Spectrum and Heat Kernel Asymptotics on General Laakso Spaces
Matthew Begue, Levi DeValve, David Miller, Benjamin Steinhurst
SSpectrum and Heat Kernel Asymptotics onGeneral Laakso Spaces
Matthew Begu´e, Levi DeValve, David Millerand Benjamin Steinhurst ∗ October 28, 2018
Contacts:[email protected] Begu´eDepartment of MathematicsUniversity of ConnecticutStorrs, CT 06269 [email protected] DeValveDepartment of MathematicsUniversity of ConnecticutStorrs, CT 06269 [email protected] MillerDepartment of MathematicsSalve Regina UniversityNewport, RI 02840 [email protected] Steinhurst Department of MathematicsUniversity of ConnecticutStorrs, CT 06269 USAt. +1 (860) 486-3923f. +1 (860) 486-4238
Abstract
We introduce a method of constructing a general Laakso space whilecalculating the spectrum and multiplicities of the Laplacian operator onit. Using this information, we find the leading term of the trace of theheat kernel and the spectral dimension on an arbitrary Laakso space. ∗ Research supported by NSF grant DMS-0505622 Corresponding author a r X i v : . [ m a t h . C A ] F e b Introduction
Much work has been done on the analysis of fractals, specifically concentratingon the spectrum of the Laplacian operator on irregular domains. One suchtopic are drums with Koch snowflake boundary, see for example [13]. Thispaper will be concerned instead with the irregular domain being a fractal itself.Some notable works with this type of domain include [3, 6, 8, 16, 17, 19] amongothers. Laakso’s spaces were introduced in [11]. They are a family of fractalswith an arbitrary Hausdorff dimension greater than one and were consideredoriginally for their nice analytic properties. Constructions of the Laakso spacesare given in [11, 16, 17] as well as in Section 2 of this paper. Theorem 6.1 in[16] gives the spectrum of the Laplacian operator on any given Laakso space, inTheorem 3.1 we give the multiplicities.An important part of the analysis of Laplacians is the heat equation andassociated heat kernel, which can reveal significant information about the oper-ator and underlying space. The information gained from studying heat kernelscan be applied in other areas of analysis as well as other fields such as physics.The papers [2, 19] are devoted to finding and analyzing the heat kernel and thetrace of the heat kernel. The notion of complex valued fractal dimensions andthe accompanying oscillating behavior of the heat kernel were studied in [2, 3].We begin by reviewing the construction of the Laakso spaces as presentedin [11, 16, 17] in Section 2. This section also contains background informationon the Hausdorff dimension, its calculation for Laakso spaces, and some specificvalues for certain Laakso spaces. In subsection 2.3 we define the Laplacianoperator that will be used throughout the rest of this paper.In Section 3 we begin by stating the spectrum of the Laplacian and themultiplicities of each eigenvalue (Theorem 3.1), while the rest of the section isdevoted to the proof of this result. In Section 3.1 we provide an analytical proofby examining, as in [16], the different “shapes” that make up the space. Sinceeach shape has a unique contribution to the spectrum counting the number ofshapes allows us to calculate the spectrum with multiplicities. In Section 3.2we verify the results computationally using MATLAB. Finally in Sections 4and 5 we use the spectrum and multiplicities obtained in Sections 2.2 and 3.1to calculate the trace of the heat kernel using the same method outlined fordiamond fractals in [2].
The spaces that will be analyzed were first defined by Laakso in [11]. Laakso’sspaces form an uncountable family of metric-measure spaces indexed by se-quences { j n } ∞ n =1 . An equivalent construction using projective limits was hintedat in [5] and fully developed in [17]. Then in [16], a more in depth descriptionof the projective limit construction was used to calculate the spectrum of theLaplacian constructed in [17]. We will be using the construction in [16] as it isalso well-suited for our calculations. 2he Laakso space can be visualized with a sequence of quantum graphs,denoted F n , n ≥
0, each an increasingly better approximation of the Laaksospace. The first of these graphs is the unit interval, denoted F . Laakso spacesare defined by a sequence { j n } ∞ n =1 of integers j n ≥
2, where each j n describedthe number of identifications at step n of the construction. To construct thegraph of F n +1 , first every cell, or interval between two nodes, of the F n graphis split evenly into j n segments by adding nodes. This graph is then duplicatedand connected at the newly-added notes. In this visualization, all nodes arearranged in columns. F F × { , } F Figure 1: Construction of F from F with j = 2.We describe a simple case, where j n = 2, n ≥
1. To obtain F , bisect F with a node. Then make a copy of this graph. Identify the new nodes “glueing”the two graphs together, represented by the arrow in Figure 1. This glueingprocess is the identification process described in [5]. This yields the graph F ,an X-shape with five nodes as seen in Figure 1.This procedure is repeated to obtain F from F . Nodes bisect each cell of F as seen in Figure 2. A duplicate copy of F is created and the two graphs are“glued” together at the newly added nodes. This is shown in Figure 2 wherethe solid line represents F and the dashed lines represent the copy of F . The j n = 2 Laakso space is the projective limit of the sequence of graphs { F n } ∞ n =0 F F Figure 2: Construction of F from F where j = 2. The dashed lines representthe second copy of F with the added nodes.As another simple example consider j n = 3, for all n ∈ N . Again startingwith the unit interval, F , F is constructed by splitting F into three subinter-vals and placing a node between each interval as shown in Figure 3. This graphis duplicated and is glued to the original graph at the newly added nodes. Thetwo nodes in the middle of the figure are connected by the middle interval andits copy, thus creating a loop shape that is not seen in the case where j = 2. Theouter thirds of the figure create a “V” shape, also seen in the j = 2 construction.These shapes, loop and “V”, will be two of those considered in Section 3. F F Figure 3: Construction of F from F with j = 3.In this paper, we deal with the general case where j n may vary at each ap-proximation level n . The sequence { j n } ∞ n =1 may be a constant integer, as seenin the previous examples. Or the sequence may alternate regularly betweentwo integers. Figure 4 shows the construction of F and F when { j n } ∞ n =1 = { , , , , ... } . The sequence { j n } ∞ n =1 could even be a completely random se-quence of integers. In any case, it is { j n } ∞ n =1 which defines the Laakso spaceand from which the properties are derived.4 =[2,3] j = [2 , , { j n } ∞ n =1 = { , , , , ... } Recall that as an inverse limit system the pair ( L, { F n } ) come with continuousprojection Φ n : L → F n . Definition 2.1.
