The spectral theory of Schrödinger operators with random or almost periodic potentials has been an area of very active study since the late 1970’s. From the beginning, it has been understood and emphasized that these two classes of models share an important property, namely that the potentials can be generated dynamically. On the one hand, this makes a unified proof of basic spectral results possible, such as the almost sure constancy of the spectrum and the spectral type, since they hold as soon as the dynamical framework is fixed and an ergodic measure is chosen. On the other hand, by the very nature of the dynamical definition of the potentials, it comes as no surprise that tools from dynamics will enter the spectral analysis of these operators.

This section is organized in the following way:

- A first part devoted to the study of the first step of the iteration described in the previous paragraph. Under some conditions on v and ω we construct the matrices U 1 , F 2 and D 1 which satisfy the estimates of Lemma 2.

- In the second part and after a suitable choice of parameters, an inductive proposition, Proposition 3 is introduced in order to prove the first result in Theorem 1, which is a simple consequence of Lemma 4.

- At the end we give the proof of Theorem 1(2).

Consider now the symmetric infinite-dimensional matrix that depends on the parameter θ , D ( θ ) + F ( θ ) with,

D ( θ ) = ( ⋱ 0 v ( θ − ω ) v ( θ ) v ( θ + ω ) 0 ⋱ ) .

For the formulation of the first step of iteration we shall assume the following:

- The rotation number ω and the potential v satisfy (2) and (3).

- { | F i j | ≤ ε e − | i − j | ρ     with     ρ > 0 | ∂ θ F i j | ≤ ε . .

- Consider the equation:

e − X ( D + F ) e X = D ′ + F ′ (7)

where the matrices X, D ′ and F ′ are defined in the following way:

Let N : = 1 ε a ρ for 0 < a < 1 4 τ .

1) The matrix X is defined by ( X i j = 0       if     i = j     or     | i − j | > N X i j = − F i j v i − v j       otherwise and satisfies the equation:

[ D , X ] = F N − D ′ + D (8)

where ( F N ) i j = ( F i j       if     | i − j | ≤ N 0       otherwise   .

2) ( D ′ − D ) i i = F i i .

3) F ′ ( θ ) = e − X ( θ ) ( D ( θ ) + F ( θ ) ) e X ( θ ) − D ′ ( θ ) .

Lemma 2. Let 0 < a τ + b < 1 4 and σ = ε b .

If ε ≤ ( κ 2 6 α ρ τ + 1 ) 1 1 − ( a τ + b ) then:

1)

a) | X i j | ≤ ε N τ α κ e − | i − j | ρ : = A e − | i − j | ρ .

b) ‖ X ‖ ≤ ε N τ α κ 2 1 − e − ρ .

c) | ( e ± X − I ) i j | ≤ 2 7 A σ ρ e − | i − j | ρ ′ where ρ ′ = ρ − σ 2 ρ .

2)

a) | F ′ i j | ≤ 16 ( 2 5 A σ ρ ) 2 e − | i − j | ρ ′ .

b) ‖ F ′ ‖ ≤ 16 ( 2 5 A σ ρ ) 2 2 1 − e − ρ ′ : = ε ′ 2 1 − e − ρ ′ .

3)

a) | ∂ θ v ′ ( θ ) | ≥ α − ε : = α ′ where D ′ ( θ ) = diag ( v ′ ( θ + n ω ) ) .

b) | ∂ θ F ′ i j ( θ ) | ≤ ε ′ .

c) | ∂ θ v ′ ( θ ) | < c + ε : = c ′ .

4) | ( D ′ − D ) i i | ≤ ε .

Proof.

1(a): Let i ≠ j and | i − j | ≤ N , we have: X i j = − F i j v i − v j therefore | X i j | ≤ ε | v i − v j | e − | i − j | ρ since | v i − v j | = | v ( θ + i ω ) − v ( θ + j ω ) | ≥ inf | ∂ θ v | | ( i − j ) ω | ≥ α κ | i − j | τ , then it follows that | X i j | ≤ ε N τ α κ e − | i − j | ρ .

1(b): Using 1(a) we obtain:

∑ i ∈ ℤ | X i j | ≤ A ∑ i ∈ ℤ     e − | i − j | ρ ≤ A ∑ i ∈ ℤ     e − | i | ρ ≤ 2 A 1 − e − ρ (9)

Thus from the generalized Young inequality [12] (page 9), (9) implies an estimate of X in the operator norm on l 2 ( ℤ ) → l 2 ( ℤ ) .

