SCHR ¨ODINGER EQUATIONS DEFINED BY A CLASS OF SELF-SIMILAR 1 MEASURES 2

. We study linear and nonlinear Schr¨odinger equations deﬁned by fractal measures. Under the assumption that the Laplacian has compact resolvent, we prove that there exists a uniqueness weak solution for a linear Schr¨odinger equation, and then use it to obtain the existence and uniqueness of weak solution of a nonlinear Schr¨odinger equation. We prove that for a class of self-similar measures on R with overlaps, the Schr¨odinger equations can be discretized so that the ﬁnite element method can be applied to obtain approximate solutions. We also prove that the numerical solutions converge to the actual solution and obtain the rate of convergence


Introduction 4
The Schrödinger operator in the fractal setting has been studied by a number of authors. model has recently been proposed by physicist . This is obtained by replacing the 1 standard Lebesgue measure on a spacetime manifold by a Borel measure which is in general not 2 absolutely continuous with respect to Lebesgue measure. Also, solution of the Schrödinger equation respectively (see, e.g., [1,2]), where Re(z) denotes the real part of a complex number z and v 7 denotes the conjugate function of v. It is known (see, e.g., [15]) that µ defines a Dirichlet Laplace 8 operator ∆ D µ (or simply ∆ µ ), if the following Poincaré inequality for a measure (PI) holds: There 9 exists some constant C > 0 such that 10 U |u| 2 dµ ≤ C U |∇u| 2 dx for all u ∈ C ∞ c (U ) (1.1) (see, e.g., [15,20,21]). The main purpose of this paper is to study the following linear Schrödinger 11 equation defined by the Dirichlet Laplacian ∆ µ : where u := u(t) is a Hilbert space valued function of t. We study the solution of equation (1.2) 13 both theoretically and numerically.
is the unique weak solution of the following non-linear schrödinger equation (1.3) As an example, let F (u) = sin u − mu, m ≥ 0 (see, e.g., [14]). Then F (·) is Lipschitz continuous 2 on dom E. 3 We call a µ-measurable subset I of U a cell (in U ) if µ(I) > 0. Clearly, U itself is a cell.

4
Two cells I, J in U are measure disjoint with respect to µ if µ(I ∩ J) = 0. Let I ⊆ U be a cell. 5 We call a finite family P of measure disjoint cells a µ-partition of I if J ⊆ I for all J ∈ P, and 6 µ(I) = J∈P µ(J). A sequence of µ-partitions (P k ) k≥1 is refining if for any J 1 ∈ P k and any 7 J 2 ∈ P k+1 , either J 2 ⊆ J 1 or they are measure disjoint, i.e., each member of P k+1 is a subset of 8 some member of P k . Throughout this paper, |E| denotes the diameter of a subset E ⊆ R n . 9 In order to discretize (1.2) and obtain numerical approximations of the weak solution, we will 10 often impose the following additional conditions on µ: there exists a sequence of refining µ-partitions (P1) there exist some constant ρ ∈ (0, 1) and some integer m 0 satisfying max{|J| : J ∈ P k } ≤ 13 ρ k−m 0 for all k ≥ 1; 14 (P2) for any k ≥ 1, each cell I ∈ P k is closed and connected; 15 (P3) for any k ≥ 2 and any 0 ≤ ≤ N (k), there exist similitudes (τ k, ,j ) j=0 of the form 16 τ k, ,j (x) = r k, ,j x + b k, ,j and constants (c k, ,j ) j=0 such that τ k, ,j (I 1,j ) ⊆ I k, , and where ∂ x u(x, t) and ∂ t u(x, t) are the weak partial derivative of u with respect to x and t, respectively. 1 Let u 1 (x, t) and u 2 (x, t) be the real and imaginary parts of u(x, t), respectively. Then (1.5) can be 2 rewritten as for all real-valued function v ∈ dom E. there exists a sequence of refining partitions (P k ) k≥1 satisfying conditions (P1)-(P3), and I x j dµ, 7 I ∈ P 1 , j = 0, 1, 2, can be evaluated explicitly. Then equations (1.6) and (1.7) can be discretized 8 and the finite element method can be applied to yield a system of first-order ordinary differential 9 equations, which has a unique solution that can be solved numerically. 10 We are mainly interested in fractal measures µ. Let X be a non-empty compact subset of R d .

