Eigenvalues and Eigenfunctions of One-Dimensional Fractal Laplacians

We study the eigenvalues and eigenfunctions of one-dimensional weighted fractal Laplacians. These Laplacians are defined by self-similar measures with overlaps. We first prove the existence of eigenvalues and eigenfunctions. We then set up a framework for one-dimensional measures to discretize the equation defining the eigenvalues and eigenfunctions, and obtain numerical approximations of the eigenvalue and eigenfunction by using the finite element method. Finally, we show that the numerical eigenvalues and eigenfunctions converge to the actual ones and obtain the rate of convergence.


Introduction
Let be a continuous, positive, finite Borel measure on ℝ with support supp( ) = [a, b] .Consider the non-negative quadratic form E(⋅, ⋅) in L 2 ((a, b), ) defined by with domain dom E equal to the Sobolev space where V ∶ [a, b] → [0, +∞) .We say is an eigenvalue of Eq. (1.1) corresponding to a non-zero eigenfunction (x) ∈ dom E if holds for all v ∈ dom E .We remark that for the case V ≡ 1 , the finite element method is used to obtain numerical solutions to the eigenproblem (1.1) for self-similar measures satisfying a family of second-order self-similar identities in [3].These identities were first introduced by Strichartz and are used in [24] to approximate the density of the infinite Bernoulli convolution associated with the golden ratio.To the best of the authors' knowledge, in the absence of second-order identities, the eigenvalue problem defined by IFSs with overlaps has not been obtained before, and this is a main motivation of this paper.
The first objective of this paper is to obtain the existence result of eigenvalues and eigenfunctions of (1.1).Theorem 1.1 Let V(x) ≥ 0 on [a, b] with c V ∶= ‖V‖ L 1 ((a,b),) > 0 .Then there exists a complete orthonormal basis { n } ∞ n=1 of L 2 ((a, b), ) such that each n is an eigen- function of Eq. (1.1) corresponding to an eigenvalue n satisfying and lim n→∞ n = ∞ .Moreover, { n (x)} ∞ n=1 also forms an orthogonal basis of dom E. (1.1) To prove Theorem 1.1, we modify the classical argument (see [21]) and replace Lebesgue measure by a more general measure .
The second objective of this paper is to study Eq.(1.1) from a numerical point of view.Two closed sub-intervals I, J of [a, b] are measure disjoint with respect to if (I ∩ J) = 0 .Let I ⊆ [a, b] be a closed interval.We call a finite family P of measure disjoint cells a -partition of I if J ⊆ I for all J ∈ P , and for any m ≥ 1 , each member of P m+1 is a proper subset of some member of P m ; (2) for any m ≥ 2 and any J ∈ P m , there exist similitudes ( IJ ) I∈P 1 of the form Intuitively, (1.2) means that the measure of each closed interval in P m for m ≥ 2 can be expressed as a linear combination of { (I) ∶ I ∈ P 1 } .By making use of (1.2), some results concerning Δ have been obtained (see, e.g., [17, 25-27]).
In order to discretize (1.1) and obtain numerical approximations of the eigenvalue and eigenfunction, we will assume that there exists a sequence of compatible -partitions (P m ) m≥1 .Thus the measure of each closed interval in the par- tition can be computed by using (1.2), making it possible to discretize the Eq.(1.1).We remark that the assumption supp( ) = [a, b] guarantees that the mass matrix that arises in the finite element method is positive definite (see [2, Proposition 3.1]), Let V ≡ 1 in (1.1).If is an eigenvalue of Eq. (1.1) corresponding to a non- zero eigenfunction u ∈ dom E , then Theorem 1.2 Let be a continuous positive finite Borel measure on ℝ with supp( ) = [a, b] .Assume that there exists a sequence of compatible -partitions (P m ) m≥1 of [a, b] and the integrals ∫ I x k d , I ∈ P 1 , k = 0, 1, 2 , can be evaluated explicitly.Then Eq. (1.3) can be discretized into a matrix Eq. (3.5) by finite element method.Moreover, Eq. (3.5) can be solved numerically.
We are mainly interested in fractal measures with overlaps.Theorem 1.2 provides a framework under which discretization can be performed.We remark that if self-similar measures satisfies a family of second-order self-similar identities, then satisfies (1.2) (see [26,Proposition 5.1]).Hence, Theorem 1.1 cannot be deduced from [3,Theorem 1.2].
The following theorem shows that the approximate eigenvalues and eigenfunctions obtained in Theorem 1.2 converge to the actual ones, and we also obtain a rate of convergence. (1.2) for all v ∈ dom E.
Theorem 1.3 Assume the hypotheses of Theorem 1.2.If there exist constants r ∈ (0, 1) and c > 0 such that max{|J| ∶ J ∈ P k } ≤ c r k for all k ≥ 1 , then the numerical eigenvalues λ(m) n and normalized eigenfunctions φ(m) n , obtained in Theorem 1.2, converge to the corresponding theoretical n and n , respectively.Moreover, for each n ≥ 1 , there exists a constant C ∶= C(n) > 0 (depending only on n) such that for all m ≥ 1, We illustrate Theorems 1.2 and 1.3 by a class of self-similar measures with overlaps that we call essentially of finite type (EFT) (see [19]).This class is used in [19] to illustrate self-similar measures satisfying EFT.These measures have been studied extensively (see, e.g., [16,25,27]).However, it is not clear whether they satisfy a family of second-order identities.
The rest of this paper is organized as follows.In Sect.2, we prove Theorem 1.1.In Sect.3, we give the proofs of Theorems 1.2 and 1.3, and apply them to a class of self-similar measures with overlaps.