1(c): By lemma A 8 (Eliasson [13] ) and for all n ∈ ℕ we deduce that: | ( X ⋯ X ︸ n   times ) i j | ≤ ( 2 5 A σ ρ ) n e − | i − j | ρ ′ , hence | ( e ± X − I ) i j | ≤ 2 7 A σ ρ e − | i − j | ρ ′ .

2(a): Let F ˜ N : = F − F N then we have:

F ′ = e − X ( D + F ) e X − D ′ = e − X ( D + F N ) e X + e − X F ˜ N e X − D ′ = [ X , F N ] − X D X − X F N X + ∑ m + n ≥ 2 ( m , n ) ≠ ( 1,1 ) ( − X ) n n ! ( D + F N ) X m m !       + ∑ m + n ≥ 0 ( − X ) n n ! F ˜ N X m m ! . .

Now we have to estimate the elements of all matrices which constitute the matrix F ′ .

(*)₁

| ( [ X , F N ] ) i j | ≤ 2 ∑ k ∈ I | X i k | | ( F N ) k j | ≤ 2 ε A ∑ k ∈ I     e − ( | k − i | + | k − j | ) ρ ≤ 2 ε A ∑ k ∈ I     e − ( | k − i | + | k − j | ) ρ ′ e − ( | k − i | + | k − j | ) σ ρ 2 ≤ ε 2 5 A σ ρ e − | i − j | ρ ′

where I = { k ∈ ℤ ; | k − i | ≤ N     and     | k − j | ≤ N } .

In the same way we get:

(*)₂

| ( X D X ) i j | ≤ ∑ k ∈ I | X i k | | ( D X ) k j | ≤ ( 2 5 A σ ρ ) 2 e − | i − j | ρ ′ .

(*)₃

| ( X F X ) i j | ≤ ( 2 5 A σ ρ ) 2 e − | i − j | ρ ′ .

(*)₄

| ( ∑ m + n ≥ 2 ( m , n ) ≠ ( 1,1 ) ( − X ) n n ! ( D + F N ) X m m ! ) i j | ≤ 3 ( 2 5 A σ ρ ) 2 ( 1 + ε ) e − | i − j | ρ ′ .

(*)₅

| ( ∑ m + n ≥ 0 ( − X ) n n ! F ˜ N X m m ! ) i j | ≤ ∑ m + n ≥ 0 1 n ! m ! ∑ k ∈ ℤ | ( X n ) i k | | ( F ˜ N X m ) k j | ≤ ∑ m + n ≥ 0 1 n ! m ! ∑ k ∈ ℤ | ( X n ) i k | ∑ | k − l | > N | ( F ˜ N ) k l | | ( X m ) l j | ≤ ε e − 1 ε a ∑ m + n ≥ 0 1 n ! m ! ∑ k ∈ ℤ | ( X n ) i k | ∑ | k − l | > N | ( X m ) l j | ≤ 3 ε e − 1 ε a ( σ ρ ) 2 e − | i − j | ρ ′ .

This gives

| F ′ i j | ≤ 16 ( 2 5 A σ ρ ) 2 e − | i − j | ρ ′ .

2(b):

∑ i ∈ ℤ | F ′ i j | ≤ 16 ( 2 5 A σ ρ ) 2 ∑ i ∈ ℤ     e − | i − j | ρ ′ ≤ 16 ( 2 5 A σ ρ ) 2 ∑ i ∈ ℤ     e − | i | ρ ′ .

therefore

∑ i ∈ ℤ | F ′ i j | ≤ 16 ( 2 5 A σ ρ ) 2 2 1 − e − ρ ′ (10)

It follows from the generalized Young inequality [12] (page 9) that (10) implies the desired estimate of F ′ in the operator norm on l 2 ( ℤ ) → l 2 ( ℤ ) .

3(a): Since ( D ′ − D ) i i = F i i then v ′ i = v i + F i i therefore ∂ θ v ′ i = ∂ θ v i + ∂ θ F i i thus

| ∂ θ v ′ i | ≥ | ∂ θ v i | − | ∂ θ F i i | ≥ α − ε : = α ′ .