The cell structure in a Laakso space, L , is determined by thepre-images under the map Φ n of the cells in the graph F n which approximates L Given a space as defined by Laakso in [11], and the construction of the spaceas given in [16], and the level of approximation, n , the cell structure has specificproperties, including number of cells, N n , and the interval length, I − n . Boththe number of cells and the interval length are dependent on the choice of j i forall i ≤ n . Proposition 2.1.
Each cell in F n has metric diameter I n = n (cid:89) i =1 j i (2.1) where I = 1 . In addition the number of cells is N n = 2 n n (cid:89) i =1 j i . (2.2) Proof. In F there is a single cell, the unit interval, with metric diameter equalto 1. At each step in the construction the diameter of each cell in F n is j − n times that of a cell in F n − . By induction the diameter of the cells in F n is I − n .There is a single cell in F . At each step of the construction there are 2 × j n cells in F n for every cell in F n − . Thus the number of cells in F n is 2 n I n .5 .2 Hausdorff dimension of the Laakso Space In order to discuss the Hausdorff dimension of Laakso spaces we fix our choiceof metric and measure. We use the path length metric. The measure usedis the probability measure that gives equal mass to all cells of a given depth.Implicitly in the given construction, we have restricted the Hausdorff dimensionto 1 ≤ Q ≤
2. In [16] the Hausdorff dimension Q of the Laakso Space associatedwith a constant j n at every level n is shown to be Q = 1 + log (2) log ( j )Here we give the Hausdorff dimension of a Laakso space associated with a generalsequence { j n } ∞ n =1 . The measure used in calculating the Hausdorff dimension isthe projective limit of Lebesgue measure on F n scaled to have total mass onefor all n . Lemma 2.1.
Given sequence { j i } ∞ i =1 the Hausdorff Dimension, Q , of the cor-responding Laakso space is given by Q j i = lim n →∞ log (cid:32) n n (cid:89) i =1 j i (cid:33) log (cid:32) n (cid:89) i =1 j i (cid:33) = lim n →∞ log (2 n I n ) log ( I n ) = lim n →∞ log (2 n ) log ( I n ) , (2.3) if the limit exists.Proof. Laakso spaces are lacunary self-similar sets as defined in [7] where thecontraction ratios at any n are equal. The number of identifications for eachcell at the i ’th iteration is j i and the formula that Igudesman gives in [7] canbe given in terms of n and j i . This formula uses the number of cells, N n andthe cell diameter, which is simply I − n . In the geodesic metric, the cell length isalso the diameter of the cell. The resulting formula is given above.While there are many sequences { j n } for which Q will not exist it is morerelevant to our interests that for every Q there exist sequence { j n } yieldinga Hausdorff dimension of Q . Different sequences { j n } can yield the same di-mension, as shown in Table 1. These values agree with the dimensions givenimplicitly in [11]. In [16, 17] Laakso spaces are described as projective limits of quantum graphsand it is shown how to extend a compatible family of self-adjoint operatorson the approximating quantum graphs to a self-adjoint operator on the limitspace, i.e. the Laakso space. It was also shown how to use the spectrum withmultiplicities of the operators on each quantum graph to determine the spectrum6 i Q j i log (6) log (4) [2 , , , , ... ] log (24) log (6) [3 , , , , ... ] log (24) log (6) Table 1: Hausdorff Dimension for Laakso Space associated with given sequenceof { j i } ∞ i =1 with multiplicities of the operator on the limit space. A quantum graph is ametric graph with a Hamiltonian operator, as described in [9, 10], the simplest ofwhich would be the Laplacian operator, i.e.: a Hamiltonian without a potential.On each metric graph, F n , consider the space of functions defined on thecollection of edges each treated as a line segment. Define an operator on thesefunction by ∆ n = − d dx . To make this a self-adjoint operator we need to alsospecify a suitable domain. A function is in Dom (∆ n ) if it is continuous every-where, continuously twice differentiable on each line segment, and has Kirchhoffmatching conditions at the nodes. Kirchhoff conditions require a function’s withdirectional first derivatives summing to zero at nodes. Definition 2.2.
