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

MEASURES

2

SZE-MAN NGAI AND WEI TANG*

3

Abstract. We study linear and nonlinear Schr¨odinger equations defined by fractal measures. Un- der the assumption that the Laplacian has compact resolvent, we prove that there exists a unique- ness 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 onRwith 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.

Contents

1. Introduction 1

2. Preliminaries 5

2.1. Dirichlet Laplacian defined by a measure 6

3. Extrapolation and weak solutions 7

4. The finite element method 12

5. Fractal measures defined by iterated function systems 15

5.1. Infinite Bernoulli convolution associated with the golden ratio 16

5.2. Three-fold convolution of the Cantor measure 16

5.3. A class of self-similar measures satisfying (EFT) 17

6. Convergence of numerical approximations 19

References 22

1. Introduction

4

The Schr¨odinger operator in the fractal setting has been studied by a number of authors.

5

Strichartz [27] studied the essential spectrum on the product of two copies of an infinite blowup of

6

the Sierpinski gasket. Chen et al. studied the spectral asymptotics of the eigenvalues and Bohr’s

7

formula on several unbouned fractals. For Schr¨odinger operators defined by measures, typically self-

8

similar measures, the authors [24] studied the bound states and Bohr’s formula. It has become more

9

and more apparent to physicists that spacetime exhibits fractal behavior. A multifractal spacetime

10

Date: July 31, 2021.

2010Mathematics Subject Classification. Primary: 28A80, 35Q41; Secondary: 74S05, 65L60, 65L20.

Key words and phrases. Fractal; Laplacian; Schr¨odinger equation; self-similar measure with overlaps.

* Corresponding author.

The authors are supported in part by the National Natural Science Foundation of China, grant 11771136, and Construct Program of the Key Discipline in Hunan Province. Also, the first author is supported in part by the Hunan Province Hundred Talents Program and a Faculty Research Scholarly Pursuit Funding from Georgia Southern University; the second author is supported in part by the National Natural Science Foundation of China, grant 11901187, and a Scientific Research Fund of Hunan Provincial Education Department (No. 19B117).

1

model has recently been proposed by physicist Calcagni [4–6]. 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 Schr¨odinger equation

3

could play an important role in studying related mathematical problems. In fact, Hu and Z¨ahle

4

obtained heat kernel upper bound in metric measure spaces by using the solution of the Schrodinger

5

equation [16]. Motivated by these, we study the Schr¨odinger equation defined by a fractal measure.

6

LetU ⊆R^d,d≥1, be a bounded open subset, and let µbe a positive finite Borel measure with supp(µ)⊆U and µ(U) >0. Let L²(U, µ) :=L²(U, µ,C) denote the space of measurable functions u:U →C such thatkuk_µ<∞ with

kuk_µ:=Z

U

|u|²dµ1/2

.

LetH¹(U) :=H¹(U,C) be the Sobolev space of complex-valued functions equipped with the norm kuk_H1(U) :=

Z

U

|u|²dx+ Z

U

|∇u|²dx 1/2

.

Let H₀¹(U) := H₀¹(U,C) denote the completion of C_c^∞(U) in the H¹-norm, where C_c^∞(U) is the space of all complex-valuedC^∞(U) functions with compact support inU. Throughout this paper, we regardL²(U, µ) and H₀¹(U) as real Hilbert spaces with the scalar product

(u, v)µ:= Re Z

U

uv dµ and (u, v)_H¹

0(U):= Re Z

U

∇u· ∇v dx,

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 µdefines a Dirichlet Laplace

8

operator ∆^D_µ(or simply ∆_µ), if the following Poincar´e inequality for a measure (PI) holds: There

9

exists some constantC >0 such that

10

Z

U

|u|²dµ≤C Z

U

|∇u|²dx 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 Schr¨odinger

11

equation defined by the Dirichlet Laplacian ∆_µ:

12







i∂_tu+ ∆_µu=f(t) on U ×[0, T],

u= 0 on ∂U×[0, T],

u=g on U × {t= 0},

(1.2) 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.

14

We will describe the construction of the Laplacian −∆_µ in (1.2), as well as the associated

15

nonnegative bilinear form (E,domE) (see (2.1)) in Subsection 2.1. Under the assumption−∆_µhas

16

compact resolvent, we give an explicit formula for the weak solution of the Schr¨odinger equation

17

(1.2) in Theorem 3.1. Let X be a Hilbert space with norm k · k, we say a map F : X → X is

18

Lipschitz continuous onXif there exists some constantC >0 such thatkF(u)−F(v)k ≤Cku−vk

19

for all u, v∈X. Using Theorem 3.1, we obtain our first main theorem.

20

Theorem 1.1. Let U ⊆R^d, d≥1, be a bounded open subset, and let µ be a positive finite Borel measure withsupp(µ)⊆U andµ(U)>0. Assume thatµsatisfies (PI),−∆_µhas compact resolvent, and F(·) is Lipschitz continuous ondomE. If g=P∞

n=1α_nϕ_n∈domE, then u(t) :=

∞

X

n=1

α_ne^−iλntϕ_n−i

∞

X

n=1

Z t 0

e^{−iλn(t−τ)} F(u(τ)), ϕ_n

µdτ ϕ_n.

is the unique weak solution of the following non-linear schr¨odinger equation

1







i∂_tu+ ∆_µu=F(u) on U×[0, T],

u= 0 on ∂U×[0, T],

u=g on U× {t= 0}.