3(b): In order to estimate | ∂ θ F ′ i j | , we have to find an upper bound of | ∂ θ X i j | . We have ∂ θ X i j = − ∂ θ F i j v i − v j + F i j ∂ θ v i − ∂ θ v j ( v i − v j ) 2 then | ∂ θ X i j | ≤ ε N τ α κ + ε ( N τ α κ ) 2 | ∂ θ v i − ∂ θ v j | ≤ ε C 1 . , which implies:

(*)₁

| ∂ θ ( [ X , F N ] ) i j | = | ∂ θ ( ∑ k ∈ ℤ     X i k ( F N ) k j − ( F N ) i k X k j ) | ≤ ∑ k ∈ ℤ | ∂ θ X i k | | ( F N ) k j | + | X i k | | ∂ θ ( F N ) k j | + | ∂ θ ( F N ) i k | | X k j |     + | ( F N ) i k | | ∂ θ X k j | ≤ ∑ k ∈ ℤ ε C 1 | ( F N ) k j | + ε | X i k | + ε | X k j | + ε C 1 | ( F N ) i k | ≤ ε 3 / 2 M 1 .

In the same way we get:

(*)₂

| ∂ θ ( X D X ) i j | ≤ ε 3 / 2 M 2 .

(*)₃

| ∂ θ ( X F X ) i j | ≤ ε 3 / 2 M 3 .

(*)₄

( X n ) i j = ∑ l n − 1 ∈ ℤ     ∑ l n − 2 ∈ ℤ     ⋯ ∑ l 1 ∈ ℤ     X i l 1 X l 1 l 2 ⋯ X l n − 1 j .

Since | ( X k ) i j | is bounded for all k < n then | ∂ θ ( X n ) i j | ≤ ε 3 / 2 C 2 therefore | ∂ θ ( ∑ m + n ≥ 2 ( m , n ) ≠ ( 1,1 ) ( − X ) n n ! ( D + F N ) X m m ! ) i j | ≤ ε 3 / 2 M 4 and | ∂ θ ( ∑ m + n ≥ 0 ( − X ) n n ! F ˜ N X m m ! ) i j | ≤ ε 3 / 2 M 5 where all constants M i and C i depend on N , α , κ and τ . It follows that | ∂ θ ( F ′ i j ) | ≤ 4 ε 3 / 2 max ( M 1 , ⋯ , M 5 ) ≤ ε ′ .

3(c):

| ∂ θ v ′ | ≤ | ∂ θ v | + ε ≤ c + ε .

4) By construction of D ′ the result follows immediately.

Let a, b such that 0 < a τ + b < 1 4 and consider ε 1 = ε , ρ 1 = ρ , α 1 = α , A 1 = A , D 1 = D , F 1 = F and for all n ≥ 1 we define the sequences

A n + 1 = ( N n + 1 ) τ κ α n + 1 ε n + 1 N n = 1 ( ε n ) a ρ n ε n + 1 = 16 ( 2 5 A n σ n ρ n ) 2 σ n = ( ε n ) b ρ n + 1 = ρ n − σ n ρ n 2 α n + 1 = α n − ε n

These parameters are defined in an iterative way and it is with which we will be able to define the matrices X n ,   F n + 1 and D n + 1 satisfying

e − X n ( D n + F n ) e X n = D n + 1 + F n + 1 (11)

where the matrices X n , D n + 1 and F n + 1 are defined in the following way:

1) The matrix X n is defined by ( ( X n ) i j = 0       if     i = j     or     | i − j | > N n ( X n ) i j = − ( F n ) i j v i n − v j n       otherwise   and satisfies the equation

[ D n , X n ] = ( F n ) N n − D n + 1 + D n (12)

where ( ( F n ) N n ) i j = ( ( F n ) i j       if     | i − j | ≤ N n 0       otherwise .

2) ( D n + 1 − D n ) i i = ( F n ) i i .

3) F n + 1 ( θ ) = e − X n ( θ ) ( D n ( θ ) + F n ( θ ) ) e X n ( θ ) − D n + 1 ( θ ) . and satisfying the property P n described in the following proposition.

Proposition 3. Let n ∈ ℕ . If ∀ m ≤ n , ε m ≤ ( κ α m ρ m 1 + τ 2 6 ) 1 1 − ( a τ + b ) then the following property P n is holds.

P n ( 1.   | ( X n ) i j | ≤ A n e − | i − j | ρ n 2.   ‖ X n ‖ ≤ 2 A n 1 − e − ρ n 3.   | ( e ± X n − I ) i j | ≤ 2 7 A n σ n ρ n e − | i − j | ρ n + 1 4.   | ( F n + 1 ) i j | ≤ ε n + 1 e − | i − j | ρ n + 1 5.   ‖ F n + 1 ‖ ≤ 2 ε n + 1 1 − e − ρ n + 1 6.   | ∂ θ v n + 1 | ≥ α n + 1 7.   | ∂ θ ( F n + 1 ) i j | ≤ ε n + 1 8.   | ∂ θ v n + 1 | < c n + 1 = c n + ε n 9.   | ( D n + 1 − D n ) i i | ≤ ε n .

