On the spectrum of the hierarchical Laplacian

Let $(X,d)$ be a locally compact separable ultrametric space. We assume that $(X,d)$ is proper, that is, any closed ball $B$ in $X$ is a compact set. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of balls (the choice function), we define the hierarchical Laplacian $L_C$ which is closely related to the concept of the hierarchical lattice of F.J. Dyson. $L_C$ is a non-negative definite, self-adjoint operator in $L^2(X,m)$. We address in this paper to the following question: How general can be the spectrum $\mathsf{Spec}(L_C)$ as a subset of the non-negative reals? When $(X,d)$ is compact, $\mathsf{Spec}(L_C)$ is an increasing sequence of eigenvalues of finite multiplicity which contains $0$. Assuming that $(X,d)$ is not compact we show that, under some natural conditions concerning the structure of the hierarchical lattice (= the tree of $d$-balls), any given closed subset $S$ of $[0,\infty)$, which contains $0$ as an accumulation point and is unbounded if $X$ is non-discrete, may appear as $\mathsf{Spec}(L_C)$ for some appropriately chosen function $C(B)$. The operator $-L_C$ extends to $L^q(X,m)$, $0<q<\infty$, as Markov generator and its spectrum does not depend on $q$. As an example, we consider the operator $\mathfrak{D}^{\alpha}$ of fractional derivative defined on the field $\mathbb{Q}_p$ of $p$-adic numbers.


Introduction
The concept of hierarchical lattice and hierarchical distance was proposed by F.J. Dyson in his famous papers on the phase transition for 1D ferromagnetic model with long range interaction [5,6].
The notion of hierarchical Laplacian L, which is closely related to the Dyson's model was studied in several mathematical papers [4], [12,13,14] and [17].
These papers contain some basic information about L (the spectrum, Markov semigroup, resolvent etc) in the case when the hierarchical lattice satisfies some symmetry conditions (homogeneity, self-similarity etc). Under these symmetry conditions, Spec(L) is pure point and all eigenvalues have infinite multiplicity. The main goal of the papers mentioned above was to introduce a class of random perturbations of L and then to justify the existence of the spectral bifurcation from the pure point spectrum to the continuous one.
A systematic study of a class of isotropic Markov semigroups defined on an ultrametric space (X, d) has been done in [1] (see also the forthcoming paper [2]). In particular, given an isotropic Markov semigroup (P t ) with Markov generator −L, one can show that the operator L is a hierarchical Laplacian on (X, d) associated with an appropriate choice function C(B) and vice versa. Then the general theory developed in [1] and [2] applies: modifying canonically the underlying ultrametric d, we call this new ultrametric d * , the set Spec(L) is pure point and can be described as In our construction the families of d-balls and d * -balls coincide, whence these two ultrametrics generate the same topology and the same hierarchical structure, and in particular, the same class of hierarchical Laplacians. The equation (1.1) leads us to the following question. In the course of study we assume that (X, d) is a locally compact and separable ultrametric space. Recall that a metric d is called an ultrametric if it satisfies the ultrametric inequality d(x, y) ≤ max{d(x, z), d(z, y)}, (1.2) that is obviously stronger than the usual triangle inequality. Usually, we also assume that the ultrametric d is proper, that is, each closed d-ball is a compact set.
The paper is organized as follows. In Section 2 we recall some basic properties of ultrametric spaces. The main original results there can be summarized in the following statement (see Proposition 2.5, Theorem 2.9, Theorem 2.16). where The operator (L C , D) acts in L 2 = L 2 (X, m), is symmetric and admits a complete system of eigenfunctions where B ⊂ B ′ are nearest neighboring balls; when m(X) < ∞, we also set f X, when m(X) < ∞, we set λ(X ′ ) = 0. In particular, we conclude that (L C , D) is an essentially self-adjoint operator in L 2 . By abuse of notation, we shell write (L C , Dom L C ) for its unique self-adjoint extension. Let B ⊂ B ′ be two nearest neighboring balls. Choosing the function for any B ∈ B. Applying Theorem 1.1, we answer the question (A). Actually, we get Theorem 1.2 not only for the particular choice function mentioned above but for more general types of them (see the proof in Section 3).
A very simple Example 2.4 shows that the condition "there exists a partition Π of X made of d-balls containing infinitely many non-singletons" in statement (2) of Theorem 1.1 and Theorem 1.2 can not be dropped: X = N and d(m, n) = max(m, n) when m = n and 0 otherwise.
In the concluding Section 4 we consider the operator D α of the p-adic fractional derivative of order α > 0. This operator related to the concept of p-adic Quantum Mechanics was introduced by V.S. Vladimirov, see [19], [20] and [21]. We prove that D α is a hierarchical Laplacian. The main novelty here is that D α admits a closed extension in L q , 1 ≤ q < ∞, call it D α q . The closed operator −D α q coincides with the infinitesimal generator of a translation invariant Markov semigroup acting in L q . The set Spec(D α q ) consists of eigenvalues p kα , k ∈ Z, each of which has infinite multiplicity and contains 0 as an accumulation point. In particular, Spec(D α q ) = Spec(D α 2 ), for all 1 ≤ q < ∞. We study also random perturbations D α (ω) of the operator D α and provide a limit behaviour of its normalized eigenvalues.