(1.3)

As an example, letF(u) = sinu−mu,m≥0 (see, e.g., [14]). ThenF(·) is Lipschitz continuous

2

on domE.

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 inU are measure disjoint with respect to µ ifµ(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 ifJ ⊆ I for all J ∈ P, and

6

µ(I) = P

J∈Pµ(J). A sequence of µ-partitions (P_k)k≥1 is refining if for any J1 ∈ P_k and any

7

J2 ∈ Pk+1, either J2 ⊆ J1 or they are measure disjoint, i.e., each member of Pk+1 is a subset of

8

some member ofPk. Throughout this paper,|E|denotes the diameter of a subsetE⊆Rⁿ.

9

In order to discretize (1.2) and obtain numerical approximations of the weak solution, we will

10

often impose the following additional conditions onµ: there exists a sequence of refiningµ-partitions

11

(P_k)_k≥1 = ({I_{k,`</sub>}<sup>N</sup><sub>`=0}^(k))_k≥1 of U such that

12

(P1) there exist some constant ρ ∈(0,1) and some integer m₀ satisfying max{|J|:J ∈ P_k} ≤

13

ρ^k−m0 for all k≥1;

14

(P2) for anyk≥1, each cellI ∈P_k is closed and connected;

15

(P3) for any k ≥ 2 and any 0 ≤ ` ≤ N(k), there exist similitudes (τ<sub>k,`,j</sub>)^N_j=0⁽¹⁾ of the form

16

τ_{k,`,j</sub>(x) =r<sub>k,`,j}x+b_{k,`,j</sub> and constants (c<sub>k,`,j})^N(1)_j=0 such thatτ_{k,`,j</sub>(I<sub>1,j</sub>)⊆I<sub>k,`}, and

17

µ|_Ik,` =

N(1)

X

j=0

c_{k,`,j</sub> ·µ|<sub>I1,j</sub> ◦τ<sub>k,`,j}⁻¹ . (1.4)

Under assumption (P3), theµmeasure of each cell in the partition can be computed by using (1.4),

18

making it possible to discretize the Schr¨odinger equation (1.2). In order to discretize (1.2) and

19

guarantee that the mass matrix that arises in the finite element method is positive definite (see

20

Proposition 4.1), we assume thatµ is a measure onR with supp(µ) = [a, b].

21

Let f(x, t) ≡ 0 in (1.2). Multiplying the first equation in (1.2) byv ∈ domE, integrating both

22

sides over [a, b] with respect todµ, and then taking the real part, we obtain

23

Re Z b

a

i∂tu(x, t)v(x)dµ= Re Z b

a

∂xu(x, t)v⁰(x)dx, (1.5)

where∂_xu(x, t) and∂_tu(x, t) are the weak partial derivative ofuwith respect toxandt, respectively.

1

Letu₁(x, t) andu₂(x, t) be the real and imaginary parts ofu(x, t), respectively. Then (1.5) can be

2

rewritten as

3

Z b a

∂tu2(x, t)v(x)dµ=− Z b

a

∂xu1(x, t)v⁰(x)dx (1.6) and

4

Z _b

a

∂_tu₁(x, t)v(x)dµ= Z _b

a

∂_xu₂(x, t)v⁰(x)dx (1.7) for all real-valued functionv∈domE.

5

Theorem 1.2. Let µ be a positive finite Borel measure on R with supp(µ) = [a, b]. Assume that

6

there exists a sequence of refining partitions (P_k)_k≥1 satisfying conditions (P1)–(P3), andR

Ix^jdµ,

7

I ∈ P₁, 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

12

similitudes{S_j}^q_j=1 defined onX, i.e.,

13

S_j(x) =ρ_jx+b_j, j= 1, . . . , q, (1.8) 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 subsetF ⊆X, called theself-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 Pq

j=1w_j = 1), there is a unique probability measure, called the self-similar measure, satisfying the following identity

µ=

q

X

j=1

w_jµ◦S_j⁻¹

(see [11, 17]). An IFS {S_j}^q_j=1 is said to satisfy the open set condition (OSC) if there exists a

14

non-empty bounded open setO such thatS

kS_k(O)⊆Oand S_k(O)∩S_j(O) =∅for allk6=j. IFSs

15

that do not satisfy (OSC), as well as all associated self-similar measures, are said to have overlaps.

16

It is worth pointing out that for general self-similar measures with overlaps, it does not seem

17

possible to discretize the Schr¨odinger equations (1.2) in the way described in the paper, and thus

18

it is not clear how numerical approximations of the weak solution can be obtain. Theorem 1.2

19

provides a framework under which discretization can be performed.

20

Based on Theorem 1.2, we solve the Schr¨odinger equation (1.2) numerically for three differen-

21

t one-dimensional self-similar measures with overlaps, namely, the infinite Bernoulli convolution

22

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

24

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 k · k_domE and k · k_2,domE, respectively.

5

Theorem 1.3. Assume the hypotheses of Theorem 1.2, let f ≡0 and g=P∞

n=1α_nϕ_n in equation (1.2), and fix t∈[0, T]. If P∞

n=1|α_n|²λ³_n <∞, then the approximate solutions u^m obtained by the finite element method converge in L²((a, b), µ) to the actual weak solution u. Moreover,

ku^m−uk_µ≤2

√

Tku_tk_2,domE+kuk_domE ρ^m/2, 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 Schr¨odinger 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.

11

2. Preliminaries

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 spaceX, we denote its topological dual byX⁰. Forv∈X⁰ and

14

u∈X, we let hv, ui=hv, ui_X⁰_,X :=v(u) denote the dual pairing of X⁰ and X.

