1 Introduction

Exact spectrum of the Laplacian on a domain in the Sierpinski gasket

footnotetext: This research was supported by the National Science Foundation of China, Grant 10901081.

HUA QIU


Abstract. For a certain domain in the Sierpinski gasket whose boundary is a line segment, a complete description of the eigenvalues of the Laplacian, with an exact count of dimensions of eigenspaces, under the Dirichlet and Neumann boundary conditions is presented. The method developed in this paper is a weak version of the spectral decimation method due to Fukushima and Shima, since for a lot of “bad” eigenvalues the spectral decimation method can not be used directly. Let , be the eigenvalue counting functions of the Laplacian associated to and respectively. We prove a comparison between and says that for sufficiently large for some positive constant . As a consequence, as , for some (right-continuous discontinuous) -periodic function with . Moreover, we explain that the asymptotic expansion of should admit a second term of the order , that becomes apparent from the experimental data. This is very analogous to the conjectures of Weyl and Berry.

Keywords. Sierpinski gasket, Laplacian, eigenvalues, spectral decimation, analysis on fractals.

Mathematics Subject Classification (2000). 28A80, 31C99


1 Introduction

The study of the Laplacian on fractals was originated by S. Kusuoka [23] and S. Goldstein [11]. They independently constructed the Laplacian as the generator of a diffusion process on the Sierpinski gasket . Later an analytic approach was developed by J. Kigami [16], who constructed the Laplacian both as a renormalized limit of difference operators and a weak formulation using the theory of Dirichlet forms.

Let be the boundary of , which consists of the three vertices of the equilateral triangle containing . Consider the following Dirichlet eigenvalue problem:

where is the standard Laplacian (with respect to the standard self-similar measure ) on . Physicists R. Rammal and G. Toulouse [29] found that an appropriate choice of a series of eigenvalues of successive difference operators produces an orbit of a dynamical system related to a quadratic polynomial, and all the eigenvalues of on should be obtained by tracking back the orbits. This is the phenomenon which M. Fukushima and T. Shima [10, 32] described from the mathematical point of view, by saying that admits spectral decimation with respect to a quadratic polynomial. Using the spectral decimation, all the eigenvalues and eigenfunctions of on have been determined exactly. This method also works for the eigenvalue problem of with Neumann boundary condition.

Later the theory of the Laplacian was developed for nested fractals and p.c.f. self-similar sets by T. Lindstrøm [25] and Kigami [17] by introducing the notion of harmonic structure. Every p.c.f. self-similar set is approximated by an increasing sequence of finite graphs and the harmonic structure determines a sequence of difference operators on the successive graphs, which converges to the Laplacian. Then some generalizations of the spectral decimation to a class of p.c.f. self-similar sets were developed by Shima [33], L. Malozemov and A. Teplyaev [26], in which some strong symmetry conditions are supposed to be satisfied to ensure the spectral decimation applies to the corresponding graph sequences. Under such strong symmetry conditions, the spectrum of the Laplacian can also be determined in terms of the iteration of a rational function. Recently, the spectrum of the Laplacian on some other fractals has been analyzed either numerically [1] or using the spectral decimation method [7, 8, 39, 41] by R. S. Strichartz (with co-authors), D. Zhou and Teplyaev. In all the references mentioned above, spectral decimation plays a key role in the theoretical study of the spectrum of the Laplacian.

The Weyl asymptotic behavior of the eigenvalue counting function of on has also been studied by Fukushima and Shima [10]. Afterwards, a general spectral distribution theory on p.c.f. self-similar sets was obtained by Kigami and M. L. Lapidus [18, 19]. Denote by the number of eigenvalues of (taking the multiplicities into account) on not exceeding , with Dirichlet boundary condition at the three vertices. As proved in [10, 18], there exist positive constant such that

(1.1)

for sufficiently large , where is the spectral dimension of . In particular, varies highly irregularly at due to the high multiplicities of localized eigenfunctions,

(1.2)

Furthermore, using a refinement of the Renewal Theorem, Kigami [19] showed that the remainder of is bounded,

(1.3)

for some (right-continuous discontinuous) -periodic function with . Exactly the same results hold for the eigenvalue counting function for the Neumann Laplacian.