Metric matters
Recall that a topological space X is totally disconnected if for any two distinct points x, y ∈ X there exists a closed and open (=clopen) subset U of X such that x ∈ U and y / ∈ U; if X has a basis consisting of clopen subsets, then X is called zero-dimensional.
Clearly, zero-dimensional spaces are totally disconnected but there are Polish spaces which are totally disconnected and not zero-dimensional (for example, the complete Erdős space [7,Section 1.4] ). Nevertheless, for locally compact Hausdorff spaces these two notions coincide, i.e. totally disconnected locally compact Hausdorff spaces are zero-dimensional.
At the beginning let us recall two classical topological characterizations which are crucial in the study of zero-dimensional separable metric spaces (see, e.g., [11, p. 35] It is well known that each non-empty locally compact, non-compact, metrizable, separable space X can be compactified by adding an extra point ω whose neighborhoods are declared to be of the form {ω}∪(X \K), where K is a compact subset of X. One can easily check that ω has a countable basis of neighborhoods of this form. It follows that the compact space X ∪ {ω} has a countable base, so it is metrizable. If, additionally, X has no isolated points (i.e., X is perfect) and is totally disconnected, then X ∪{ω} is homeomorphic to C by Proposition 2.1. Thus we get the following characterization.
Proposition 2.2 Each metrizable locally compact, non-compact, separable, perfect, totally disconnected space is homeomorphic to C \ {p}, where p is an arbitrary point of C.
Let us now list some basic properties of ultrametric spaces (X, d) (see [3, p. 227], [15]).   If an ultrametric space (X, d) is separable, then the following facts also hold. (f) All distinct balls of a given radius r > 0 form at most countable partition of X.
If, in addition, X is compact, then It is easy to see that properties (b) and (c) are in fact characteristic for an ultrametric: • if d is a metric on X satisfying either one of them then d is an ultrametric.
It follows from the above properties that each ultrametric space has a basis of clopen sets, i.e., it is zero-dimensional. Conversely, • each zero-dimensional separable metrizable space X is metrizable by an ultrametric d.

Definition 2.3 A metric d on a set X is called proper if every closed
Notice that any proper metric is complete. It is known that any metrizable, locally compact, separable space admits a proper metric [18].
The proper ultrametric on N given in the next example is, in a sense, generic for metrizable, locally compact, non-compact, separable totally disconnected spaces.
Then any d max -ball is either a singleton or is of the form {1, 2, . . . , n}.
Proposition 2.5 If X is a metrizable, locally compact, separable, totally disconnected space, then X admits a proper ultrametric that generates the topology of X.
Proof. There is nothing to prove if X is compact, since any ultrametric metrizing X is automatically proper. Assume that X is not compact. Then there is a partition Π = {P 1 , P 2 , . . . } of X made of non-empty, compact-open subsets of X. Let d be an ultrametric on X that generates the topology of X. We get our proper ultrametric by the following formula: , if x, y ∈ P k for some k; max(m, n), if x ∈ P m , y ∈ P n and m = n.
Example 2.6 One of the most known example of a proper ultrametric space is the field Q p of p-adic numbers endowed with the p-adic norm x p and the This ultrametric space is locally compact, non-compact, separable, perfect and totally disconnected, so it is homeomorphic to the Cantor set minus a point (see Proposition 2.2).