15

Definition 2.1. Let X be a Banach space, u: (a, b)⊆R→ X, and t0 ∈(a, b). Then u is said to be differentiable att0 in the norm k · k_X if there exists v0∈X such that

h→0lim

u(t₀+h)−u(t₀)

h −v₀

X = 0.

v0 is called the strong derivativeof u att0, and we write v₀ =u_t(t₀) = lim

h→0

u(t₀+h)−u(t₀)

h .

Higher-order strong derivatives are defined similarly.

16

Note that ifu is differentiable at t0 in the normk · k_X, then it is continuous at t0 in the norm

17

k · k_X.

18

Definition 2.2. Let X be a separable Banach space with normk · k_X. Denote by L^p([0, T], X) the

19

space of all measurable functions u: [0, T]→X satisfying

20

(1) kuk_Lp([0,T],X) :=

RT

0 ku(t)k^p_Xdt 1/p

<∞, if 1≤p <∞, and

21

(2) kuk_L∞([0,T],X):= esssup_0≤t≤Tku(t)k_X <∞, ifp=∞.

22

If the interval [0, T] is understood, we will abbreviate these norms as kuk_p,X and kuk∞,X, respec-

1

tively.

2

LetU ⊆R^d,d≥1, be a bounded open subset. For a functionϕ:U →R, we letϕ⁰ denote both

3

its classical and weak derivatives. Ifu∈L²([0, T], X), whereX isH₀¹(U) or L²(U, µ) etc., then for

4

each fixedt, we denote byux(x, t) (or∇u) the classical or weak derivatives ofuwith respect tox.

5

Remark 2.1. For each 1 ≤p ≤ ∞, L^p([0, T], X) is a Banach space; moreover, L^p2([0, T], X) ⊆

6

L^p1([0, T], X) if 1≤p1 ≤p2 ≤ ∞. Let X be a separable Banach space with inner product(·,·)_X. If

7

(X,(·,·)_X) is a separable Hilbert space, thenL²([0, T], X) is a Hilbert space with the inner product

8

(u, v)_L²_([0,T],X):=

Z T 0

(u(t), v(t))Xdt.

Definition 2.3. Let X be a Banach space. We define C([0, T], X) to be the vector space of all

9

continuous functionsu: [0, T]→X such that

10

kuk_C([0,T],X):= max

0≤t≤Tkuk_X <∞.

2.1. Dirichlet Laplacian defined by a measure. LetU ⊆R^d,d≥1, 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. In [15], (PI) implies thatµ defines a Dirichlet operator in

L²(U, µ,R) :=n

u:U →R: Z

U

|u|²dx <∞o .

Similarly, we can define a Dirichlet operator in L²(U, µ) := L²(U, µ,C), as follows. (PI) implies

11

each equivalence classu∈H₀¹(U) contains a unique (in theL²(U, µ) sense) member ˆu that belongs

12

toL²(U, µ) and satisfies both conditions below:

13

(1) there exists a sequence{u_n}inC_c^∞(U) such thatu_n→uˆinH₀¹(U) andu_n→uˆinL²(U, µ);

14

(2) ˆu satisfies inequality (1.1).

15

We call ˆu the L²(U, µ)-representative of u. Define a mapping ι:H₀¹(U) → L²(U, µ) by ι(u) = ˆu.

16

ι is a bounded linear operator, but not necessarily injective. Consider the subspace N of H₀¹(U)

17

defined as N :=

u ∈H₀¹(U) :kι(u)k_µ = 0 .Now let N^⊥ be the orthogonal complement of N in

18

H₀¹(U). Then ι:N^⊥→L²(U, µ) is injective. Unless explicitly stated otherwise, we will denote the

19

L²(U, µ)-representative ˆu simply byu.

20

Consider the non-negative bilinear form E(·,·) in L²(U, µ) defined by

21

E(u, v) := Re Z

U

∇u· ∇v dx (2.1)

with domain domE = N^⊥, or more precisely, ι(N^⊥). (PI) implies that (E,domE) is a closed quadratic form in L²(U, µ). Hence there exists a non-negative self-adjoint operator A in L²(U, µ) such that

E(u, v) = A^1/2u, A^1/2v

µ and domE = dom (A^1/2)

(see, e.g., [12, Theorem 1.3.1]). We write ∆^D_µ = −A, and call it the (Dirichlet) Laplacian with

1

respect toµ. If no confusion is possible, we denote ∆^D_µ simply by ∆_µ.

2

Let u ∈ domE. Then u ∈ dom ∆_µ if and only if there exists a unique f ∈ L²(U, µ) such that E(u, v) = (f, v)_µ for all v∈domE.In this case, −∆_µu=f. Throughout this paper, we let

domE:=N^⊥ and k · k_domE :=k · k_H1 0(U). 3. Extrapolation and weak solutions

3

In this section, we consider the existence and uniqueness of weak solution of equation (1.2). Let U ⊆R^d,d≥1, be a bounded open subset andµbe a positive finite Borel measure with supp(µ)⊆U and µ(U) >0. We assume that (PI) holds. Let (E,domE) be defined as in Section 2.1, and −∆_µ is the Dirichlet operator with respect to µ. By identifying L²(U, µ) with (L²(U, µ))⁰, we have the following Gelfand triple (see, e.g., [13, 30]):

domE ,→L²(U, µ)∼= (L²(U, µ))⁰,→(domE)⁰,

where all the embeddings are continuous, injective, and dense. The embeddingL²(U, µ),→(domE)⁰ is given by

w∈L²(U, µ)7→(w,·)_µ∈(L²(U, µ))⁰ ⊂(domE)⁰. It follows that for any u∈domE, there exists a unique w∈(domE)⁰ such that

E(u, v) =hw, vi for all v∈domE,

where throughout this paper,h·,·i denotes the pairing between (domE)⁰ and domE. On the other hand, we note that the form E is coercive by (PI). Hence by the Lax-Milgram theorem, for every w∈(domE)⁰, there exists a unique u∈domE such that

E(u, v) =hw, vi for all v∈domE.