In this paper, we are mainly concerned with eigenvalue problems for a domain in . Although analysis on fractals has been made possible by the definition of the Laplacian, there has been little research into differential equations on bounded subsets of fractals. Recall that is the attractor of the iterated function system with where are the vertices of an equilateral triangle in the plane,

In Kigami’s theory the boundary of consists of the three points , and the space of harmonic functions (solutions of ) is three dimensional, with determined explicitly by its boundary values . (Note that this boundary is not a topological boundary.) Thus the harmonic function theory on is more closely related to the theory of linear functions on the unit interval than to harmonic functions on the disk. To get a richer theory we should take an open set in and restrict the Laplacian on to functions defined on . Hence we believe it is appropriate to begin the study of differential equations related to a bounded domain in .

Fig. 1.1. and .

For simplicity, here we particularly focus on the certain domain which is a triangle obtained by cutting with a horizontal line at any vertical height ( if we suppose that the height of is equal to .) below the top vertex . See Fig. 1.1. An important motivation for studying this kind of domains is that they are the simplest examples which could serve as a testing ground for questions and conjectures on analysis of more general fractal domains with fractal boundaries. These domains were first introduced by Strichartz [34] and later studied by J. Owen and Strichartz [27], where they gave an explicit analog of the Poisson integral formula to recover a harmonic function on from its boundary values. It is also natural to calculate an explicit Green’s function for the Laplacian on . This was studied by Z. Guo, R. Kogan and Strichartz in [12] which is completely similar to the construction of the Green’s function on given by Kigami in [16, 17, 20]. For some other analytic topics related to this kind of domains, see [14, 15, 21, 22].

In the present paper, we study the spectral properties of the Laplacian on , which is an open problem posed in [27]. For the simplicity of description, we mainly concentrate our attention to a particular domain (We drop the subscript on in all that follows without causing any confusion.) which is the complement of , where is the line segment joining and (in this case ). We give a complete description of the Dirichlet and Neumann spectra of the Laplaician on .

In our context, for a number of “bad” eigenvalues (whose associated eigenfunctions have supports touching the bottom boundary line ) the spectral decimation method can not be used directly, which makes things much more complicated. By choosing a sequence of appropriate graph approximations, we describe a phenomenon on those eigenvalues called weak spectral decimation which approximates to spectral decimation when the levels of the successive graphs go to infinity. And we use this weak spectral decimation to replace the role of spectral decimation in the original Fukushima and Shima’s work [10]. Actually, similarly to the standard case, weak spectral decimation can also produce a “weak” orbit related to the same quadratic polynomial by an appropriate series of eigenvalues of successive difference operators on graph approximations. We can then trace back those “weak” orbits to capture all the “bad” eigenvalues. More precisely, we classify the eigenvalues of on into three types, the localized eigenvalues, primitive eigenvalues and miniaturized eigenvalues. The localized eigenfunctions associated to localized eigenvalues on are just a subspace of the localized eigenfunctions on , whose supports are disjoint from . This type of eigenvalues can be dealt with in a same way as the case, for which the spectral decimation can apply. The primitive and miniaturized eigenvalues are the so-called “bad” eigenvalues. They are the eigenvalues need to be paid particular attention to. We will give a precise description of the structure of the Dirichlet and Neumann spectra of on in Section 3, before giving out the technical proofs.

Now what happens to the asymptotic behavior of the eigenvalue counting function (with Dirichlet boundary condition on ) on ? A natural analogue of holds. Namely, there exists some positive constant such that for sufficiently large ,

(1.4)

which can be proved by first considering the asymptotic behavior of the eigenvalue counting function for each type of eigenvalues separately, then adding up them together. In fact, can be even easily proved without involving the structure of the Dirichlet spectrum on , as follows: the Dirichlet eigenvalue counting function on the top cell is given by by the self-similarity of both the Dirichlet form and the measure . Therefore it follows from the minmax principle that , which together with , also yields . Moreover, the high multiplicities of localized eigenfunctions immediately imply that does not vary regularly at , similarly to . Thus,

Since most eigenvalues are localized, and are very close. We are interested in the difference . More precisely, is there some power such that ? For this question, we have the following partial result:

Theorem 3.10. There exists some constant such that for sufficiently large ,

As a consequence, it then follows from that

(1.5)

The same argument also works for the Neumann Laplacian.

