential role in the context of semiclassical limits of abstract Schro¨dinger semigroups. This seems to be a new observation. Another difficulty of our generalized setting is that the Hunt process associated with the regular Dirichlet form that stems from p(t,x,y) has in general càdlàg (and possibly explosive) paths, which makes some arguments more subtle. This is of course the price for working with possibly nonlocal Dirichlet forms. Acknowledgements: Iwould like tothankBernardHelffer for manyvery helpful remarks on the literature. In addition, I would like to thank Klaus Mohnke, Ralf Rueckriemen and Eren Ucar for helpful discussions. This research has been financially supported by the project SFB 647: Raum - Zeit - Materie of the German research foundation (DFG). 2. Main results In the sequel, all function spaces are understood to be complex-valued. Let X be a seper- able metrizable locally compact space and let µ be a Radon measure on X with a full support. We fix once for all a strictly positive pointwise consistent µ-heat kernel p : (0, ) X X (0, ), (t,x,y) p(t,x,y). ∞ × × −→ ∞ 7−→ By definition, p is a jointly Borel measurable function such that for all t > 0, x,y X one ∈ has (5) p(t+s,x,y) = p(t,x,z)p(s,z,y)dµ(z), Z (6) p(t,x,y) = p(t,y,x), (7) p(t,x,z)dµ(z) 1, Z ≤ such that with P f(x) := p(t,x,y)f(y)dµ(z) t Z it holds that P f f 2dµ 0 as t 0+, for all f L2(X,µ). Z | t − | → → ∈ ON THE GEOMETRY OF SEMICLASSICAL LIMITS 5 Itfollowsthat(P ) isastronglycontinuousself-adjointcontractionsemigroupinL2(X,µ), t t>0 so let H 0 denote its self-adjoint generator, and Q 0 the densely defined symmetric p p ≥ ≥ sesquilinear form associated with H . p Definition 2.1. p is called regular, if C (X) Dom(Q ) is dense in C (X) with respect cpt p cpt ∩ to the uniform norm, and dense in Dom(Q ) with respect to the norm Q (f,f)+ f 2dµ. p p | | R If p is regular, then Q becomes a regular Dirichlet form [4] in L2(X,µ), a fact that should p justify this notion. Given t > 0 let Ω(X,t) denote the space of càdlàg paths γ : [0,t] X. For every 0 s t let F (X,t) denote the sigma-algebra on Ω(X,t) which is generat→ed by the coordina≤te m≤ap s on Ω(X,t) including the time s, and let the sub-σ-algebra F−(X,t) F (X,t) be defined s ⊂ s by F−(X,t) := smallest-σ-algebra on Ω(X,t) which contains F (X,t). s u [ 0≤u0 in our case). This immediately gives the first part of the following result, where Px0 is just t the restriction of the law of this process with initial point x to paths that do not explode 0 until t: Proposition and definition 2.2. Assume that p is regular. a) For every x X, the Wiener measure Px0 from x with terminal time t w.r.t. p is defined to be th0e∈unique sub-probability measurte on F (X0 ,t) which satisfies t (8) Px0 γ : γ(t ) A ,...,γ(t ) A t { 1 ∈ 1 n ∈ n} = p(δ ,x ,x ) p(δ ,x ,x )dµ(x ) dµ(x ) Z ···Z 0 0 1 ··· n−1 n−1 n 1 ··· n A1 An for all n N , all partitions 0 = t < t < < t < t = t and all Borel sets ≥1 0 1 n−1 n ∈ ··· A ,...,A X, where δ := t t . 