Thus we can define a bijective operatorL from domE to (domE)⁰ by

4

Lu=w, (3.1)

and equip (domE)⁰ with the scalar product

(u, v)_(domE)⁰ :=E(L⁻¹u, L⁻¹v).

Note that domL= domE andhw, vi= (w, v)_µ for all w∈(domE)⁰ and v∈domE. It follows that L is an extension of−∆_µ. Throughout this paper, we equip (domE)⁰ with the norm

kwk_(domE)0 :=kL⁻¹wk_domE for w∈(domE)⁰.

We remark that this norm is the standard norm in (domE)⁰, which is equivalent to the general

5

norm in (domE)⁰ (see, e.g., [2]).

6

Definition 3.1. Use the notation above. Let 0 < T < ∞. Assume f ∈ L^∞([0, T],domE) and

7

g∈domE. A function u∈L^∞([0, T],domE) with ∂tu∈L^∞([0, T],(domE)⁰) is aweak solution of

8

the Schr¨odinger equation (1.2) if the following conditions are satisfied:

9

(1) hi∂_tu, vi − E(u, v) = (f(t), v)_µ for each v∈domE and Lebesgue a.e. t∈[0, T];

1

(2) u(0) =g.

2

Remark 3.1. Here are some comments on Definition 3.1.

3

(a) The boundary condition u|_∂U = 0 in (1.2) is included in the assumption u(t) ∈ domE. If

4

u∈L^∞([0, T],domE)and∂tu∈L^∞([0, T],(domE)⁰), thenu∈C([0, T], L²(U, µ)), and thus

5

the initial condition u(0) =g makes sense.

6

(b) Condition (1) above is equivalent to

i∂_tu−Lu=f(t) in (domE)⁰ for Lebesgue a.e. t∈[0, T], 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 L²(U, µ) so that −∆_µ such that −∆_µϕ_n=λ_nϕ_n for all n≥1, where 0< λ₁ ≤ · · · ≤λ_n ≤λ_n+1 ≤

· · · and limn→∞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 suchµ. We remark that

domE = ^∞

X

n=1

anϕn:

∞

X

n=1

|a_n|²λn<∞

and dom ∆µ= ^∞

X

n=1

anϕn:

∞

X

n=1

|a_n|²λ²_n<∞

.

Since domE ,→ L²(U, µ) ,→ (domE)⁰, {ϕ_n}^∞_n=1 is also a complete orthonormal basis of (domE)⁰. Note that w =P∞

n=1a_nϕ_n ∈ (domE)⁰ if and only if there exists a uniqueL⁻¹w = P∞

n=1b_nϕ_n ∈ domE such that E(L⁻¹w, v) = hw, vi for all v ∈ domE. Substituting v = ϕ_n for n ≥ 1, we get a_n = hw, ϕ_ni = E(L⁻¹w, ϕ_n) = b_nλ_n, and so w = P∞

n=1a_nϕ_n ∈ (domE)⁰ if and only if kwk²_(domE)0 =kL⁻¹wk²_domE =P∞

n=1a²_n/λ_n < ∞. Therefore, for every u =P∞

n=1a_nϕ_n ∈ domE, we have Lu=P∞

n=1anλnϕn∈(domE)⁰, and (domE)⁰ =

∞

X

n=1

anϕn:

∞

X

n=1

a²_n/λn<∞

.

We will describe the construction of the Laplacian −∆_µ in (1.2), as well as the associated

8

nonnegative bilinear form (E,domE) (see (2.1)) in Subsection 2.1. To give an explicit formula for

9

the weak solution of the Schr¨odinger equation (1.2), we assume that −∆_µ has compact resolvent.

10

Then there exists a complete orthonormal basis{ϕ_n}^∞_n=1 ofL²(U, µ) such that−∆_µϕn=λnϕnfor

11

all n≥1, where the eigenvalues satisfy 0< λ1 ≤ · · · ≤λn ≤λn+1 ≤ · · · with limn→∞λn=∞. If

12

g∈L²(U, µ) and f(t)∈L²([0, T], L²(U, µ)), we can write

13

g=

∞

X

n=1

αnϕn and f(t) =

∞

X

n=1

βn(t)ϕn, (3.2)

whereβn(t) = (f(t), ϕn)µ forn≥1. Let

14

u(t) :=

∞

X

n=1

α_ne^−iλntϕ_n−i

∞

X

n=1

Z t 0

e^{−iλn(t−τ)}β_n(τ)dτ

ϕ_n (3.3)

and

1

K(t) :=−i

∞

X

n=1

αnλne^−iλntϕn−if(t)−

∞

X

n=1

λn

Z t 0

e^{−iλn(t−τ)}βn(τ)dτ

ϕn. (3.4)

Theorem 3.1. Let U ⊆R^d, d≥1, be a bounded open subset, and let µ be a positive finite Borel

2

measure with supp(µ) ⊆ U and µ(U) > 0. Assume that µ satisfies (PI) and −∆_µ has compact

3

resolvent. Letg,f(t)u(t) andK(t)be defined as above. Ifg∈domE andf(t)∈L^∞([0, T],domE),

4

then

5

(a) ∂_tu=K(t) in (domE)⁰ for Lebesgue a.e. t∈[0, T].