Nevertheless, this should not be the entire story for the Weyl asymptotic behavior of . Recall that in the classical case. Suppose is an arbitrary nonempty bounded open set in with smooth boundary , then Weyl’s classical asymptotic formula can be stated as follows:

as , where depends only on . See details in [28, 30, 31]. The above remainder estimate constitutes an important step on the way to H. Weyl’s conjecture [40] which states that if is sufficiently “smooth”, then the asymptotic expansion of admits a second term, proportional to . Extending Weyl’s conjecture to the fractal case, M. V. Berry [3, 4] conjectured that if has a fractal boundary with Hausdorff dimension (which later was revised into Minkowski dimension in [6, 24]) , then the order of the second term should be replaced by . See further discussion and a partial resolution of the conjectures of Weyl and Berry in Lapidus’s work [24]. Hence it is natural to ask that whether there is an analogue result in or setting. For case, Kigami [19] showed that the remainder is bounded, see . Note that this is consistent with the fact that the boundary of consists of three points, hence has dimension zero. This was refined by Strichartz in [38], where an exact formula was presented with no remainder term at all, provided we restrict attention to almost every . As for case, can be viewed as a weak analog of the Weyl-Berry’s conjecture. Moreover, We will show that although we are unable to prove, it becomes apparent there is a second term of order in the expansion of the eigenvalue counting function on from observing the experimental data.

We note that our work deals with the vibrations of “drums with fractal membrane” since the domain itself is a fractal. The order of the second term should has a close connection with the dimension of the boundary due to Weyl-Berry’s conjectures. Moreover, when consider a more general domain , we will meet “drums with fractal membrane” with also fractal boundary.

The paper is organized as follows. In Section 2 we will briefly introduce some key notions from analysis on fractals and give a concise description of the Dirichlet and Neumann spectra of the Laplacian for the standard case, which will be used in the rest of the paper.

In Section 3, we will present the exact structure of the Dirichlet spectrum of on in a self-contained and precise way before going into the technical details. We will find an appropriate sequence of graph approximations for the fractal domain , and describe the exact structures of the discrete Dirichlet spectra of the corresponding successive difference operators on them. Accordingly, for each graph all the graph eigenvalues are also divided into three types, localized, primitive and miniaturized. By using an eigenspace dimensional counting argument, we will show that they should make up the whole discrete Dirichlet spectrum. We will also briefly describe how to relate the spectra of consecutive levels and how to pass the graph approximations to the limit by using spectral decimation for localized eigenvalues and weak spectral decimation for other types of eigenvalues. We will also present analogous results for Laplacians with Neumann boundary conditions.

In Section 4, we will describe the discrete graph primitive Dirichlet eigenvalues on the graph approximations for each level. We will divide our discussion into symmetric case and skew-symmetric case. In each case, we will prove that for each level the primitive graph eigenvalues are exactly the total roots of a high degree polynomial. And we will describe the weak spectral decimation phenomenon by studying the relation between roots of consecutive polynomials. Moreover, we will prove that for each level, the complete discrete spectrum is made up of the three types of eigenvalues as expected.

In Section 5, we will discuss the primitive Dirichlet eigenvalues of on by passing the results of Section 4 on graph approximations to the limit. Since we can only use weak spectral decimation which is essentially based on estimates, comparing to the case, some trivial results become nontrivial and need to be proved in this section.

In Section 6, first we will prove that the whole Dirichlet spectrum on is made up of the three types of eigenvalues as expected, following the basic idea of Fukushima and Shima’s work. Then we will give a comparison concerning the eigenvalue asymptotics of the eigenvalue counting functions between case and case.

In Section 7, we will give a brief discussion on how to deal with the Neumann spectrum. We will find a similar weak spectral decimation for primitive eigenvalues by establishing a relation between symmetric (or skew-symmetric) primitive graph eigenvalues with some high degree polynomials, but the proof is quite different from that in the Dirichlet case.

Then in Section 8, we will list some conjectures concerning eigenvalue asymptotics (especially the existence of the second term of the expansion of the eigenvalue counting function), gaps in the ratios of consecutive eigenvalues and eigenvalue clusters, which become apparent from observing the experimental data.

We will also give a brief discussion on how to extend our method from to with in Section 9.

The purpose of this paper is to work out the details for one specific example. We hope this example will provide insights which will inspire future work on a more general theory.