Example 2.7 The other example which we have in our mind is a discrete Abelian group
is the minimal value of n such that x and y belong to the same coset of the Proof. Let us take any infinite, countable partition R of X consisting of compact-open subsets of X and let . . }, then there are mutually disjoint two-point sets P 1 , P 2 , . . . such that ∞ n=1 R n = ∞ n=1 P n and we can put The proof is finished.
Notice that if (X, d) is a proper ultrametric space then, for any increasing function φ : Range(d) → [0, ∞), metric d ′ = φ•d is again a proper ultrametric having the same collection of balls as d. Hence, by Proposition 2.5, each nondegenerate, metrizable, locally compact, separable, totally disconnected space admits infinitely many proper equivalent ultrametrics.
Let (X, d) be a compact separable ultrametric space. It follows from property (g) that if X is finite, then Range(d) is finite and if X is infinite, we can arrange the values of d in a sequence decreasing to 0, i.e., Range(d) = {c 1 , c 2 , . . . }, where c n ց 0. Since, for any other sequence c ′ n ց 0, there is an increasing surjection φ : The idea of the proof of Theorem 2.9 is based on a specific tree-structure of the family of balls in an ultrametric space. So, let us first introduce necessary notions.
Let T (X, d) be the collection of all balls in a proper ultra-metric space (X, d). Consider T (X, d) with a partial order Observe that, by properties (c) and (g), each ball B ∈ T (X, d) has a unique immediate predecessor with respect to and if B is not a singleton, then it has at most finitely many immediate successors. If A ∧ B = inf{A, B}, the infimum taken with respect to (which is the smallest, with respect to the inclusion, ball containing balls A and B), then (T (X, d), ∧) is a semilattice. We prefer to view T (X, d) geometrically as a graph with vertices being elements of T (X, d) and edges being pairs of d-balls (B, B ′ ) such that B is an immediate successor or predecessor of B ′ .
A path in T (X, d) from B 1 to B n is a finite sequence B 1 , B 2 , . . . , B n of mutually distinct vertices such that, for each i = 1, 2, . . . , n − 1, either (B i , B i+1 ) or (B i+1 , B i ) is an edge. Given two vertices A, B, there are unique paths from A to A ∧ B and from B to A ∧ B and the concatenation of these two paths gives the unique path from A to B. Thus, T (X, d) is a countable, locally finite, path-connected tree. Vertices with no successor are called end-points of the tree; they represent singleton balls.
Let 2 Y be the family of compact nonempty subsets of a Hausdorff topological space Y . We consider 2 Y with the Vietoris topology (which is generated by the subbase of sets of the form a) : a ∈ A} (see, e.g., [10]).
Whitney maps for 2 Y exist for metric separable spaces Y [10, p. 205]. It is easy to see that the diameter function diam is never a Whitney map for The following proposition will be used in Section 3.
denotes the smallest ball containing x and y) and d * (x, y) = 0 for x = y defines a proper ultrametric in X which induces the same topology and the same collection of balls as d.
Proof. It is easy to verify that d * is an ultrametric. It is equivalent to d by the continuity of w at each singleton. In order to show that d * is proper, let B * r (x) be a closed d * -ball of radius r centered at x. Notice that (2.1) implies that there is a d-ball B(x), centered at x, containing B * r (x). Since B(x) is compact and B * r (x) is closed, we conclude that the ball B * r (x) is compact.
Proof of Claim 2.12. Clearly, we can assume that B(x) is nondegenerate.
The balls {x} ∧ {y n } are contained in a branch of T (X, d), so we can choose a subsequence n k such that balls {x} ∧ {y n k } form a decreasing family of sets, in view of (2.2). So d * (x, y n k ) → w(B(x)) if k → ∞ and points x, y n k , k ∈ N, belong to a compact set {x} ∧ {y n 1 }. It means that Range(d * ) has an accumulation point different from 0 on a compact set, contrary to property (g). It remains to show that each nondegenerate Indeed, the inclusion ⊂ in (2.3) is obvious for any metric. The inclusion ⊃ is also trivial in the case r ′ / ∈ Z. But the case r ′ ∈ Z cannot occur because Proof of Theorem 2.9. We are going to construct a countable, locally finite, path-connected tree T ⊂ 2 X without the least element and a Whitney map w for T ∪ F 1 such that Range(w) = M (recall that F 1 is the set of all singletons in X).
For each P ∈ Π, let T (P, d p ) be the rooted (at P ) tree of closed d p -balls of X which are contained in P . Observe that if P is a doubleton, then T (P, d p ) splits into two singletons, otherwise T (P, d p ) has an infinite branch contained in the set T (P, d p ) of non-singleton vertices. Trees T (P, d p ) extend the tree φ(T (N, d max )) and as a result we obtain a tree T ⊂ 2 X .
To each branch L of T there corresponds a point x L ∈ X such that {x L } = L (since (X, d p ) is a complete space) and this correspondence is a bijection between the set of all branches and X. Observe that the bijection locally (inside of each P ∈ Π) preserves d p -balls. The semi-lattice operation ∧ on T can now be extended over the set of all singletons of X by We will now construct a Whitney map w for T ∪F 1 such that Range(w) = M. Enumerate positive numbers in M as m 1 , m 2 , . . . and choose a sequence (k n ) ∈ M such that k 1 > max{m 1 , m 2 }, k n+1 > max{k n , m n+1 } (this can be done since M is unbounded). Notice that w(B) = diam B in the metric d and T = T (X, d).