6

(b) u(t) is the unique weak solution of the Schr¨odinger equation (1.2).

7

(c) if f ≡0, then

8

ku(t)k_µ=kgk_µ and E(u, u) =E(g, g) for allt∈[0, T].

If, in addition, P∞

n=1|α_n|²λ³_n<∞, then ∂_tu(t)∈domE for Lebesgue a.e. t∈[0, T].

9

Proof. Letg,f(t),u(t) andK(t) be defined as in (3.2), (3.3) and (3.4), respectively. Sinceg∈domE and f(t) ∈ L^∞([0, T],domE), u(t) ∈ C([0, T],domE) and K(t) ∈ L^∞([0, T],(domE)⁰). In fact, using H¨older’s inequality, we obtain

ku(t)k²C([0,T],domE) = max

t∈[0,T]ku(t)k²_domE ≤2 ^∞

X

n=1

|α_n|²λ_n+T

∞

X

n=1

λ_n Z T

0

β_n(τ)

2dτ

= 2

kgk²_domE+Tkf(t)k²_2,domE

<∞,

and

kK(t)k²_∞,(domE)0 = esssup

t∈[0,T]

kK(t)k²_(domE)0

≤3 ^∞

X

n=1

|α_n|²λ_n+kf(t)k²_∞,(domE)0 + esssup

t∈[0,T]

∞

X

n=1

λ_n

Z t 0

e^{−iλn(t−τ)}β_n(τ)d τ

2

≤3

kgk²_domE+kf(t)k²_∞,(domE)0+Tkf(t)k²_2,domE

<∞.

(a) Let δ satisfy 0 <2δ < T and u(t) =:P∞

n=1cn(t)ϕn. For all t∈[δ, T −δ] and each h∈(−δ, δ), we have

u(t+h)−u(t) =

∞

X

n=1

αn e^−iλn(t+h)−e^−iλnt ϕn−i

∞

X

n=1

Z t+h t

e^{−iλn(t+h−τ)}βn(τ)dτ

ϕn

−i

∞

X

n=1

Z t 0

e^{−iλn(t+h−τ)}−e^{−iλn(t−τ)}

β_n(τ)dτ ϕ_n.

It follows that

1

|c_n(t+h)−c_n(t)|² =

α_n e^−iλn(t+h)−e^−iλnt

−i Z t+h

t

e^{−iλn(t+h−τ)}β_n(τ)dτ

−i Z t

0

e^{−iλn(t+h−τ)}−e^{−iλn(t−τ)}

βn(τ)dτ

2

≤3 |α_n|²·

e^−iλn(t+h)−e^−iλnt

2

+

Z t+h t

e^{−iλn(t+h−τ)}β_n(τ)dτ

2

+

Z t 0

e^{−iλn(t+h−τ)}−e^{−iλn(t−τ)}

βn(τ)dτ

2!

≤3 h²|α_n|²λ²_n+h Z t+h

t

|β_n(τ)|²dτ

+T Z T

0

e^{−iλn(t+h−τ)}−e^{−iλn(t−τ)}

2· |β_n(τ)|²dτ

!

≤3h²

|α_n|²λ²_n+ esssup

t∈[0,T]

|β_n(t)|²+T λ²_n Z T

0

|β_n(τ)|²dτ

=: 3h²λ_nM_n,

(3.5)

where the fact|e^−iθ−1| ≤θ forθ >0 is used in the second and third inequalities. We remark that

2

P∞

n=1Mn= kgk²_domE +kf(t)k²_∞,(domE)0 +Tkf(t)k²_2,domE <∞. Let K(t) =: P∞

n=1dn(t)ϕn. Using

3

(3.4) and H¨older inequality, we have

4

|d_n(t)|² =

−iα_nλ_ne^−iλnt−iβ(t)−λ_n Z t

0

e^{−iλn(t−τ)}β_n(τ)dτ

2

≤3

|α_n|²λ²_n+ esssup

t∈[0,T]

|β(t)|²+T λ²_n Z T

0

|β_n(τ)|²dτ

= 3λ_nM_n.

(3.6)

Note that the classical derivativec⁰_n(t) =dn(t) for Lebesgue a.e. t∈[0, T]. It follows that

5

s_n(t, h) := c_n(t+h)−c_n(t)

h −d_n(t)→0 as h→0 (3.7)

for Lebesgue a.e. t ∈[δ, T −δ] andh ∈(−δ, δ). Combining (3.5) and (3.6), we have for Lebesgue

6

a.e. t∈[δ, T −δ] and each h∈(−δ, δ),

7

|s_n(t, h)|² λ_n ≤ 2

λ_n

|c_n(t+h)−c_n(t)|²

h² +|d_n(t)|²

≤12M_n. (3.8)

Using (3.8) and Weierstrass’ M-test, we see the series P∞

n=1|s_n(t, h)|²/λ_n converges uniformly for all h∈(−δ, δ) and Lebesgue a.e. t∈[δ, T −δ]. Thus for Lebesgue a.e. t∈[δ, T −δ],

h→0lim

u(t+h)−u(t)

h −K(t)

2

(domE)⁰ = lim

h→0

∞

X

n=1

|s_n(t, h)|²/λ_n=

∞

X

n=1

h→0lim|s_n(t, h)|²/λ_n= 0, where (3.7) is used in the last equality. It follows that∂tu(t) =K(t) in (domE)⁰ for Lebesgue a.e.