2 Spectral decimation on

First we collect some key facts from analysis on that we need to state and prove our results. These come from Kigami’s theory of analysis on fractals, and can be found in [16, 17, 20]. An elementary exposition can be found in [35, 37]. The fractal will be realized as the limit of a sequence of graphs with vertices . The initial graph is just the complete graph on , the vertices of an equilateral triangle in the plane, which is considered the boundary of . See Fig. 2.1. The entire fractal is the only -cell, which has as its boundary. At stage of the construction, all the cells of level lie in triangles whose vertices make up . Each cell of level splits into three cells of level , adding three new vertices to .

Fig. 2.1. The first graphs, in the approximations to the Sierpinski gasket.

We define the unrenormalized energy of a function on by

The energy renormalization factor is , so the renormalized graph energy on is

and we can define the fractal energy . We define as the space of continuous functions with finite energy. Then extends by polarization to a bilinear form which serves as an inner product in this space. The energy gives rise to a natural distance on called the effective resistance metric on , which is defined by

(2.1)

for . It is known that is bounded above and below by constant multiples of , where is the Euclidean distance. Furthermore, the definition implies that functions on are Hölder continuous of order in the effective resistance metric.

We let denote the standard probability measure on that assigns the measure to each cell of level. The standard Laplacian may then be defined using the weak formulation: with if is continuous, , and

(2.2)

for all , where . There is also a pointwise formula (which is proven to be equivalent in [37]) which, for nonboundary points in (not in ) computes

where is a discrete Laplacian associated to the graph , defined by

for not on the boundary.

The Laplacian satisfies the scaling property

and by iteration

for .

Although there is no satisfactory analogue of gradient, there is normal derivative defined at boundary points by

the limit existing for all . The definition may be localized to boundary points of cells. For each point , there are two cells containing as a boundary point, hence two normal derivatives at . For , the normal derivatives at satisfy the matching condition that their sum is zero. The matching condition allows us to glue together local solutions to .

The above matching condition property follows easily from a local version of the following Gauss-Green formula, which is an extension of to the case when doesn’t vanish on the boundary:

The local version of the Gauss-Green formula is

where is any finite union of cells and is the restriction of the energy bilinear form to , which can also be defined directly by

Now we come to a brief recap of the spectral decimation on . Our goal is to find all solutions of the eigenvalue equation

as limits of solutions of the discrete version

In the case, we are lucky that we may take , which is necessarily convenient for the spectral decimation. We should emphasize that this is not true for case.

The method of spectral decimation on was invented by Fukushima and Shima [10] to relate eigenfunctions and eigenvalues of the discrete Laplacian ’s on the graph approximation ’s for different values of to each other and the eigenfunctions and eigenvalues of the fractal Laplacian on . In essence, an eigenfunction on with eigenvalue can be extended to an eigenfunction on with eigenvalue , where for an explicit function defined by

(2.3)

except for certain specified forbidden eigenvalues, and all eigenfunctions on arise as limits of this process starting at some level which is called the generation of birth. This is true regardless of the boundary conditions, but if we specify Dirichlet or Neumann boundary condition we can describe explicitly all eigenspaces and their multiplicities.

Denote the real valued inverse functions of by . That is

(2.4)

We describe the procedure briefly here. First, there is a local extension algorithm that shows how to uniquely extend an eigenfunction defined on to a function defined on such that the -eigenvalue equations hold on all points of . For , the extension algorithm is: Suppose is an eigenfunction on with eigenvalue . Let . Consider an -cell with boundary points and let denote the points in in that cell, with opposite . Extend to a function on by defining (for simplicity of notation, we drop the subscripts on )

(2.5)

Then we have the following proposition taken from [37].

Proposition 2.1. Suppose or , and . If is a -eigenfunction of and is extended to a function on by , then is a -eigenfunction of , Conversely, if is a -eigenfunction of and is restricted to a function on , then is a -eigenfunction of .

The forbidden eigenvalues are singularities of the spectral decimation function . It is “forbidden” to decimate to a forbidden eigenvalue. Because forbidden eigenvalues have no predecessor, we speak of forbidden eigenvalues being “born” at a level of approximation .

Next we want to take the limit as . We assume that we have an infinite sequence related by with all but a finite number of ’s. Then we may define

It is easy to see that the limit exists since

(2.6)

as . We start with a -eigenfunction of on , and extend to successively using , assuming that none of is a forbidden eigenvalue. Since implies as , it is easy to see that is uniformly continuous on and so extends to a continuous function on . Moreover, it satisfies the -eigenvalue equation for .

A proof in [10] guarantees that this spectral decimation produces all possible eigenvalues and eigenfunctions (up to linear combination).