A Laakso space, L , is a projective limit of the F n . Also thereexist projection maps Φ n : L → F n for all n . Thus any function on F n canbepulled back to a function on L by writing f ◦ Φ n = ˜ f : L → R . The pullingback is under the projections Φ n . By Theorem 7.1 in [17] those functions in
Dom (∆) that are pull backs aredense. so a complete set of eigenfunctions can be taken from this set. A con-sequence of this is that we can numerically approximate the spectrum of theLaplacian on the Laakso spaces working on some F n . Computations of theseapproximations are described in Section 3.2 along with calculations describedin Section 3.1. Tables 2 and 3 show calculated values of the spectrum of theLaplacian for Laakso space. ∆ Theorem 3.1 gives the spectrum and associated multiplicities of the Laplacianoperator by considering ∆ n on F n and on any Laakso space. We devote the restof the section to proving the theorem. Following the analytic arguments aredetails of computational experiments carried out before the analytic results wereavailable. We use an iterative, computer-assisted process to find the bottom endof the spectrum on a number of specific Laakso spaces. In all cases the computedresults and analytic results agree within the precision of the computations.7 = 3 n = 4 n = 5 n = 6 n = 7 Expectedλ m λ m λ m λ m λ m .
87 3 9 .
87 3 9 .
87 3 9 .
87 3 9 .
87 3 π = 9 . .
58 1 39 .
38 1 39 .
45 1 39 .
48 1 39 .
48 1 (2 π ) = 39 . .
35 8 88 .
32 8 88 .
70 8 88 .
81 8 88 .
82 8 (3 π ) = 88 . .
32 1 157 .
51 1 157 .
87 1 157 .
90 1 (4 π ) = 157 . .
46 3 242 .
85 3 245 .
76 3 246 .
63 3 246 .
71 3 (5 π ) = 246 . .
26 26 353 .
28 26 355 .
08 26 355 .
25 26 (6 π ) = 355 . .
78 3 479 .
86 3 483 .
19 3 483 .
51 3 (7 π ) = 483 . .
41 1 625 .
27 1 630 .
94 1 631 .
48 1 (8 π ) = 631 . .
18 8 789 .
22 8 798 .
30 8 799 .
15 8 (9 π ) = 799 . .
89 1 971 .
40 1 985 .
22 1 986 .
53 1 (10 π ) = 986 . . . . . π ) = 1194 . . . . π ) = 1421 . . . . . π ) = 1668 . . . . . π ) = 1934 . . . . . π ) = 2220 . . . . . π ) = 2526 . . . . . π ) = 2852 . . . . π ) = 3197 . . . . . π ) = 3562 . Table 2: Calculated Values of the first 20 Eigenvalues for { j n } ∞ n =1 = { , , , , ... } with multiplicity, m , the iteration value, n , and the expected value, λ . As n increases, the observed eigenvalues converge to the expected result. Theorem 3.1.
Given any Laakso space, L , with associated sequence { j i } ni =1 ,the spectrum of ∆ on Dom (∆) is ∞ (cid:91) k =0 { π k } ∪ ∞ (cid:91) n =1 ∞ (cid:91) k =0 { ( k + 1 / π I n } ∪ ∞ (cid:91) n =1 ∞ (cid:91) k =1 { k π I n }∪ ∞ (cid:91) n =2 ∞ (cid:91) k =1 { k π I n } ∪ ∞ (cid:91) n =2 ∞ (cid:91) k =1 (cid:26) k π I n (cid:27) (3.1) with associated multiplicities: , n , n − ( j n − I n − , n − ( I n − − , n − ( I n − −
1) (3.2) respectively.
This does correct a typographical error in the similar statement given in [16].
In order to determine the spectrum of the Laplacian on the Laakso space, theapproximating quantum graph is considered as a collection of simpler parts. In8
B C D
Expectedλ m λ m λ m λ m .
87 3 9 .
87 1 9 .
87 1 9 .
87 1 π . . . π ) .
48 1 39 .
48 1 39 .
48 1 39 .
48 3 (2 π ) .
82 8 88 .
82 8 88 .
82 2 88 .
82 1 (3 π ) .
91 1 157 .
91 1 157 .
91 1 157 .
91 3 (4 π ) . . .
85 2 (4 . π ) .
71 3 246 .
71 1 246 .
71 1 246 .
71 1 (5 π ) .
25 26 355 .
25 8 355 .
25 8 355 .
25 10 (6 π ) .
51 3 483 .
51 1 483 .
51 1 483 .
51 1 (7 π ) . . . π ) .
48 1 631 .
48 1 631 .
48 1 631 .
48 3 (8 π ) .
15 8 799 .
15 36 799 .
15 2 799 .
15 1 (9 π ) .
53 1 986 .
53 1 986 .
53 1 986 .