8

t∈[δ, T −δ]. The desired results follows by letting δ →0⁺.

9

(b) We first note thatu(0) =g, and

1

Lu=

∞

X

n=1

αnλne^−iλntϕn−i

∞

X

n=1

λn

Z t 0

e^{−iλn(t−τ)}βn(τ)dτ

ϕn, (3.9)

whereLis defined as in (3.1). Combining (3.9) and (a), we havei∂tu(t)−Lu(t) =f(t) on (domE)⁰

2

for Lebesgue a.e. t ∈ [0, T]. It follows from Remark 3.1, that u(t) is a weak solution of (1.2). It

3

suffices to show the only solution of (1.2) with f(t) ≡ g ≡ 0 is that u(t) ≡ 0. Let u be a weak

4

solution of (1.2) with f(t)≡g≡0. To verify this, note that

5

hi∂_tu,−iui+E(u,−iu) = 0 for Lebesgue a.e. t∈[0, T]. (3.10) Since E(u,−iu) = 0,u(0) =g≡0, and

hi∂_tu,−iui= 1 2

d

dtku(t)k²_µ,

we obtainku(t)k²_µ= 0 for Lebesgue a.e. t∈[0, T]. It follows thatu= 0, which proves (b).

6

(c) Sincef ≡0, by parts (a) and (b), we haveu(t) =P∞

n=1α_ne^−iλntϕ_nand∂_tu(t) =−iP∞

n=1α_nλ_ne^−iλntϕ_n.

7

It follows that

8

ku(t)k²_µ=

∞

X

n=1

|α_n|² =kgk²_µ and E(u, u) =

∞

X

n=1

|α_n|²λn=E(g, g) for allt∈[0, T].

Note thatE(∂_tu, ∂_tu) =P∞

n=1|α_n|²λ³_nfor Lebesgue a.e. t∈[0, T]. Thus the assumptionP∞

n=1|α_n|²λ³_n<

9

∞ implies∂_tu(t)∈domE for Lebesgue a.e. t∈[0, T].

10

Now we prove Theorem 1.1. The main ingredients are Banach’s fixed point theorem and Theorem

11

3.1.

12

Proof of Theorem 1.1. Given a function u ∈ L^∞([0, T],domE), set h(t) := F(u(t)). By the as-

13

sumption on F(·), we see thath∈L^∞([0, T],domE). Consequently, Theorem 3.1 ensures that the

14

linear Schr¨odinger equation

15







i∂_tw+ ∆_µw=h onU ×[0, T],

w= 0 on∂U×[0, T],

w=g onU × {t= 0},

(3.11)

has a unique weak solution w(t)∈L^∞([0, T],domE) given by

16

w(t) :=

∞

X

n=1

α_ne^−iλntϕ_n−i

∞

X

n=1

Z t 0

e^{−iλn(t−τ)} h(τ), ϕ_n

µdτ

ϕ_n. (3.12)

Define A:L^∞([0, T],domE)→L^∞([0, T],domE) by A[u] =w.

17

We now claim that ifT >0 is small enough, thenAis a contraction mapping fromL^∞([0, T],domE)

18

toL^∞([0, T],domE). Letu(t), v(t)∈L^∞([0, T],domE). We first get from (3.12) that for Lebesgue

19

a.e. 0≤t≤T,

1

A[u(t)]−A[v(t)]

2 domE =

∞

X

n=1

λ_n

Z _t

0

e^{−iλn(t−τ)} F(u(τ))−F(v(τ)), ϕ_n

µdτ

2

≤t Z t

0

∞

X

n=1

λn

F(u(τ))−F(v(τ)), ϕn

µ

2

dτ

=t Z t

0

F(u(τ))−F(v(τ))

2 domEdτ

≤CT Z _t

0

u(τ)−v(τ)

2

domEdτ ≤CT²ku−vk²_∞,domE,

(3.13)

where the assumption that F(·) is Lipschitz is used in the second inequality. It follows that kA[u]−A[v]k∞,domE ≤√

CTku−vk∞,domE. ThusA[·] is a strict contraction, provided T >0 is so small that √

CT =γ <1.

2

Given any T >0, we select T₁ >0 so small that √

CT₁ <1. We can then apply Banach’s fixed

3

point theorem to obtain a weak solution u of the nonlinear Schr¨odinger equation (1.3) that exists

4

on the time interval [0, T₁]. Sinceu(t)∈domEfora.e.0≤t≤T₁, we can find someT₂ ∈(T₁/2, T₁)

5

such thatu(T₂)∈domE. We can then repeat the argument above to extend our solutionu to the

6

time interval [T₂, T₃] such that u(T₃) ∈domE and T₃ ∈[2T₂, T₁+T₂]. Repeating this process for

7

a finite number of steps, we obtain a weak solution that exists on the entire interval [0, T].

8

To prove the uniqueness, suppose thatuandvare two weak solutions of the nonlinear Schr¨odinger

9

equation (1.3). Then we have A[u] =u and A[v] =v. It follows from the uniqueness of the fixed

10

point of A that u(t) = v(t) in L^∞([0, T1],domE). Combining this argument with the extension

11

argument above shows thatu(t) =v(t) in L^∞([0, T],domE).