To describe the explicit Dirichlet and Neumann spectra, we have to describe all possible generations of birth and values for , and describe the multiplicity of the eigenvalue by giving an explicit basis for the -eigenspace of . For each , we have to add up the dimensions of eigenspaces with generation of birth , extended to in all allowable ways. This total must be (Neumann) or (Dirichlet), the dimension of the space on which the symmetric operator acts. Now we give a brief description of the structure of the Dirichlet and Neumann spectra of on respectively.

Dirichlet spectrum.

We denote by the Dirichlet spectrum of on and by the discrete Dirichlet spectrum of on for . Due to the above discussion, we only need to make clear the spectrum for each level . There are two kinds of eigenvalues, initial and continued. The continued eigenvalues will be those that arise from eigenvalues of by the spectral decimation. Those that remain, the initial eigenvalues, must be some of the forbidden eigenvalues by Proposition 2.1.

In [32], it is proved that consists of two eigenvalues and with multiplicities 1 and 2 respectively, and for , the only possible initial eigenvalues in are the two forbidden eigenvalues and with multiplicities and respectively. Hence we may classify eigenvalues into three series, which we call the -series, -series, and -series, depending on the value of . The eigenvalues in the -series all have multiplicity , while the eigenvalues in the other series all exhibit higher multiplicity. Also, if is an eigenvalue in the -series or -series, then is also an eigenvalue, corresponding to a generation of birth , with the same choice of relations (suitably reindexed).

Neumann spectrum.

We impose a Neumann condition on the graph by imagining that it is embedded in a larger graph by reflecting in each boundary vertex and imposing the -eigenvalue equation on the even extension of . This just means that we impose the equation

at for . Then the Neumann -eigenvalue equations consist of exactly equations in unknowns. Similarly to the Dirichlet case, we also only need to make clear all the discrete spectra. The result is very similar to the Dirichlet spectrum, with only a few changes. We omit it.

It should be emphasized here that those eigenfunctions which are simultaneously Dirichlet and Neumann play an important role in the spectral analysis of . Here we call them localized eigenfunctions since all of them have small supports. (Here this definition of localized eigenfunctions is slightly different from that of [2, 19, 37] for the convenience of further discussion for case.) Similarly to , to describe the structure of localized eigenfunctions, we only need to make clear the structure of all initial localized eigenvalues, which consists of -series and -series eigenvalues. In fact, the multiplicity of a -series eigenvalue with generation of birth , is with an eigenfunction associated to each -level loop (a -level circuit around an empty upside-down triangle in the graph ). The eigenfunction associated to each loop takes value on all -level points not lying in that loop. Moreover, the support of is exactly the union of all -cells intersecting that loop. The multiplicity of a -series eigenvalue with generation of birth , is with an eigenfunction associated to each point in . Each such eigenfunction takes value on all points in except . Moreover, is supported in the union of two -level cells containing . The existence of localized eigenfunctions is unprecedented in all of smooth mathematics. However, for a class of p.c.f. self-similar sets, including , localized eigenfunctions dominate global eigenfunctions. See more details in [2, 19]. See also Section 4 of Kigami’s book [20], where most results are explained in detail.

3 The structures of Dirichlet spectrum and Neumann spectrum on

To give the readers an intuitive perception of the structure of the spectrum of on in advance, in this section we describe all Dirichlet and Neumann eigenvalues and eigenfunctions on avoiding involving technical proofs. We will go to the details in the remaining sections.

3.1 Dirichlet spectrum

We begin with the Dirichlet case. First we formulate the eigenvalue problem of on with Dirichlet boundary condition(for short, the Dirichlet Laplacian).

Definition 3.1. Let . The Dirichlet Laplacian on with domain is formulated as follows: for and ,

If we replace by in the above definition, then we get the standard Dirichlet Laplacian which is introduced in [20].

Definition 3.2. For and if

then is called an eigenvalue of on (or, a Dirichlet eigenvalue of on ), and is called an associated (Dirichlet) eigenfunction.

Let denote the spectrum of on ( is also called the Dirichlet spectrum of on ). We will consider three kinds of Dirichlet eigenfunctions, localized, primitive, and miniaturized eigenfunctions.

In the following, we will always use to denote an eigenfunction of on and to denote the associated eigenvalue of .

Definition 3.3. is called a localized eigenfunction if it is a localized eigenfunction on whose support is disjoint from (the line segment joining and ).