53 3 (10 π ) . . . π ) . . . . π ) . . π ) . . . . π ) . . . π ) . . . . π ) . . π ) . . π ) . . π ) . . π ) . . π ) Table 3: Calculated Values of the first 20 eigenvalues for given sequences of j i ’swith multiplicity, m and the expected value, λ . A= { } B= { } C= { } D= { } [16] it was determined that three distinct shapes with appropriate boundaryconditions could be used to construct any quantum graph representations ofLaakso spaces, save F , which is treated as a special case. Definition 3.1 definesthese three shapes shown in Figure 5 with their respective boundary conditionswhich are forced by the Kirchhoff matching conditions and the orthogonalityrequirements that assign an eigenfunction to a given representation level. Theseorthogonality conditions were discussed in detail in [16]. In short they allow thecounting arguments to count an eigenfunction only once. Definition 3.1. (a) A shape is a connected quantum sub-graph, as shown inFigure 5. In that figure the “D” denotes a Dirichlet boundary condition at thatnode and an “N” the Neumann condition.(b) A V is the shape consisting of three nodes: two nodes in a column andthe third node a second. The two nodes in the first column are degree one andthe node in the second column is degree two. When a V is in F n , it shares itsdegree two node with another shape thus making it a degree four node, as seen N
00 00 “ V ” “ Loop ” “
Cross ”Figure 5: Constructions of v’s, loops, and crosses along with associated bound-ary conditions in Figure 4.(c) A loop is the shape that consists of two nodes each of degree two. Thenodes are connected to each other by two cells, creating a loop. When a loop isin F n , both degree two nodes are shared as degree two nodes for another shapethus making them degree-four nodes, Figure 4.(d) A cross is the shape consisting of six nodes four of degree two and two ofdegree four. The degree two nodes each have a cell connecting the node to eachof the degree four nodes . Notice the cross is the only shape containing nodesof degree four in the subgraph. When a cross is in F n the degree two nodes areshared with degree two nodes of another shape thus making them degree fournodes, as in Figure 4. Before determining the spectrum of ∆ n on the three shapes, we must firstestablish the following proposition which describes how these three shapes areinvolved in the construction of F n and L . Proposition 3.1. (a) Any node in any quantum graph approximating a Laaksospace is either of degree one or degree four.(b) For any degree one node in F n , a V is produced in F n +1 .(c) For any degree four node in F n , a cross is produced in the construction of F n +1 .(d) Any cell in F n produces j n +1 − in F n +1 between the V ’s or crosses produced by the nodes in F n .(e) For n ≥ the number of nodes in F n is N n = 2 n − ( I n + 3) .Proof. (a) A degree one node in F n − gives rise to two degree one nodes in F n as an immediate consequence of the construction. Similarly a degreefour node gives rise to a single degree four node in F n . The new nodesin F n that are not nodes in a copy of F n − are the identification of twodegree two nodes, hence of degree four.(b) In the construction of F n +1 , the cell connected to a degree one node issplit into j n +1 intervals by adding ( j n +1 −
1) nodes. Then the graph is10uplicated yielding two rows of cells connected between j n +1 columns withtwo nodes in each, all of degree two except for the nodes at the end of thecell. The original and duplicated cells are connected at the newly addednodes. Thus the original node of degree one from F n remains degree onein F n +1 . The graph around the original node and it’s duplicate is a “V.”(c) In the construction of F n +1 , the four cells connected to the degree fournode in F n will be split into j n +1 intervals. To construct F n +1 , F n isduplicated, new nodes inserted, and connected at the newly added nodes.Thus the original degree-four node remains degree four and is duplicated,creating two degree-four nodes. The graph around the original node andit’s duplicate is a “cross.”(d) Parts b and c account for two of the j n +1 intervals. The rest produceloops. So, there are j n +1 − F n +1 for every cell in F n .(e) We will induct on n . The unit interval, F , has two nodes. Suppose that F n − has N n − = 2 n − ( I n − +3) nodes. Then N n = 2 × N n − +2 n − ( j n − I n − , the nodes from the two copies of F n − plus the new nodes of whichthere are j n − F n − and there are 2 n − I n − cellsin F n − . This simplifies to the claimed formula.We now generalize the results from [16] in three lemmas that give the eigen-values and multiplicites (counts) for each of the three shapes. Lemma 3.1.
For any n ≥ , the number of V ’s in F n is n . The eigenvaluesfor this shape at this level are: { [ I n ( k + 1 / π ] : k = 0 , , . . . } . (3.3) Proof.
We prove the count by induction. F is constructed out of F , which isa single cell connecting two degree one nodes. This implies by Proposition 3.1that F will have 2 V ’s. Now assume that for some arbitrary n ≥
1, the numberof V ’s in F n is 2 n . From Definition 3.1, the V is the only shape that has adegree one node. Furthermore, it has two degree one nodes. From Proposition3.1 we know that each degree one node in F n produces a V in F n +1 . From [16]the shapes defined in Definition 3.1 are all the possible shapes in the graphs, sothere cannot be any degree one nodes from any other shape. So the number of V ’s in F n +1 is twice the number of V ’s in F n . So F n +1 has 2 n +1 V ’s.In order to get the spectrum of ∆ n restricted to a V we look at the functionsin this domain that are orthogonal to the functions expressible on an interval.These functions have the property that the values on the top branch are thenegative of the values on the lower. We therefore need only consider the topbranch, as it fully determines the behavior on the bottom branch. This topbranch is one interval, and has Neumann boundary conditions at one end andDirichlet boundary conditions at the other. The length of the cell is I − n . So weare looking for eigenfunctions on intervals of length I − n with zero derivative at11ne end and zero value at the other. These come in the form cos( I n ( k + 1 / πx )where k = 0 , , . . . and x ∈ [0 , I − n ]. The eigenvalues in (3.3) are now obtainedin the usual way. Lemma 3.2.
For any n ≥ , the number of loops in F n is n − ( j n − I n − ) . (3.4) The eigenvalues for this shape at this level are { [ I n kπ ] : k = 1 , , . . . } . (3.5) Proof.