12

4. The finite element method

13

In this section, we letf ≡0 in equation (1.2), and use the finite element method to solve the ho-

14

mogeneous Schr¨odinger equation (1.2). Letµbe a positive finite Borel measure onRwith supp(µ) =

15

[a, b]. Assume that there exists a sequence of refining partitions (P_k)k≥1 = ({I_{k,`</sub>}<sup>N</sup><sub>`=0}^(k))k≥1 satisfy-

16

ing conditions (P1)–(P3) in Section 1. Without loss of generality, we can writeI_{m,`</sub>= [x<sub>m,`}, x_m,`+1]

17

form≥1 and 0≤`≤N(m)−1. It is easy to see thatx_m,0=a andx_m,N(m)=bfor all m≥1.

18

We first note that the equations (1.6) and (1.7) are the integral form of the homogeneous

19

Schr¨odinger equation (1.2). Now, we apply the finite element method to approximate the solu-

20

tion u(x, t) satisfying (1.6) and (1.7) by

21

u^m(x, t) :=

N(m)

X

j=0

(w1j(t) +iw2j(t))φj(x), (4.1) where, forj = 0,1, . . . , N(m),w1j(t) :=w^m_1j(t) andw2j(t) :=w^m_2j(t) are real-valued functions to be

22

determined, and φj(x) := φm,j(x) are the standard piecewise linear finite element basis functions

23

(also called tent functions) defined by

1

φ_j(x) :=















x−xm,j−1

x_m,j−xm,j−1

ifx∈Im,j−1, j= 1,2, . . . , N(m), x−xm,j+1

x_m,j−x_m,j+1 ifx∈Im,j, j= 0,1, . . . , N(m)−1,

0 otherwise.

(4.2)

Letu^m₁ (x, t) andu^m₂ (x, t) be the real and complex part ofu^m(x, t), respectively. We requireu^m(x, t)

2

to satisfy equations (1.6) and (1.7) as follows:

3

Z b a

∂tu^m₂ (x, t)φj(x)dµ=− Z b

a

∂xu^m₁ (x, t)φ⁰_j(x)dx (4.3) and

4

Z b a

∂tu^m₁ (x, t)φj(x)dµ= Z b

a

∂xu^m₂ (x, t)φ⁰_j(x)dx. (4.4) Moreover, we require u^m(x, t) to satisfy the Dirichlet boundary conditionu^m(a, t) =u^m(b, t) = 0.

5

We note that φ`(a) = φ`(xm,0) = 0 and φj(b) = φj(x_m,N(m)) = 0 for all ` = 1, . . . , N(m) and

6

j= 0,1, . . . , N(m)−1. Thus w_k0(t) =w_kN(m)(t) = 0 for allt∈[0, T] and k= 1,2. Using this and

7

substituting (4.1) into (4.3) and (4.4) gives

8

N(m)−1

X

`=1

w⁰_2`

Z _b

a

φ_`(x)φ_j(x)dµ=−

N(m)−1

X

`=1

w_1`

Z _b

a

φ⁰_`(x)φ⁰_j(x)dx (4.5) and

9

N(m)−1

X

`=1

w⁰_1`

Z b a

φ_`(x)φj(x)dµ=

N(m)−1

X

`=1

w_2`

Z b a

φ⁰_{`</sub>(x)φ<sup>0</sup><sub>j</sub>(x)dx (4.6) for 1 ≤ j ≤ N(m)−1. We define the mass matrix M = M<sup>(m)</sup> = (M<sub>`j}^(m)) and stiffness matrix K=K^(m) = (K_`j^(m)), respectively, by

M_`j^(m) = Z b

a

φ_{`</sub>(x)φj(x)dµ and K<sub>`j}^(m)= Z b

a

φ⁰_`(x)φ⁰_j(x)dx,

where 1 ≤ `, j ≤ N(m)−1. It follows from the definition of φj(x) that both M and K are tridiagonal. Let

w₁(t) =w_1m(t) :=







w₁₁(t) ... w_1,N(m)−1(t)





 and w₂(t) =w_2m(t) :=







w₂₁(t) ... w_2,N(m)−1(t)





. Define

Mˆ :=

M 0 0 M

, Kˆ :=

0 −K K 0

, and wˆ :=

w₁ w₂

. Then (4.5) and (4.6) can be expressed in matrix form as

10

Mˆ dwˆ

dt =−K ˆˆw. (4.7)

This gives us a system of first-order linear ODEs with constant coefficients. To solve it, we need to

11

impose initial conditions. The initial conditionu(x,0) =g(x) for a≤x≤b can be approximated

12

by its linear interpolanteg(x) =PN(m)−1

j=1 g(x_m,j)φ_j(x). Letg₁(x) andg₂(x) be the real and complex

13

part of g(x). Therefore, we set w_1j(0) =g₁(x_m,j) and w_2j(0) = g₂(x_m,j) for 1 ≤ j ≤ N(m)−1.

1

This leads to the initial condition

2

ˆ

w(0) =wˆ_m(0) :=

g1(xm,1),· · · , g1(xm,N(m)−1), g2(xm,1),· · ·, g2(xm,N(m)−1)

. (4.8) Consequently, we get the linear system

3

Mˆ dwˆ

dt =−K ˆˆw, t >0, and w(0) =ˆ wˆm,0. (4.9) Since supp(µ) = [a, b], [29, Proposition 4.1] implies that Mis invertible. It follows that Mˆ also is

4

invertible. Thus the system in (4.9) has a unique solution. More precisely, the following proposition

5

holds.