Remark 2.13
As we have already remarked in the proof of Theorem 2.9, the trees T (X, d p ) and T (X, d) locally coincide, i.e., the collections of d pballs and d-balls are the same within each P ∈ Π. Whether one can build an ultrametric d on X which satisfies conditions of the theorem and such that collections of all d p -balls and d-balls coincide is an interesting on its own and useful in applications question, see Section "Hierarchical Laplacian". Example 2.4 shows that, in general, the answer is negative. On the other hand, the answer is positive under the following extra condition: There is a partition Π of X consisting of d p -balls and infinitely many of the balls are non-singletons. In terms of the order : there is an infinite antichain in T (X, d p ) (i.e., a subset of T (X, d p ) whose elements are pairwise incomparable by ) which contains at most finitely many end-points.
Notice that a maximal antichain in T (X, d p ) is a partition of X.
The above condition evidently holds if the ultrametric space X is perfect (or contains at most finitely many isolated points).
The following example (a particular case of Example 2.7) is a good illustration of the condition in case of discrete X.

Example 2.14 Consider the infinite countable discrete group
with the standard ultrametric All d p -balls are either finite subgroups G k or their cosets G k + g. The balls form a binary tree T (X) without the least element and with singletons as its end-points. Proof. By Lemma 2.15 there is a partition Π such that each nondegenerate element P ∈ Π either contains an accumulation point or all immediatesuccessors of P are singletons. Let {B 1 , B 2 , . . . } ⊂ Π be the family of all nondegenerate elements of Π.
We slightly modify the proof of Theorem 2.9 by considering the original tree of d p -balls over partition Π instead of tree φ(T (N, d max )).
Let 0 = l 0 < l 1 < l 2 · · · → ∞ be a sequence such that Consider a function κ : Let us define a Whitney map w for T (X, d p ) ∪ F 1 . Put w = 0 for all singletons and let w(B i ) = m i . Each d p -ball B preceding some P ∈ Π uniquely decomposes into the union of distinct elements of Π (one of them is P itself): B = P i 1 ∪ · · · ∪ P in for some i 1 < · · · < i n . We can also observe that d(x, y) ≤ d p (x, y) for x, y ∈ B and each d p -ball B properly contained in P ∈ Π.