By Proposition 3.1 every cell in F n − produces j n − F n . Inorder to know how many loops are in F n , the number of cells in F n − are countedand multiplied by j n −
2. The number of cells in F n were already counted inProposition 2.1 and shown to be 2 n ( I n ). Substituting in n − n in thisexpression and multiplying by j n − n restricted to a loop we look at thefunctions in this domain that are orthogonal to the functions expressible on aninterval. Again, these functions have the property that the vales on the topbranch are the negative of those on the lower. As was the case with the V above, the orthogonality condition imposed on the functions reduces the ques-tion to only considering the top interval of length I − n with Dirichlet boundaryconditions. The eigenfunctions that fit these conditions are sin( I n kπx ) with k = 1 , , . . . and x ∈ [0 , I − n ]. These result in the set defined in (3.5). Lemma 3.3.
For any n ≥ , the number of crosses in F n is n − ( I n − − . (3.6) There are two sets of eigenvalues for this shape at this level. They are (cid:40)(cid:20)
12 ( I n kπ ) (cid:21) : k = 1 , , . . . (cid:41) (3.7) with multiplicity one and { [ I n kπ ] : k = 1 , , . . . } (3.8) with multiplicity two.Proof. From Proposition 3.1 crosses in F n appear only where there were degreefour nodes in F n − . Therefore, to find the number of crosses in F n , we willcount the number of degree four nodes in F n − . By Proposition 3.1 every nodein a quantum graph approximating a Laakso space is either of degree one ordegree four. Therefore, subtracting the number of degree one nodes from thetotal number of nodes will give the number of degree four nodes. From thesame proposition, the total number of nodes is 2 n − ( I n + 3). We have seen12lready that in F n degree one nodes only occur in V ’s and that for every vthere are two degree one nodes. From Lemma 3.1 that there are 2 n V’s in F n .Therefore, there are 2 n +1 degree one nodes in F n and 2 n − ( I n + 3) − n +1 =2 n − ( I n + 3 − ) = 2 n − ( I n −
1) degree four nodes. Substituting n − n in this last expression gives (3.6). We note that this lemma is stated only for n ≥ F never has a cross since there are only degree one nodes in F .To obtain the spectrum of ∆ n restricted to the cross, we must consider thefunctions in the domain of the Laplacian on the cross. We can think of the crossas two X-shapes, (such as F in Figure 1) connected at their four outer nodes.The orthogonality conditions from [16] force the function on the bottom X toequal the negative of the function on the top X. The value of the function on thetop of the X determines the value of the function on the bottom. The width ofthe X shape is 2 I − n and will have Dirichlet boundary conditions at the degreetwo nodes. Any function can be decomposed as symmetric and anti-symmetricwith respect to the upper and lower branches of the X. We consider the twocases in turn.In the symmetric case, the function is the same along the top and bottombranches of the X. Therefore we need only to look at the top branch as it fullydetermines the bottom branch. Here we are looking for eigenfunctions on aninterval of length 2 I − n and zero at the boundaries. These are sin( I n kπx ) with k = 1 , , . . . and x ∈ [0 , I − n ]. The associated eigenvalues to these functions arethose given in (3.7).In the antisymmetric case, the function value horizontally along the bottombranch of the X is the negative of the value along the top branch. At thecentral node, where the two branches meet, these two values must equal, sothey must be zero. We then effectively have the X broken up into two V ’s oflength I − n but with Dirichlet boundary conditions at either end. Looking atone of these V ’s, we still have the value along the bottom branch equal to thenegative of the value along the top, so we consider only the top branch. Herewe look for functions of length I − n with Dirichlet boundary conditions at bothends. This has already been done in Lemma 3.2 for the loop shape. Therewe got sin( I n kπx ) with k = 1 , , . . . and x ∈ [0 , I − n ] as the eigenfunctions and { [ I n kπ ] : k = 1 , , . . . } as the spectrum. This spectrum has multiplicity twobecause there are two halves of in the cross.Now we must consider the graph F and the eigenvalues it contributes tothe spectrum. This graph is just the unit interval, and has Neumann boundaryconditions forced by the Kerchoff matching conditions. So we are looking foreigenfunctions on intervals with length one and zero derivative at either end.These come in the form cos( kπx ) where k = 0 , , . . . . This results in the followingspectrum with multiplicity one: { [ kπ ] : k = 0 , , . . . } (3.9)Table 4 we summarizes the results of these lemmas. In order to obtainthe full spectrum with multiplicities, these sets must be combined with themultiplicities over all n ≥
0. Hence, Theorem 3.1 holds.13 hape Count Spectrum n Value F { [ kπ ] : k = 0 , , . . . } n = 0V 2 n { [ I n ( k + ) π ] : k = 0 , , . . . } n ≥ n − ( j n − I n − ) { [ I n kπ ] : k = 1 , , . . . } n ≥ n − ( I n − − { [ ( I n kπ )] : k = 1 , , . . . } n ≥ { [ I n kπ ] : k = 1 , , . . . } × Table 4: Summary of Lemmas 3.1 through 3.3