6

Proposition 4.1. Assume thatsupp(µ) = [a, b]. Then the mass matrix Mand the stiffness matrix

7

Kare positive definite. Consequently, (4.9) has a unique solutionw(t). Moreover,ˆ w_1j(t)∈C(0, T)

8

and w_2j(t)∈C(0, T) for j= 1, . . . , N(m)−1.

9

Proof of Theorem 1.2. The assertions hold by combining the derivations above and Proposition 4.1.

10

11

We note that the matrix K can be computed directly. The authors have given the explicitly

12

formula for M in [29], as follows. Let c_{m,`,j</sub> and τ<sub>m,`,j} be the constants and similitudes from

13

condition (P3) in Section 1, respectively. Then for 1≤`≤N(m)−1, we have

14

M_`,`^(m) = (x<sub>m,`</sub>−xm,`−1)⁻²·

N(1)

X

j=0

cm,`−1,j

Z

I1,j

τm,`−1,j(x)−xm,`−12

dµ

+ (xm,`−xm,`+1)⁻²·

N(1)

X

j=0

cm,`,j

Z

I1,j

τm,`,j(x)−xm,`+1

2

dµ.

(4.10)

For 2≤`≤N(m)−1, we obtain

15

M_`,`−1^(m) =−(x<sub>m,`</sub>−xm,`−1)⁻²·

N(1)

X

j=0

cm,`−1,j

Z

I1,j

τm,`−1,j(x)−xm,`−1

τm,`−1,j(x)−x<sub>m,`</sub>

dµ, (4.11) and M_`−1,`^(m) =M_`,`−1^(m) .

16

Define

17

J_k,j :=

Z

I1,j

x^kdµ, k= 0,1,2, and j= 0, . . . , N(1). (4.12) Since eachτk,`,j is of the formτk,`,j(x) =rk,`,jx+bk,`,j, we can see that the matrixMis completely

18

determined by the integrals J_k,j, wherek= 0,1,2 andj= 0, . . . , N(1). Hereafter, we assume that

19

the constantJ_k,j can be evaluated explicitly fork= 0,1,2 andj = 0, . . . , N(1).

20

Next, we discuss the solution of the linear system (4.7). Let wˆ_n := w(tˆ _n), n≥ 0, and use the

21

central difference method to solve the equation (4.9). We approximate the derivative as

22

ˆ

w⁰(t_n)≈ wˆ_n+1−wˆ_n

∆t and w(tˆ _n)≈ wˆ_n+1+wˆ_n

2 . (4.13)

Substituting (4.13) into (4.7) yields

Mˆ wˆ_n+1−wˆ_n

∆t =−Kˆwˆ_n+1+wˆ_n

2 .

Since M and K are positive definite, so is 2Mˆ + (∆t)Kˆ for all ∆t > 0. Thus, wˆ_n+1 = (2Mˆ +

1

(∆t)K)ˆ ⁻¹(2Mˆ −(∆t)K)ˆ wˆ_n.Therefore, equation (4.7) becomes

2





 ˆ

wn+1 = (2Mˆ + (∆t)K)ˆ ⁻¹(2Mˆ −(∆t)K)ˆ wˆn, n= 0,1,2, . . . , ˆ

w0=w(tˆ 0) =w(0),ˆ

t_n=n∆t. .

(4.14)

To solve this system, fix ∆tand substitute the initial conditionwˆ0 from (4.8) into the first equation

3

in (4.14) to get wˆ₁. Then wˆ_n+1 can be computed recursively.

4

5. Fractal measures defined by iterated function systems

5

In this section, we solve the homogeneous Schr¨odinger equation (1.2) numerically for three d-

6

ifferent measures. These measures are defined by IFSs with overlaps and satisfy the conditions

7

(P1)-(P3) in Section 1 (see [29, Proposition 5.1 and Section 5.3]). In the first and second cases,

8

the measures satisfy a family of second-order self-similar identities. These identities were first in-

9

troduced by Strichartz et al. [28] to approximate the density of the infinite Bernoulli convolution

10

associated with the golden ratio.

11

Let {S_j}^q_j=1 be an IFS of contractive similitudes on R, and letµ be the associated self-similar measure. Assume that supp(µ) = [a, b]. Define an auxiliary IFS

T_j(x) =r_jx+d_j, j= 1,2, . . . , N,

wherenj ∈N,dj ∈R. We assume thatµ satisfies a family of second-order (self-similar) identities

12

with respect to{T_j}^N_j=1 (see [18]), that is, the following conditions hold:

13

(1) supp(µ)⊆SN

j=1Tj(supp(µ)), and

14

(2) for each Borel subsetA⊆supp(µ) and 0≤`, j≤N,µ(T`TjA) can be expressed as a linear

15

combination of {µ(T_kA) :k= 1, . . . , N}. In matrix form,

16

µ(T_{`</sub>T<sub>j</sub>A) =e<sub>`}M_j







µ(T₁A) ... µ(T_NA)





, `, j= 1, . . . , N, (5.15) where e<sub>`</sub> is the `th row of the N ×N identity matrix and M_j is some N ×N matrix

17

independent of A.

18

Furthermore, assume that{T_j}^N_j=1 satisfies (OSC). Define P_k:=

n

T_j([a, b]) :j ∈ {1, . . . , N}^ko

fork≥1.