1 n j j j−1 b) For ever⊂y x ,y X, the pi−nned Wiener measure Px0,y0 from x to y with terminal time t w.r.t. p 0is d0efi∈ned to be the unique probability meatsure on F−0(X,t)0which satisfies t 1 (9) Px0,y0(A) = p(t s,γ(s),y )dPx0(γ) t p(t,x ,y ) Z − 0 t 0 0 A for all 0 s < t and all A F (X,t). s ≤ ∈ Note that Px0 is in general only a sub-probability measure, as we have made no conser- t vativeness assumption. The second part of Proposition 2.2 follows from the results of [3]. Note the subtlety that the pinned Wiener measure is only defined F−(X,t), which is in t general not equal to F (X,t), as the paths are not left-continuous. t Now we can give the following definition: 6 B. GU¨NEYSU Definition 2.3. If p is regular, then it is said to satisfy the principle of not feeling the boundary, if for all compact subsets K X having a nonempty interior K˚ and all x K˚, ⊂ ∈ the pinned Wiener measures satisfy lim Px,x γ : γ(s) K for all s [0,t) = 1. t→0+ t { ∈ ∈ } Note that in the situation of Definition 2.3 one indeed has γ : γ(s) K for all s [0,t) F−(X,t), { ∈ ∈ } ∈ t as can be seen from Ω(X,t) γ : γ(s) K for all s [0,t) = γ : γ(s) X K . \{ ∈ ∈ } { ∈ \ } [ s∈[0,t)∩Q It is shown below thatthe principle of not feeling the boundaryholds onarbitrarycomplete Riemannian manifolds and weighted graphs. We believe that an abstract investigation of this property should be of an independent interest. Let us now take a look at the perturbations of H by potentials w : X R that we will p → consider in the sequel. The next (well-known) definition is motivated by the fact that we want to investigate the behaviour of operators of the form exp( (tH +w)), where t > 0 p − is small. In order to use the Feynman-Kac formula, which is valid for expressions of the form exp( t(H +w′)), we have to factor p − exp( (tH +w)) = exp( t(H +w/t)) p p − − and this shows that we have to guarantee that H +w/t is semibounded from below for all p small t′s. This clearly requires some control on the negative part of w, which is the simple idea behind the following definition, that will implement the above in a general quadratic form sense: Definition 2.4. We say that a Borel function w : X R is in the infinitesimal Q -class p M (X,µ), if for all ǫ > 0 there exists an C 0 such t→hat for all f Dom(Q ) one has p ǫ p ≥ ∈ w f 2dµ ǫQ (f,f)+C f 2µ < . Z | || | ≤ p ǫZ | | ∞ Clearly, this property depends only on the µ-equivalence class of w. It is also important to note that bounded functions are in M (X,µ). In case p is regular, the results from [21] also p show that (X,µ) M (X,µ), where (X,µ) stands for the Kato class with respect to p p p K ⊂ K p and µ. Recall here that a Borel function w : X R is by definition in (X,µ), if and p → K only if t lim p(s,x,y) w(y) dµ(y)ds= 0. t→0+Z Z | | X 0 Finally, (possibly weighted) Lq-conditions on w that imply w (X,µ) can be found in p ∈ K [17, 6]. Given a function w : X R let w± = max( w,0) denote its positive and negative part → ± respectively, so that w = w+ w−. − ON THE GEOMETRY OF SEMICLASSICAL LIMITS 7 Proposition and definition 2.5. Let p be regular and assume we are given a potential w : X R with 1 w L1(X,µ) for all compact K X and w− M (X,µ). Then the K p → ∈ ⊂ ∈ symmetric sesqui-linear form (10) Q (w)(f ,f ) := Q (f ,f )+ wf f dµ p 1 2 p 1 2 Z 1 2 with domain of definition given by all f Dom(Q ) such that w f 2dµ < , is densely p ∈ | || | ∞ defined, semibounded (from below) and closed in L2(X,µ). TRhe semibounded self-adjoint operator in L2(X,µ) associated with the form Q (w) will be denoted with H (w). p p Proof. The claim for the case w = 0 follows immediately from the KLMN theorem. For + the general case, one just has to note that the maximally defined form given by w is + densely defined and closed (using Fatou’s lemma). Thus (10) is just the sum of closed symmetric forms and thus has these properties, too. In addition, the form is densely defined, as it clearly