The associated eigenvalue is called a localized eigenvalue. Denote by the set consisting of all such eigenvalues. Obviously, all the eigenvalues in have generation of birth (the ones with all have supports intersecting ) and or .

Comparing to the case, instead of the eigenfunctions associated to the -series eigenvalues, there is also a type of global eigenfunctions in case, which will be sorted into symmetric and skew-symmetric parts according to the reflection symmetry fixing .

Definition 3.4. is called a symmetric primitive eigenfunction if it is symmetric under the reflection symmetry fixing and also local symmetric in each cell under the reflection symmetry fixing with word taking symbols only from .

Fig. 3.1. gives a symbolic picture of the above mentioned symmetries, indicated by dotted lines. The associated eigenvalue is called a symmetric primitive eigenvalue. Denote by the set consisting of all such eigenvalues.

Similarly,

Definition 3.5. If is skew-symmetric under the reflection symmetry fixing , but still local symmetric in small cells, then it is called a skew-symmetric eigenfunction.

The associated eigenvalue is called a skew-symmetric eigenvalue. Denote by the set consisting of all such eigenvalues.

Both the symmetric and skew-symmetric primitive eigenfunctions are called primitive eigenfunctions. All the associated eigenvalues are called primitive eigenvalues. Let denote the set consisting of all of them. Namely,

The primitive eigenfunction (either the symmetric or skew-symmetric case) is uniquely determined by the values denoted by of on vertices by using the eigenfunction extension algorithm described in (2.5). Due to the Dirichlet boundary condition, , for the associated eigenfunction of , we always have and . We call a skeleton of since it plays a critical role in the study of primitive eigenfunctions.

Theorem 3.1. All the primitive eigenvalues are of multiplicity .

This theorem will be proved in Section 6.

Fig. 3.1. The first level symmetries and the skeleton of .

The following argument will show that there is another type of eigenfunctions. For each skew-symmetric eigenvalue , there is a family of eigenfunctions with eigenvalue and multiplicity for . To get such an eigenfunction, just take the -eigenfunction , contract it times, place it in any one of the bottom cells of level , and take value elsewhere. See Fig 3.2. The reason we can do this is that on the boundary point , and which make the matching condition holds automatically.

Definition 3.6. We call all the above obtained eigenfunctions miniaturized eigenfunctions.

If is a miniaturized eigenfunction obtained by contracting a skew-symmetric primitive eigenfunction times, then we call a -contracted miniaturized eigenfunction. Let denote all the eigenvalues associated to them. Obviously, is determined by .

Fig. 3.2. The first level miniaturized eigenfunctions.

We will prove in Section 6 that all the eigenfunctions of on fall into one of these three types, and there are no coincidences of eigenvalues among different types.

For the purpose of studying the exact structure of the spectrum , the first thing we should consider is to describe the Dirichlet spectra of ’s on the associated graph approximations. Similarly to the case, the fractal domain can be realized as the limit of a sequence of graphs . More precisely, , let be a subset of with all vertices lying along removed. Let be the subgraph of restricted to . Denote by the boundary of the finite graph . It is easy to find that and approximate to and as goes to infinity respectively. See Fig. 3.3.

A routing argument shows that the Dirichlet Laplaician could be viewed as the limit of suitably scaled graph Laplaicians on , as is done in [16, 17, 20, 35, 37] for the standard case. Hence, there is also a pointwise formula which, for nonboundary points in , computes

where is a discrete Laplacian associated to the graph , defined by

for in .

We denote by the discrete Dirichlet spectrum of on for . On the Dirichlet -eigenvalue equations consist of exactly equations in unknowns. We start from m=2 since there is no Dirichlet -eigenvalue equation. For simplicity, let . It is easy to check that , , and more generally,

Proposition 3.1. , .

Proof. Notice that , where , is the number of points lying on the bottom boundary of , and is the number of points in lying on or .

Fig. 3.3. The first graphs, in the approximations to with inside points and boundary points represented by dots and circles respectively.

Due to different types of eigenvalues of on , we should consider the associated different types of graph Dirichlet eigenvalues of . We now describe how to define , and respectively. For simplicity, in the rest of this subsection, without causing any confusion, we omit the term “Dirichlet” for graph eigenfunctions and eigenvalues.

In the following, we will always use to denote an eigenfunction of on satisfying the Dirichlet boundary condition, and to denote the associated eigenvalue of .