A MATLAB script described in [16] calculated the spectrum of the Laplacianfor constant j n for all n ∈ N by producing the incidence matricies of the ap-proximating graphs. We modified this script to handle general Laakso spaces.As in the original script, the eigenvalues are calculated using the eigs function,which is based on ARPACK (see users guide [15]).Quantum graphs approximating the Laakso space associated with the se-quence j i =[2,3,2,3,...] is shown in Figure 4 and the first twenty eigenvalues ofthe Laplacian on this space are shown in Table 2. The first twenty eigenvaluesof Laplacians on other Laakso spaces are shown in Table 3. These computationsagree with the calculations found in Lemmas 3.1 through 3.3. Given a Laplacian on a Laakso space, the trace of its heat kernel will be obtainedfollowing the same procedure used in [2] where the heat kernel’s trace was foundfor the diamond fractal. From [19] the heat kernel of the Laplacian is p ( t, x, y ) = (cid:88) k,l,m ψ k,l ( y ) ψ k,m ( x ) e − tE k where ψ k,l and ψ k,m are L − normalized eigenfunctions of ∆. The heat kernelon Laakso spaces will be further studied in [18] where continuity and boundswill be proved. The trace of the heat kernel Z ( t ) is defined in [2] as Z ( t ) = (cid:90) p ( t, x, x ) dx = (cid:88) k g k e − E k t , (4.1)where E k are the eigenvalues of the Laplacian on the fractal and g k are therespective multiplicities associated with those eigenvalues. Associated with theheat kernel is the spectral zeta function also defined in [2] from the heat kernelas ζ ( s, γ ) = 1Γ( s ) (cid:90) ∞ t s − Z ( t ) e − γt dt (4.2)14here Γ( s ) = (cid:82) ∞ t s − e − t dt is the gamma function. We set γ = 0 throughoutthe rest of this paper. Substitute (4.1) into (4.2) to obtain ζ ( s, γ ) = 1Γ( s ) (cid:90) ∞ t s − (cid:88) k g k e − E k t e − γt dt = (cid:88) k g k ( E k + γ ) s . (4.3)The next step in any specific example is to simplify the spectral zeta function byrecognizing Riemann zeta functions, ζ R ( s ) = ∞ (cid:88) n =0 n s , and identifying the otherterms as geometric series. Definition 4.1.
Define r = lim n →∞ ( I n ) /n when this limit exists. In the case ofself-similar spaces r is the contraction ratio since I − n is the diameter of eachcell. There is for any value of r a sequence j n that will produce that value. Once all of the series are simplfied, the poles of ζ ( s,
0) in the complex planecan be calculated. The poles of ζ ( s ) for the diamond fractals are given in [2] as s m = d h d w + 2 iπmd w ln r , m ∈ Z , (4.4)where d h and d s are the Hausdorff and walk dimensions respectively.Since the spectral zeta function was expressed as an integral of Z ( t ), applyingan inverse Mellin transform [4] allows the heat kernel to be expressed as Z D ( t ) = 12 πi (cid:90) a + i ∞ a − i ∞ ζ D ( s )Γ( s ) t − s ds. (4.5)By the Residue Theorem, Z D ( t ) is the sum of the residues of ζ D ( s )Γ( s ) t − s . Theresidue must also be calculated at s = 0 (a pole for Γ( s )) and at s = 1 / ζ R (2 s ) term in ζ ( s, γ )).It is known that ζ R ( s ) = ζ R ( s ) and Γ( s ) = Γ( s ) for all complex s . Thus,the residues from s m and s − m are complex conjugates of one another; thereforetheir sums equal twice the real part of the residue of s m . The complex valuesof the trace of the heat kernel yield oscillatory behavior in the heat kernel. Weshall observe what happens to the heat kernel as t →
0. Therefore, only theleading term with the most negative real power of t as well as any constants areincluded. For example, a result of [2] shows that for the diamond fractals, thetrace of the heat kernel is Z D ( t ) ∼ ζ D (0) + r d h − − r d w t d s / ( a + 2 Re ( a t − iπ/ ( d w log r ) )) + .... (4.6)This shows that the dominating power of t in the leading term as t → − d s /
2. This incidentally is the complex dimension introduced in [12]. We shallnow perform the same calculation for general Laakso spaces.15
The Trace of the Heat Kernel on Laakso Spaces
In Laakso spaces the dimensions and value of r are not always known by othermeans. Therefore the poles will be calculated analytically and the results willprovide information about the Hausdorff, walk, and spectral dimensions. Wenow give the leading powers of the trace of the heat kernel for the Laplacian ona general Laakso space. Theorem 5.1.
For the Laakso space associated to the sequence { j i } ∞ i =1 the traceof the heat kernel is Z ( t ) = ∞ (cid:88) n =2 n − ( I n − − ∞ (cid:88) k =1 e − k π I n t + ∞ (cid:88) n =2 n − ( I n − − ∞ (cid:88) k =1 e − k π I n t + ∞ (cid:88) n =1 n ∞ (cid:88) k =0 e − ( k +1 / π I n t + ∞ (cid:88) n =1 n − I n − ( j n − ∞ (cid:88) k =1 e − k π I n t + ∞ (cid:88) k =0 e − k π t (5.1) with an associated spectral zeta function ζ L ( s ) = ζ R (2 s ) π s (cid:34)(cid:32) ∞ (cid:88) n =2 n − ( I n − )(2 s − + j n −
1) + 2 n − ( s − I sn (cid:33) + 2 s +1 − j j s + 1 (cid:21) . (5.2) Proof.