11
Throughout this paper, an iterated function system (IFS) refers to a finite family of contractive where 0 < ρ j < 1, and b j ∈ R d . It is well-known that for each IFS {S j } q j=1 , there exists a unique non-empty compact subset F ⊆ X, called the self-similar set, such that F = q j=1 S j (F ); moreover, associated to each set of probability weights {w j } q j=1 (i.e., w j > 0 and q j=1 w j = 1), there is a unique probability measure, called the self-similar measure, satisfying the following identity [11,17]). An IFS {S j } q j=1 is said to satisfy the open set condition (OSC) if there exists a associated with the golden ratio, the three-fold convolution of the Cantor measure, and a class 23 of self-similar measures that we call essentially of finite type (EFT) (see [23]). These measures share the common property that the support can be partitioned into a sequence of arbitrarily small 1 intervals whose measures can be computed explicitly.

2
The following theorem shows that the approximate solutions converge to the actual weak solution 3 and we also obtain a rate of convergence. See Subsection 2.1 and Definition 2.2 for the definitions 4 of · dom E and · 2,dom E , respectively. 5 Theorem 1.3. Assume the hypotheses of Theorem 1.2, let f ≡ 0 and g = ∞ n=1 α n ϕ n in equation (1.2), and fix t ∈ [0, T ]. If ∞ n=1 |α n | 2 λ 3 n < ∞, then the approximate solutions u m obtained by the finite element method converge in L 2 ((a, b), µ) to the actual weak solution u. Moreover, where ρ is the constant in condition (P1).

6
The rest of this paper is organized as follows. Section 2 summarizes some notation, definitions 7 and results that will be needed throughout the paper. We give the existence and uniqueness of 8 solution of Schrödinger equation (1.2) in Section 3. Section 4 is devoted to the proof of Theorem 1.2. 9 In Section 5, we apply Theorem 1.2 to three different self-similar measures with overlaps. The proof 10 of Theorem 1.3 is given in Section 6.

12
In this section, we summarize some notation, definitions and facts that will be used throughout 13 the rest of the paper. For a Banach space X, we denote its topological dual by X . For v ∈ X and 14 u ∈ X, we let v, u = v, u X ,X := v(u) denote the dual pairing of X and X.

15
Definition 2.1. Let X be a Banach space, u : (a, b) ⊆ R → X, and t 0 ∈ (a, b). Then u is said to be differentiable at t 0 in the norm · X if there exists v 0 ∈ X such that v 0 is called the strong derivative of u at t 0 , and we write Higher-order strong derivatives are defined similarly.

16
Note that if u is differentiable at t 0 in the norm · X , then it is continuous at t 0 in the norm If the interval [0, T ] is understood, we will abbreviate these norms as u p,X and u ∞,X , respec- be a bounded open subset. For a function ϕ : U → R, we let ϕ denote both 3 its classical and weak derivatives. If u ∈ L 2 ([0, T ], X), where X is H 1 0 (U ) or L 2 (U, µ) etc., then for 4 each fixed t, we denote by u x (x, t) (or ∇u) the classical or weak derivatives of u with respect to x.

13
(1) there exists a sequence {u n } in C ∞ c (U ) such that u n →û in H 1 0 (U ) and u n →û in L 2 (U, µ);

Extrapolation and weak solutions
3 In this section, we consider the existence and uniqueness of weak solution of equation ( be a bounded open subset and µ be a positive finite Borel measure with supp(µ) ⊆ U and µ(U ) > 0. We assume that (PI) holds. Let (E, dom E) be defined as in Section 2.1, and −∆ µ is the Dirichlet operator with respect to µ. By identifying L 2 (U, µ) with (L 2 (U, µ)) , we have the following Gelfand triple (see, e.g., [13, 30]): where all the embeddings are continuous, injective, and dense. The embedding L 2 (U, µ) → (dom E) is given by

It follows that for any
where throughout this paper, ·, · denotes the pairing between (dom E) and dom E. On the other hand, we note that the form E is coercive by (PI). Hence by the Lax-Milgram theorem, for every w ∈ (dom E) , there exists a unique u ∈ dom E such that Thus we can define a bijective operator L from dom E to (dom E) by and equip (dom E) with the scalar product Note that dom L = dom E and w, v = (w, v) µ for all w ∈ (dom E) and v ∈ dom E. It follows that L is an extension of −∆ µ . Throughout this paper, we equip (dom E) with the norm We remark that this norm is the standard norm in (dom E) , which is equivalent to the general 5 norm in (dom E) (see, e.g., [2]).