Remark 2.17
We can compare Theorem 2.9 with a result in [15,Theorem 2] which says that by a slight change of an ultrametric d in an arbitrary separable ultrametric space (X,d) one can get an equivalent ultrametric r ≤ d that assumes only dyadic rational values. No preservation of balls is discussed in [15].

Hierarchical Laplacian
The aim of this section is to justify the properties of the hierarchical Laplacian Given a choice function C(B) and a measure m as above we consider the hierarchical Laplacian (L C , D) defined pointwise by the equation (1.3), that is, .
Proof. Since the intersection L 1 ∩ L ∞ is a subset of each L p , p > 1, it is enough to prove the claim if p equals 1 and ∞. For any ball T of positive measure we set f T = 1 T /m(T ) and compute L C (f T )(x), Next observe that for any ball B centered at x, It follows that Clearly we have For the second term in (3.1), call it u 2 , we have Let T := T 0 ⊂ T 1 ⊂ T 2 ⊂ ... ⊂ X be an increasing sequence of balls such that each T l+1 is the immediate predecessor of T l . We set T −1 = ∅ and write Applying the Abel transformation we obtain It is easy to see that all functions f B,B ′ ∈ D and that for any two distinct balls B ′ and C ′ the functions f B,B ′ and f C,C ′ are orthogonal in L 2 = L 2 (X, m). Proposition 3.3 In the above notation the following properties hold.
for any x ∈ X and B ′ ∈ B.
In particular, (L C , D) is a non-negative definite essentially self-adjoint operator in L 2 . By abuse of notation, we shell write (L C , Dom L C ) for its unique self-adjoint extension.
Proof. For the first claim, consider f B = 1 B /m(B) for any ball B of positive measure and observe that the series converges pointwise, and since the series (3.2) converges in L 2 as well. This evidently proves the claim. For the second claim, fix a couple of closest neighbors T ⊂ T ′ and write the equation (3.1) for both T and T ′ . Subtracting the T ′ -equation from the T -equation we obtain as desired.
The operator (L C , D) acts in L 2 by Lemma 3.2, its symmetry follows by inspection. Since (L C , D) has a complete system of eigenfunctions, it is essentially self-adjoint, i.e. it admits a unique self-adjoint extension. The proof is finished.
The modified ultrametric d * associated with the operator (L C , D) is defined by Observe that the function w : T (X, d) ∪ F 1 → [0, ∞), w(B) := 1/λ(B) and w = 0 at each singleton, is a Whitney map. By Proposition 2.11, d * is a proper ultrametric which induces the same topology and the same collection of balls as d and, as one easily verifies, .
which completes the proof of Theorem 1.2.
Let P t = exp(−tL C ), t ≥ 0, be a symmetric contraction semigroup generated by the self-adjoint operator (L C , Dom L C ).

Proposition 3.4 The semigroup {P t } has the following representation
where σ t (r) = exp(−t/r) and B r (x) is the d * -ball of radius r centered at x.
In particular, {P t } is an isotropic Markov semigroup on the ultrametric measure space (X, d * , m) as defined and studied in [2].
Proof. We choose f = f B and compute P t f (x). Using the identity (3.2) we obtain Next observe that for any ball T centered at x, With this observation in mind we write the equality from above as Applying the Abel transformation and the definition (3.3) of the modified ultrametric d * we get the desired equality with f = f B . The set spanned by the functions f B is dense in L 2 , the result follows.
L p -Spectrum of the hierarchical Laplacian Consider the semigroup P t = exp(−tL C ). As {P t } is symmetric and Markovian, it admits an extension to L q , 1 ≤ q < ∞, as a continuous contraction semigroup, call it {P q t }, Let (−L, Dom L ) be the infinitesimal generator of the semigroup {P q t } . Since the operator (L C , D) acts in L q and {P q t } extends {P t }, the operator L defines a closed extension of L C , call it L q C . Applying Theorem 7.8 in [2] we obtain