Existence of Eigenvalues and Eigenfunctions
In this section, we mainly prove Theorem 1.1.The technique used here is the same in spirit as in classical setting.

Proof of Theorem 1.1
Step 1.We show the existence of the smallest eigenvalue 1 .We remark that for each u ∈ dom E, Then  1 ≥ 1∕ > 0 .We claim that 1 is the smallest eigenvalue of Eq. (1.1).We choose a minimising sequence {u m } ⊂ A such that By the definition of A , we have the sequence {u m } is bounded in dom E .
Combining it with the fact dom E is compactly embedded into the space For w ∈ dom E , we define g(t) ∶= R( 1 + tw) .Then It follows from (2.2) and (2.5) that g(t) reaches its minimum at t = 0 .Then g � (0) = 0 .By virtue of (2.4) and (2.5), we have for all w ∈ dom E , which implies that 1 is an eigenvalue of Eq. (1.1).We remark that 1 is the smallest eigenvalue following from (2.2). (2.1) in dom E and 1 = E( 1 , 1 ).
Journal of Nonlinear Mathematical Physics (2023) 30:996-1010 Step 2. We prove the existence of the smallest eigenvalues 2 ≤ 3 ≤ ⋯ .Since the existence of 1 has been obtained, we can assume that there exists m − 1 eigenval- ues 1 ≤ 2 ≤ ⋯ ≤ m−1 with corresponding to the eigenfunctions 1 , 2 , ⋯ , m−1 , and (V k , k ) = 1 for all k = 1, 2, ⋯ , m − 1 , where m ≥ 2 .We define Similarly, we can prove that there exist some positive constant m and For each w ∈ dom E , we can write w as It follows that for all w ∈ dom E .Thus m is an eigenvalue of Eq. (1.1).It is easy to see that m ≥ m−1 .Therefore, Eq. (1.1) has a sequence of positive eigenvalues such that Step 3. We show that for any distinct eigenvalues i and j , the corresponding eigenfunctions i and j are orthogonal in dom E , that is Step 4. We claim that the dimension of the space consisting of eigenfunctions corresponding to a fixed eigenvalue is finite.Suppose, on the contrary, that there exists countably infinite sequence of eigenfunctions { k } ∞ k=1 corresponding to the same eigenvalue , which are linearly independent in dom E .So we can renormalize such that ) .However, we can see that which is a contradiction.Thus the claim holds.
Step 4. We show that Suppose, on the contrary, that there exists some positive constant M such that 0 <  n ≤ M for all n ≥ 1 .Without loss of generality, we assume that the sequence of corresponding eigenfunctions and hence there exists a convergent subsequence of Thus (2.7) holds.
Step 5. We show that the eigenfunctions { m } corresponding to { m } form a basis of dom E .Suppose that there exists some non-zero w However, m → ∞ as m → ∞ , and then (2.8) is impossible.Hence, the desired result holds.◻ Since dom E is dense in C 0 (a, b) , we see that { m } ∞ m=1 is also an orthogonal basis of L 2 ((a, b), Vd ) .We remark that (V m , m ) = 1 .Thus { m } ∞ m=1 is an orthonormal basis of L 2 ((a, b), Vd ) .It is well known that can also define a Neumann Laplace operator Δ N in L 2 ((a, b), ) .We remark that the eigenproblem (1.1) replacing Δ by Δ N also has a sequence of orthogonal eigenfunctions in L 2 ((a, b), ) , but the first eigenvalue is zero, corresponding to the constant eigen- function.The proof is similar to that of Theorem 1.1.We omit the details.