The spectrum of the Laplacian on various Laakso spaces was given inTable 4 as σ (∆ L ) = ∞ (cid:91) n =2 ∞ (cid:91) k =1 (cid:8) k π I n (cid:9) ∪ ∞ (cid:91) n =2 ∞ (cid:91) k =1 (cid:26) k π I n (cid:27) ∪ ∞ (cid:91) n =1 ∞ (cid:91) k =0 { ( k + 1 / π I n }∪ ∞ (cid:91) n =1 ∞ (cid:91) k =1 { k π I n } ∪ ∞ (cid:91) k =0 π k (5.3)with respective multiplicities2 n − ( I n − − , n − ( I n − − , n , n − I n − ( j n − , . (5.4)Direct substitution of the these values into (4.1) gives the heat kernel. By(4.3) the associated spectral zeta function is ζ L ( s ) = ∞ (cid:88) n =2 ∞ (cid:88) k =1 n − ( I n − −
1) + 2 n − s ( I n − − I n k π ) s + ∞ (cid:88) n =1 ∞ (cid:88) k =0 n +2 s ( I n (2 k + 1) π ) s + ∞ (cid:88) n =1 ∞ (cid:88) k =1 n − I n − ( j n − I n k π ) s + ∞ (cid:88) k =1 k π ) s . (5.5)16his can be simplified by identifying Riemann zeta functions and, in certaincases, a Dirichlet Lambda function [1]. Then the function can be manipulatedin to one sum. ζ L ( s ) = ζ R (2 s ) π s (cid:34)(cid:32) ∞ (cid:88) n =2 n − ( I n − )(2 s − + j n −
1) + 2 n − ( s − I sn (cid:33) + 2 s +1 − j j s + 1 (cid:21) (5.6)as claimed in the proposition.This expression of the spectral zeta function cannot be simplified further fora general Laakso space with an arbitrary sequence of j i ’s. However, it provides acommon starting point for Laakso spaces in which the sequence of j i ’s is known.The next step is to locate the poles of the spectral zeta function. Recall thatonly poles which yield the most negative real power of t are considered sincethey produce the dominating behavior as t → Proposition 5.1.
Of all the poles of the spectral zeta function in the complexplane, the poles that yield the most negative real power of t in the leading termof the trace of the heat kernel are located at s m = log 2 r + 2 πim log r (5.7) for integer m .Proof. The trace of the heat kernel requires the most negative power of t whichcorresponds to the poles with the greatest real component due to the t − s termin (4.5). The proof of the proposition relies on analyzing the series in (5.2)to find the poles. Note that the in the series in (5.2), the numerator has twoterms; values of s will be calculated that will make the denominator grow atthe same rate as the numerator. The value of s with the greatest real part arethe poles that will be used. First rewrite the denominator of (5.2) as I s n I s n where 2 s = s + s . Select s to match the rate of growth for the first term inthe numerator. To make I s n grow at the same rate as I n − , s should equal 1.To have I s n grow at the same rate as 2 n − s should equal log r
2. Therefore, forthe first term, 2 s = 1 + log r
2. It can be verified that any poles from the secondterm in (5.2) will not have a real component as large as this pole. Therefore,the real part of the poles that will yield the desired leading term in the traceof the heat kernel are s = + log 22 log r = log 2 r log r . Including 2 πim in the numeratorgives all of the complex values of this pole Corollary 5.1.
The dominating t term in the trace of the heat kernel has power − ds/ − Re ( s m ) . roof. Recall from Section 4 that the dominating power of t in the trace of theheat kernel as t → − d s /
2. A result of Proposition 5.1 is that the greatestreal component of the poles yield the dominating power of t . Therefore, whencalculating the residue of ζ L ( s )Γ( s ) t − s at the pole obtained from (5.7), the realpower of t will be precisely − Re ( s m ). But as stated at the beginning of theproof, it is also equal to − d s /
2. Thus, d s / Re ( s m ) Corollary 5.2.
The spectral dimension of any Laakso space with { j i } such that r exists is d s = log 2 r log r . (5.8) The walk dimension, d w , is 2 for any Laakso space which implies d h = d s .Proof. This follows directly from Proposition 5.1 and Corollary 5.1 since d s / Re ( s m ) = log 2 r log r which implies d s = log 2 r log r . The walk dimension is a result in[18]. It does agree with d h and d s via the Einstein relation 2 d h /d w = d s .The next two subsections give the trace of the heat kernel for two specificLaakso spaces where the sum in the spectral zeta function can be evaluatedexactly: j = 2, and j = { , , , ... } . Proposition 5.2.