contains Dom(Q ) C (X), and the latter set is dense in L2(X,µ), p cpt ∩ by the regularity of p. (cid:4) In order to formulate our main result, we add: Definition 2.6. A pair (̺ ,̺ ) of Borel functions 1 2 ̺ : (0,1) (0, ), ̺ : X [0, ) 1 2 −→ ∞ −→ ∞ is called an asymptotic control pair for p, if the following assumptions are satisfied: the limit lim p(t,x,x)̺ (t) exists for all x X. t→0+ 1 • ∈ there exists a Borel function φ : (0,1) (0, ) such that • → ∞ p(t,x,x) . ̺ (x)φ(t) for all (t,x) (0,1) X, sup φ(t)̺ (t) < . 2 1 ∈ × ∞ 0 1, let r (x,b) be the supremum of Eucl,g ∈ all r > 0 such that the open ball B (x,r) with respect to the geodesic distance admits a g coordinate system ~x = (x1,...,xm) : B (x,r) U Rm g −→ ⊂ with ~x(x) = 0 and with respect to which one has one has the following inequality for all y B (x,r), g ∈ (11) (1/b)(δ ) (g (y)) := (g(∂ ,∂ )(y)) b(δ ) as symmetric bilinear forms. ij ij i j ij ≤ ≤ Then r (x,b) is called the Euclidean radius of X at x with accuracy b. It is shown in Eucl,g [6] that for all b > 1 the function (12) ̺ (x) := 1/min(r (x,2),1)m, x X, g Eucl,g ∈ turns (̺(m),̺ ) into an asymptotic control function for p , without any curvature assump- g g tions on (X,g). This result relies on a parabolic L1-mean value inequality for solutions of the heat equation. In case the Ricci curvature Ric is bounded from below by a constant, then the function g ̺′(x) := 1/µ (B (x,1)), x X, g g g ∈ turns (̺(m),̺′) into an asymptotic control function for p (cf. Example 2.7 in [6]). This g g follows from the Li-Yau heat kernel estimate and volume doubling. As for graphs: Example 2.8. Recall that a weighted graph is a triple (X,b,µ), with X is a countable set, b a symmetric function b : X X [0, ) with b(x,x) = 0, b(x,y) < for all x X, × −→ ∞ ∞ ∈ X y∈X and µ : X (0, ) is an arbitrary function. One equipps X with its discrete topology → ∞ (which is thus induced by the discrete metric) and writes x y if b(x,y) > 0, referring to b ∼ such points as the edges of graph. In this sense, the points of X become the vertices and µ a vertex weight function. The function µ defines a Radon measure having a full support in the obvious way: µ(A) = µ(x), A X. ⊂ X x∈A In particular, the scalar product on L2(X,µ) is given by f (x)f (x)µ(x). x∈X 1 2 We assume also that (X,b) is connected in the graph theoPretic sense, meaning that for any x,y X there is a finite sequence x ,...,x X such that x = x, x = y. 1 n 0 n ∈ ∈ ON THE GEOMETRY OF SEMICLASSICAL LIMITS 9 Denoting the space of complex-valued functions on X with C(X), where an index ‘c’ now simply means ‘finitely supported’, and a set of functions F (X) given by all ψ C(X) b ∈ such that b(x,y) ψ(y) < for all x X, we define a Laplace type formal difference y∈X | | ∞ ∈ operator ∆P by b,µ 1 (13) ∆ : F (X) C(X), ∆ ψ(x) = b(x,y) ψ(x) ψ(y) . b,µ b b,µ −→ −µ(x) − X (cid:0) (cid:1) {y:y∼bx} Then for all fixed y X, there exists [16] a pointwise minimal element p ( , ,y) of the b,µ ∈ · · set given by all bounded functions u : (0, ) X [0, ) ∞ × → ∞ that satisfy ∂ u(t,x) = ∆ u(t,x), lim u(t,x) = δ (x) for all (t,x) (0, ) X. t b,µ y t→0+ ∈ ∞ × The function (t,x,y) p (t,x,y) is strictly positive (this property requires the above b,µ 7→ notion of connectedness) and defines a strictly positive pointwise consistent and regular µ -heat kernel [16]. In addition, one has the trivial estimates µ p (t,x,y) 1/µ(x) for all (t,x,y) (0, ) X X, b,µ ≤ ∈ ∞ × × and the convergence lim p (t,x,x)µ(x) = 1 for all x X. b,µ t→0+ ∈ The first property follows from p(t,x,y)µ(y) 1, and the second one from the strong x ≤ continuity of P (noting by discPreteness, L2-convergence implies pointwise convergence). t To see that p satisfies the principle of not feeling the boundary, note that in this case b,µ Px,x is concentrated on pure jump paths, and one has t Px,x γ : γ(s) x,x ,...,x for all s [0,t) Px,x γ : γ has not jumped before t t { ∈ { 1 l} ∈ } ≥ t { } t exp b(x,y) p(t,x,x)−1µ(x)−1, ≥ (cid:16)− µ(x) X (cid:17) y where the first estimate is trivial and the second one is well-known (cf. page 10 in [5] and the references therein). Finally, in this case the operator H := H is a restriction of b,µ pb,µ ∆ [16, 10, 5], and we set M(X,b,µ) := M (X,µ). b,µ pb,µ Here is our main result: Theorem 2.9. Assume that there exists a metric on X which induces the original topology in a way that for every x X there is an r > 0 such that the open metric ball B(x,r) is ∈ relatively compact. Assume also that p satisfies the principle of not feeling the boundary. Then for every asymptotic control pair (̺ ,̺ ) for p, and every continuous potential w : 1 2 X R with w− M (X,µ) and e−w̺ dµ < , one has p 2 → ∈ ∞ R (14) lim ̺ (t)tr e−tHp(w/t) = e−w(x) lim p(t,x,x)̺ (t)dµ(x) < . t→0+ 1 Z t→0+ 1 ∞ (cid:0) (cid:1) 10 B. GU¨NEYSU The proof of Theorem 2.9 will be given in Section 3 below. A few remarks about the formulation of Theorem 2.9 are in order: Remark 2.10. 1. The integrability e−w(x) lim p(t,x,x)̺ (t)dµ(x) < Z t→0+ 1 ∞ is not assumed, but is a consequence of the assumptions. Clearly, this implies that for all sufficiently small t one has tr e−tH(w/t) < , showing that H (w/t) has a discrete p ∞ spectrum for small t’s. (cid:0) (cid:1) 2. The continuity of w is only used to prove the lower bound liminf̺ (t)tr e−tHp(w/t) e−w(x) lim p(t,x,x)̺ (t)dµ(x), t→0+ 1 ≥ Z t→0+ 1 (cid:0) (cid:1) and for this to hold an inspection of the proof shows that it is enough to assume that w is continuous away from a closed set N with µ(N) = 0. The upper bound limsup̺ (t)tr e−tHp(w/t) e−w(x) lim p(t,x,x)̺ (t)dµ(x) t→0+ 1 (cid:0) (cid:1) ≤ Z t→0+ 1 remains true under much weaker local assumptions on w than continuity. For example, an assumption of the form 1 w L1(X,µ) (X,µ) for all compact K X would do. K p ∈ ∩K ⊂ Let us specialize our main result to the Riemannian setting (cf. Example 2.7 for the corresponding notation). Recall that ̺(m)(t) := (2πt)m/2. Corollary 2.11. Assume that (X,g) is a smooth geodesically complete connected Rie- mannian m-manifold. Then for every Borel function ̺ : X [0, ) which makes (̺(m),̺) → ∞ an asymptotic control pair2 for p, and for every continuous potential w : X R with w− M(X,g) and e−w̺dµ < , one has → g ∈ ∞ R lim(2πt)m/2tr e−tHg(w/t) = e−w(x)dµ (x) < . t→0+ Z g ∞ (cid:0) (cid:1) IncaseRic isboundedfrombelowbyaconstant, onecanturntheintegrabilityassumption g from Corollary 2.11 on w into a natural pointwise one: Corollary 2.12. Let (X,g) be a smooth geodesically complete connected Riemannian m- manifold with Ric A for some constant A 0, and let w : X R be a continuous g ≥ − ≥ → potential with inf w > and3 X −∞ ∞ (15) exp inf w(x) kme2k√(m−1)A < for some x X. 0 X (cid:16)−x∈X,k−1 0 satisfies (15). g 0 A similar linear growth cond→ition from below≥suffices, too, as long as∈A′ >A.

On the geometry of semiclassical limits on Dirichlet spaces

Similar On the geometry of semiclassical limits on Dirichlet spaces