In fact, by the spectral decimation recipe, each localized eigenfunction of on whose generation of birth can be restricted to to get a graph eigenfunction of , with the Dirichlet boundary condition of on holding automatically.

Definition 3.7. Let be a localized eigenfunction of on with generation of birth , then its restricted graph function on is called a -level localized graph eigenfunctions of on .

All the associated eigenvalues are called -level localized graph eigenvalues. We use to denote the set consisting of all these type eigenvalues.

We can not imitate the above process to get the -level primitive graph eigenfunctions since the Dirichlet boundary condition would be destroyed if we do the similar restriction. But we can define -level primitive graph eigenfunctions on directly in the following way.

Definition 3.8. A Dirichlet eigenfunction on is called a -level symmetric primitive graph eigenfunction if it is symmetric under the reflection symmetry fixing and also local symmetric in under the reflection symmetry fixing for each word taking symbols only from .

The associated eigenvalue is called a -level symmetric primitive graph eigenvalue. Denote by the set of all this type of eigenvalues.

Similarly,

Definition 3.9. If is skew-symmetric under the reflection symmetry fixing , but still local symmetric in small cells, then it is called a -level skew-symmetric primitive graph eigenfunction.

can be defined in a similar way. Let denote all the -level primitive graph eigenvalues. Namely,

Similarly to the limit case, the primitive graph eigenfunction (either the symmetric or skew-symmetric case) is uniquely determined by the values denoted by of on vertex points by using the eigenfunction extension algorithm (2.5). Due to the Dirichlet boundary condition, we always have for a -level primitive graph eigenfunction . We call a skeleton of . It also plays a critical role in the study of -level primitive graph eigenfunctions.

Miniaturized graph eigenfunctions on can be defined in a similar way by using miniaturization of skew-symmetric primitive graph eigenfunctions whose level strictly less than .

Definition 3.10. For a Dirichlet eigenfunction on , if there exists an integer and a -level skew-symmetric primitive graph eigenfunction such that after contracting times, placing it in one of the bottom copies of in , and taking value elsewhere, one can obtain , then is called a -level miniaturized graph eigenfunction.

The associated eigenvalue is called a -level miniaturized graph eigenvalue. is called the type of . Denote by the set of all such eigenvalues. Obviously, is determined by all ’s with .

It is not difficult to make clear all the localized graph eigenvalues in , since they are almost the same as the case. There are two kinds of eigenvalues in , initial and continued.

Theorem 3.2. Let , then . Moreover, the initial eigenvalues in are and with multiplicity and respectively.

Proof. Similarly to the case, the initial eigenvalues are and . For the -eigenfunctions of on , comparing to the -eigenfunctions of on , the only difference is those eigenfunctions whose support intersecting the boundary should be removed. A similar analysis shows that they are indexed by points in . Hence the multiplicity of is

Similarly, the -eigenfunctions of on are indexed by -level loops except those loops touching . Hence the multiplicity of is

The continued eigenvalues will be those that arise from eigenvalues of by the spectral decimation. Note that every eigenvalue of bifurcates into two choices of of by , except , which just yields the single choice since the other is a forbidden eigenvalue . We know that of correspond to eigenvalue of , while the remaining of them correspond to other eigenvalues, leading to a space of continued eigenfunctions of dimension . If we add to this and , we should obtain . Hence we have

Combining this with , we can easily get

As for primitive graph eigenvalues , things become more complicated. We consider and respectively. We will show in the next section the spectral decimation recipe for this type of eigenvalues can not be used directly. In fact there is even not an analytic relation between elements in (or ) and elements in (or ). A rough but intuitive explanation of why does this “bad” thing happen is that the Dirichlet boundary condition will be destroyed when we use the eigenfunction extension algorithm to extend a -eigenfunction from to or restrict a -eigenfunction from to . However, a weak but useful relation between (or ) and (or ) will be found in the next section, which will take the place of spectral decimation in the further discussion. Let be the same functions as defined in . We will prove that:

Theorem 3.3. For each , consists of distinct eigenvalues with multiplicity 1, between and strictly, denoted by in increasing order, satisfying

Moreover, and

Similar properties hold for with replaced by .

In order to study the relation between (or ) and (or ), we introduce the following notations. In symmetric case, let denote the -level eigenvalue between and . Let denote the -level eigenvalue between and for . Let denote the -level eigenvalue between and for . Let denote the -level eigenvalue between and