6
Definition 3.1. Use the notation above.
the Schrödinger equation (1.2) if the following conditions are satisfied: Here are some comments on Definition 3.1.
, and thus 5 the initial condition u(0) = g makes sense.
where L is defined as in (3.1). 7 We now assume that −∆ µ has compact resolvent and let {ϕ n } ∞ n=1 be an orthonormal basis of L 2 (U, µ) so that −∆ µ such that −∆ µ ϕ n = λ n ϕ n for all n ≥ 1, where 0 < λ 1 ≤ · · · ≤ λ n ≤ λ n+1 ≤ · · · and lim n→∞ λ n = ∞. Some sufficient conditions for which −∆ µ has compact resolvent can be found in [9,15,20]. In particular, if n = 1, then −∆ µ has compact resolvent for any such µ. We remark that is also a complete orthonormal basis of (dom E) . Note that w = ∞ n=1 a n ϕ n ∈ (dom E) if and only if there exists a unique Substituting v = ϕ n for n ≥ 1, we get a n = w, ϕ n = E(L −1 w, ϕ n ) = b n λ n , and so w = ∞ n=1 a n ϕ n ∈ (dom E) if and only if w 2 (dom E) = L −1 w 2 dom E = ∞ n=1 a 2 n /λ n < ∞. Therefore, for every u = ∞ n=1 a n ϕ n ∈ dom E, we have Lu = ∞ n=1 a n λ n ϕ n ∈ (dom E) , and We will describe the construction of the Laplacian −∆ µ in (1.2), as well as the associated 2), we assume that −∆ µ has compact resolvent.

(4.2)
Let u m 1 (x, t) and u m 2 (x, t) be the real and complex part of u m (x, t), respectively. We require u m (x, t) 2 to satisfy equations (1.6) and (1.7) as follows: where 1 ≤ , j ≤ N (m) − 1. It follows from the definition of φ j (x) that both M and K are tridiagonal. Let . . .  g(x m,j )φ j (x). Let g 1 (x) and g 2 (x) be the real and complex part of g(x). Therefore, we set w 1j (0) = g 1 (x m,j ) and w 2j (0) = g 2 (x m,j ) for 1 ≤ j ≤ N (m) − 1.
[28] showed that µ satisfies a family of second-order identities with respect to the following auxiliary IFS:  5.2. Three-fold convolution of the Cantor measure. We consider the following three-fold convolution of the Cantor measure studied in [18,22,23]. The three-fold convolution of the Cantor measure µ is the self-similar measure defined by the following IFS with overlaps (see [22]):
Let µ be a positive finite Borel measure on R with supp(µ) = [a, b]. In this case, there exists a complete orthonormal basis {ϕ n } ∞ n=1 of L 2 (U, µ) such that −∆ µ ϕ n = λ n ϕ n for all n ≥ 1, where the eigenvalues satisfy 0 < λ 1 ≤ · · · ≤ λ n ≤ λ n+1 ≤ · · · with lim n→∞ λ n = ∞. Assume that there exists a sequence of refining partitions (P k ) k≥1 satisfying conditions (P1)-(P3) in Section 1. Let V m be the set of end-points of all level-m sub-intervals, and arrange its element so that is called the Rayleigh-Ritz projection with respect to V m . Let Proof. The proof can be found in [26].

3
Throughout the rest of this section, let g = ∞ n=1 α n ϕ n ∈ dom E satisfy ∞ n=1 |α n | 2 λ 3 n < ∞ and 4 let f = 0, and u be the solution of the corresponding homogeneous Schrödinger equation (1.2). Proof. We first note that the functions e, e t and (P m u) t = P m u t all belong to S m D . Thus substituting ie for v in (6.2) and for v m in (6.3), we get (iu t , ie) µ + E(u, ie) = 0 and (iu m t , ie) µ + E(u m , ie) = 0. Subtracting these equations gives (i(u t − u m t ), ie) µ + E(u − u m , ie) = 0. Using the fact that (i(u t − u m t ), ie) µ = ((u t − u m t ), e) µ , we get (u t − P m u t + P m u t − u m t , e) µ + E(u − P m u + P m u − u m , ie) = 0, which, together with the fact E(u − P m u, ie) = 0 (Lemma 6.1), yields (P m u t − u m t , e) µ + E(P m u − u m , ie) = (P m u t − u t , e) µ . The desired result follows from the equalities E(P m u − u m , ie) = E(e, ie) = 0 and P m u t − u m t = 9 e t . Proof. The proof is similar to that of [29,Theorem 6.4]; we include it here for completeness. The left side of (6.4) can be rewritten as (e t , e) µ = 1 2 d dt e 2 µ = e µ · d dt e µ .
Thus (6.4) leads to e µ · e µ t = (P m u t − u t , e) µ ≤ P m u t − u t µ · e µ , and hence 1 e µ t ≤ P m u t − u t µ . (6.6) Integrating the left side of (6.6) with respect to τ from 0 to t, we get g(x m, )φ (x) = 0 is used in the last equality. Combining (6.6), (6.7), Lemma 6.3 and Hölder's inequality, we have e(t) µ ≤