For the Laakso space where at each iteration j = 2 , written L , the trace of the heat kernel is Z L ∼ ζ L (0) + t log 2 (cid:18) Re (cid:18) ζ R (2+ πi log 4 )Γ(1+ πi log 4 ) t πi log 4 π
2+ 4 πi log 4 (cid:19) + ... (cid:19) .Proof. For Laakso spaces with a fixed j = 2, I n = 2 n . Substituting these into(5.1) and (5.2) gives the following two equations Z L ( t ) = ∞ (cid:88) n =1 (cid:2) n − − n − (cid:3) ∞ (cid:88) k =1 e − t n k π + ∞ (cid:88) n =2 (cid:2) n − − n − (cid:3) ∞ (cid:88) k =1 e − tk π n − + ∞ (cid:88) n =1 n ∞ (cid:88) k =0 e − t (2 k +1) π n − (5.9)and ζ L ( s,
0) = ζ R (2 s ) π s (cid:18) s − + 1)4 s (4 −
4) + 6(2 s − − s (4 s −
2) + 2 s +1 − s s (cid:19) (5.10)which has poles z = (cid:18)
12 + 2 πim log 4 (cid:19) , z = (cid:18) πim log 4 (cid:19) , ∀ m ∈ Z . (5.11)18igure 6: Heat kernel Z L , normalized by the leading non-oscillating term forthe j = 2 Laakso space. The variable s is on the horizontal axis.By Proposition 5.1, only the second term of the above equation with thegreatest real part contributes to the leading t term. Then an inverse Mellintransform is applied just as in Theorem 5.1. Table 5 shows the residues of theintegrand after the inverse Mellin transform at the poles given in (5.11) as wellas s = 0 from the Γ( s ) term. Again we add complex conjugates and take onlythe most negative powers of t. Notice that when adding the residue from thepoles in (5.11) only the poles with real part one contribute to the heat kernel,the others with the exception of zero and one half have residue zero. Once all ofthe residues are simplified we obtain the expression for the trace heat kernel’sleading term as given in the proposition.The power of 1 of t in the denominator of the leading term in the propositionimplies that the spectral dimension d s for L is 2. Knowing that the Hausdorffdimension d h = 2 from Table 1, we conclude that the walk dimension d w =2 since d s = 2 d h /d w . Notice the poles and dimensions of this fractal wereexplicitly calculated. But Corollary 5.2 and Proposition 5.1 yield the sameresult once r = lim n →∞ I /nn = lim n →∞ (2 n ) /n = 2 is known.19 m Residue0 ζ L (0))1/2 √ π √ t t log 2 ± πim log 4 116 t log 2 6 ζ R (2+ πim log(4) )Γ(1+ πim log(4) ) t πim log(4) π
2+ 4 πim log(4) + iπ log 4 m (cid:54) = 0Table 5: Residues of the integrand of the inverse Mellin transfrom for givenpoles of the spectral zeta function for Laakso spaces with a fixed j = 2 { } Laakso Space
Proposition 5.3.
For the Laakso space with j k = 3 and j k − = 2 where k ≥ , the trace of the heat kernel is Z L ( t ) ∼ ζ L (0)+ (5.12)124 t + log(2)log(6) log(6) Γ (cid:16) + log(2)log(6) (cid:17) ζ R (cid:16) (cid:17) π + ∞ (cid:88) m =1 Re Γ (cid:16) + log(2)log(6) + πim log(6) (cid:17) ζ R (cid:16) + πim log(6) (cid:17) π + πim log(6) t − πim log(6) + πim log(6) + 10 · + πim log(6) + 122 + πim log(6) . Proof.
In this case r = lim n →∞ I /nn = lim n →∞ (2 n/ n/ ) /n = √
6. This locates thepoles with largest real part at s m = + log(2)log 6 + πim log 6 .The next step is to obtain and simplify the spectral zeta function associatedwith the trace of heat kernel given in (5.2). Since j i alternates between 2 and3, the following values can be directly substituted in, for any k we have: I = 2 I = 6 I k − = 6 k − I k − = 2 × k − I k = 6 k In preparation for substituting these values into (5.2) the sum is split intotwo sums, one over even n and the other over odd20 L ( s ) = ∞ (cid:88) n =2 n − ( I n − −
1) + 2 n − s ( I n − −
1) + 2 n +2 s − n + 2 n − I n − ( j n − I sn = 2( I −
1) + 2 s ( I −
1) + 2 s − + 2 I I s + ∞ (cid:88) k =2 (cid:18) k − ( I k − −
1) + 2 k − s ( I k − −
1) + 2 k − s − k − I s k − + 2 k ( I k −
1) + 2 k − s ( I k − −
1) + 2 k +2 s − k + 2 k − ( I n − ) I s k (cid:19) . (5.13) Substituting the known values of I k , I k +1 , j k , and j k +1 we obtain ζ L ( s ) = ζ R (2 s ) π s (cid:20)(cid:18) s + 412 + s s − + 12 s (cid:19) s −
24 + (cid:18) · s − · s − s (cid:19) s − (cid:21) . (5.14)To apply the inverse Mellin transform as was done in (4.5) to obtain anexpression for Z L ( t ). Then Z L ( t ) can be calculated using the sum of the residuesof ζ L ( s )Γ( s ) t − s using the poles obtained in (5.7) as well as s = 1 / ζ R (2 s ) and s = 0 (from Γ( s )). Table 6 liststhe residues for those poles. s m Residue0 ζ L (0)
12 23 √ πt s m = + log(2)log 6 + πim log(6) Γ( s m ) ζ R (2 s m ) t sm π sm − sm
24 log(6) (cid:0) s m + 10 · s m + 12 (cid:1) s k = log(2)log(6) + πim log(6) Γ( s k ) ζ R (2 s k ) t sk π sk
38 log(6) 4 sk − sk Table 6: Residues of the integrand of the inverse Mellin transfrom for givenpoles of the spectral zeta function for the { } Laakso spacesThe sum of the residues in Table 6 expresses the trace of the heat kernelwith the leading terms as shown in the statement of the proposition. Note thatthe general terms in Z L ( t ) are shown in the proposition to indicate the behaviorof the trace of the heat kernel in more detail. Corollary 5.3.
The exponent of t , log(2 √ , in the leading term as t goes to-wards zero implies that the spectral dimension of this Laakso space is d s =log 24 / log 6 . The Hausdorff dimension for this fractal is given in Table 1 as d h = log 24 / log 6 . Again, the same results are obtained by simply applying Corollary 5.2 andProposition 5.1. 21 cknowledgments
The authors would like to thank Alexander Teplyaev, Luke Rogers, RobertStrichartz, Shotaro Makisumi, Grace Stadnyk, Jun Kigami, and Naotaka Ka-jino.
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