Corollary 2.1 Assume the hypotheses of Theorem 1.1 and let 1 be the first eigenvalue of eigenproblem (1.1). Then there exists a non-negative eigenfunction corresponding to 1 .
Proof Let 1 be an eigenfunction of (1.1) corresponding to 1 .It suffices to prove that | 1 | is also an eigenfunction corresponding to 1 .We remark that ) .Combining it with (2.4) and (2.5), we can see that 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:996-1010 On the other hand, by the definition of 1 , we have Thus the desired result follows by combining (2.9) and (2.10).◻

The Finite Element Method and Convergence of Numerical Solutions
In this section, we mainly prove Theorems 1.2 and 1.3.Let V ≡ 1 in Eq. (1.1).We first use the finite element method to solve the equation, and then prove the convergence of numerical approximations of the eigenvalue and eigenfunction.Let be a continuous positive finite Borel measure on ℝ with supp( ) = [a, b] , and } be the set of end- points of all level-m subintervals in P m , and S m be the space of continuous piece- wise linear functions on [a, b] with nodes W m , and let be the subspaces of S m consisting of functions satisfying the Dirichlet boundary con- dition.Then We choose the basis of S m consisting of the following tent functions: and choose the basis Proof of Theorem 1.2 Use the notation above.We use the finite element method to discretize Eq. (1.3).Let u(x) be an eigenfunction of Eq. (1.1) corresponding to an eigenvalue , that is, u(x) satisfies Eq. (1.3).We approximate u(x) by (2.10) where each w m,i is a constant to be determined.Fix any m ≥ 1 .We require u m (x) to satisfy the integral form of the eigenvalue equation 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:996-1010 We remark that numerical algorithms for general eigenproblem as (3.5) have been well developed.We can get good approximations for eigenvalues and eigenfunctions by choosing m sufficiently large (see Theorem 1.3 for details).
We now prove Theorem 1.3.Define the linear map We call F m the Rayleigh-Ritz projection with respect to W m .Using [23, Theo- rem 1.1], we have for all u ∈ dom E and v ∈ S m D .We remark that for all u ∈ dom E and m ≥ 1 (see [2, Lemma 5.3]), where is the norm of W m for m ≥ 1 .Recall that E n is the subspace spanned by eigenfunc- tions 1 , ⋯ , n of Eq. (1.1) with V ≡ 1 .Let e n be the set of unit vectors in E n and let Then provided that  (m) n < 1 , the approximated eigenvalues are bounded above by (see [3,Lemma 5.4]).
Proof of Theorem 1. 3 We want to estimate (m) n .Fix any n ≥ 1 and let u ∈ e n .Then ‖u‖ L 2 ((a,b), ) = 1 and u can be expressed as u = ∑ n i=1 a i i , where each i is an eigen- function corresponding to eigenvalue i .It follows that Combining it with Hölder inequality and (3.8), we have for all m ≥ 1 .According to (3.8), (3.9), and (3.10), we have u ∈ dom E and m ≥ 1.