p-Adic Fractional Derivative
Consider the field Q p of p-adic numbers endowed with the p-adic norm x p and the p-adic ultrametric d(x, y) = x − y p . Let m be the normalized Haar measure on Q p , that is, m(Z p ) = 1, where Z p is the set of p-adic integers.
In the ultrametric space (Q p , d) all d-balls are either compact subgroups p k Z p or their cosets p k Z p + a; diam(p k Z p + a) = p −k and m(p k Z p + a) = p −k . The balls form a regular tree T p (X) of forward degree p without the least element and without end-points.
The notion of p-adic fractional derivative related to the concept of padic Quantum Mechanics was introduced in several mathematical papers Vladimirov [19], Vladimirov and Volovich [20], Vladimirov, Volovich and Zelenov [21]. In particular, in [19] a one-parametric family {(D α , D)} α>0 of operators (called operators of fractional derivative of order α) have been introduced. Recall that D is the set of all locally constant functions having compact support.
The operators D α were defined via Fourier transform available on locally compact Abelian group Q p , Moreover, it was shown that each operator D α can be written as a Riemann-Liouville type singular integral operator The aim of this section is to illustrate the results of Section 3 showing that the operator (D α , D) is in fact a hierarchical Laplacian. More precisely, we claim that (D α , D) is a hierarchical Laplacian corresponding to the choice function 2) or equivalently, the eigenvalues λ(B) are of the form In particular, the set Spec(D α ) consists of eigenvalues p kα , k ∈ Z, each of which has infinite multiplicity and contains 0 as an accumulation point.
To prove the claim observe that the Fourier transform F : f → f on the locally compact Abelian group Q p is a linear isomorphism from D onto itself. This basic fact and (4.1) imply that (D α , D) is an essentially self-adjoint and non-negative definite operator in L 2 = L 2 (Q p , m). Next we claim that the spectrum of the symmetric operator (D α , D) coincides with the range of the function ξ → ξ α p , that is, the eigenspace H(λ) corresponding to the eigenvalue λ = p kα , is spanned by the function and all its shifts f k (· + a) with a ∈ Q p /p k Z p . Indeed, the ball B s (0), p l ≤ s < p l+1 , is the compact subgroup p −l Z p of Q p , whence the measure ω s = 1 Bs(0) m/m(B s (0)) coincides with the normed Haar measure of that compact subgroup. Since for any locally compact Abelian group, the Fourier transform of the normed Haar measure of any compact subgroup is the indicator of its annihilator group and, in our particular case, the annihilator of the group p −l Z p is the group p l Z p , (see [8]), we obtain Computing now the Fourier transform of the function f k , Finally, we apply Proposition 3.3 to conclude that the essentially selfadjoint operator (D α , D) coincides with the hierarchical Laplacian (L C , D) with C(B) given by the equation (4.2).
At last, applying Proposition 3.5 we obtain the following result. In the general setting of Propositions 3.3 and 3.5, some eigenvalues may well have finite multiplicity and some not. Indeed, attached to each ball B of d * -diameter 1/λ there are the eigenvalue λ and the corresponding finite dimensional eigenspace H B . This eigenspace is spanned by the finitely many functions where C runs through all balls whose predecessor is It follows that in general, if there exists only a finite number of distinct balls of d * -diameter 1/λ then the eigenvalue λ has finite multiplicity. This is clearly not the case for the ultrametric measure space (Q p , d, m) and the operator D α . Indeed, every d * -ball has its diameter in the set Λ α = {p kα : k ∈ Z}, and each ball B 1/λ (0) centered at the neutral element 0 and of diameter 1/λ has infinitely many disjoint translates {a i + B 1/λ (0) = B 1/λ (a i ), i = 1, 2, ..., which cover Q p and are balls of the same diameter. Thus, all eigenvalues have infinite multiplicity.