where the last inequality used the assumption max{|J| ∶ J ∈ P k } ≤ c r k for all k ≥ 1 , and M 0 is a constant (depending only on n).We first assume that m is sufficiently large so that  (m) n < 1∕2 .Then by (3.9), we get Thus there exists a constant C > 0 (depending only on n) such that for all m ≥ 1, On the other hand, every eigenvalue is approximated from above(see [3]): Together with (3.11), it yields the convergence of the eigenvalues as the first inequality in (1.4).It suffices to prove the convergence of the numerical eigenfunctions.Let B ∶= { φ(m) 1 , ⋯ , φ(m) N(m) } be the set of normalized approximate eigenfunctions.Then B forms an orthonormal basis for S m D , which implies k and n are eigenfunctions, the two sides of this equation can be rewritten as E(F m  n , φ(m) k ) and E( n , φ(m) k ) , respectively.It follows from (3.7) that We remark that for any fixed n , there exists a constant  > 0 such that for all m suf- ficiently large, Then the size of the remaining sum is given by where the claim has been used in the second equality and  ∶= (F m  n , φ(m) k )  .It fol- lows that Since n and φn are normalized eigenfunctions, we have Combining (3.8), (3.13) and (3.14), we see that which completes the proof of Theorem 1.3.◻ Based on Theorems 1.2 and 1.3, we solve the eigenproblem (1.1) for a class of self-similar measures satisfying (EFT), which is defined by the following family of IFSs: where r 1 , r 2 ∈ (0, 1) satisfy r 1 + 2r 2 − r 1 r 2 ≤ 1 , i.e., S 2 (1) ≤ S 3 (0) .Recently, the self-similar measures defined by IFSs in (3.15) have been studied extensively (see, e.g., [5,13,18,19]).These papers mainly study the multifractal properties and spectral dimension of the corresponding self-similar measures.
Let be a self-similar measure defined by an IFS in (3.15) and a probability vector (p i ) 3
We remark that, I • is a connected component of S M k (0, 1) ∶= ⋃ i∈M k S i (0, 1) for all level-k islands I, (see Fig. 1).For k ≥ 1 , define
there exists a subsequence (also denoted by {u m } , for convenience) and a function 1 ∈ C 0 (a, b) such that Thus we can deduce from the Lebesgue dominated convergence theorem that Combining Eqs.(2.2)-(2.4)and the fact that we can see that which implies that {u m } is a Cauchy sequence in dom E .Hence

for j = 1 ,
… , N(m) − 1 , and the Dirichlet boundary condition u m (x) = 0 on {a, b} .It follows that w m,0 = w m,N(m) = 0 .Using this and substituting (3.2) into (3.3),we can deduce that for 1 ≤ j ≤ N(m) − 1 .We define the mass matrix M = M (m) = (M (m) ij ) and stiffness matrix K = K (m) = (K (m) ij ) , respectively, by where 1 ≤ i, j ≤ N(m) − 1 .Thus M and K are tridiagonal following from the defini- tion of m,j (x) .Hence, (3.4) can be expressed into matrix form as where We remark that the matrix K can be computed directly.We claim that the matrix M also can be computed.Recall that (P m ) m≥1 is a sequence of compatible -partitions of [a, b].Then (1.2) holds, which implies that the matrix M is completely deter- mined by Thus the claim holds, since the integrals in (3.6) can be evaluated explicitly for k = 0, 1, 2 and I ∈ P 1 .Since supp( ) = [a, b] , M is invertible (see [2, Proposition 3.1]).It follows that the eigenvalues and eigenfunctions of Eq. (3.5) can be solved numerically.◻ (3.1)