Proof. A direct application of Lemma 2 allows us to obtain the desired result for each n.

Now we will deal with the study of the convergence of our iteration. We will therefore look at the conditions and the size of ε with which we will have the convergence, this will be the goal of the next lemma. Finally, we conclude with the proof of Theorem 1 which is a simple deduction of Proposition 3 and Lemma 4.

Lemma 4. Suppose that

( 1 2 9 α 3 ) 1 − ( a τ + b ) < α κ ρ τ + 1 < 2 τ + 7 3 ( 2 6 α 2 ) b ≥ 2 ;   0 < a τ + b < 1 4 .

Then for ε < 1 2 37 ( κ ρ τ + 1 2 τ 3 ) 4 α we have for all n:

1) ε n ≤ ( 1 2 6 α 2 ) ( 3 / 2 ) n ∀ α ≥ 11 2 4 .

In particular lim n → ∞ ε n = 0 .

2) ε n ≤ ( κ α n ρ n τ + 1 2 6 ) 1 1 − ( a τ + b ) .

Proof: 1) The result is holds for n = 1 . Suppose that the result remain holds for 1,2, ⋯ , n thus α 1 > α 2 > ⋯ > 2 3 α . Now we shall prove that the result is also true for n + 1 .

Let M = ∑ j = 1 + ∞ ( 1 2 ( 1 − ( a τ + b ) ) ) j , we have ε n + 1 = 16 ( 2 5 A n σ n ρ n ) 2 = 2 14 ( 1 κ α n ρ n τ + 1 ) 2 ε n 2 ( 1 − ( a τ + b ) ) = ⋯ = [ 2 14 ∑ k = 1 n ( 2 ( 1 − ( a τ + b ) ) ) − k ∏ k = 1 n ( 1 κ α k ρ k τ + 1 ) 2 ( 2 ( 1 − ( a τ + b ) ) ) − k ε ] ( 2 ( 1 − ( a τ + b ) ) ) n .

Since ∏ k = 1 n ( 1 κ α k ρ k τ + 1 ) 2 ( 2 ( 1 − ( a τ + b ) ) ) − k ≤ ( 1 κ α n ρ n τ + 1 ) 2 ∑ k = 1 n ( 2 ( 1 − ( a τ + b ) ) ) − k , hence ε n + 1 ≤ [ 2 14 M ( 1 κ α n ρ n τ + 1 ) 2 M ε ] ( 2 ( 1 − ( a τ + b ) ) ) n ≤ [ 2 28 ( 2 τ 3 κ α ρ τ + 1 ) 4 ε ] ( 3 / 2 ) n , which proves that lim n → + ∞ ε n = 0 .

2) By 1) we have ∀ n ,   ε n ≤ 1 then ε n + 1 = 2 2 ( 2 6 ε n 1 − ( a τ + b ) κ α n ρ n τ + 1 ) 2 ≤ 1 , thus 2 6 ε n 1 − ( a τ + b ) κ α n ρ n τ + 1 ≤ 1 hence ε n ≤ ( κ α n ρ n τ + 1 2 6 ) 1 1 − ( a τ + b ) .

Remark. 1) One can assume without loss of generality that α = 1 and we have the same result, in fact the operators H θ and α H θ have the same spectral properties.

2) The real b exists and satisfying all conditions.

Proof of Theorem 1. 1) The operator H θ is identified to matrix D + F with | F i j | ≤ ( e ε ) e − | i − j | ρ . Then for ρ = 1 and ε < κ α ρ τ + 1 3 e 2 τ + 37 we have the existence of matrices X n and D n + 1 for all n ∈ ℕ such that for all θ , ( e X 1 ( θ ) ⋯ e X n ( θ ) ) * ( D ( θ ) + F ( θ ) ) e X 1 ( θ ) ⋯ e X n ( θ ) = D n + 1 ( θ ) + F n + 1 ( θ ) , where D n + 1 ( θ ) is a diagonal matrix, ‖ F n + 1 ‖ ≤ ε n + 1 1 1 − e − ρ n + 1 , | ( e ± X n − I ) i j | ≤ 2 7 A n σ n ρ n e − | i − j | ρ n + 1 and | ( D n + 1 − D n ) i i | ≤ ε n .

Therefore F n ( θ ) → 0 and D n ( θ ) → D ∞ ( θ ) with D ∞ ( θ ) is a diagonal matrix. All convergence are fulfilled for all θ .

On the other hand e X 1 ( θ ) ⋯ e X n ( θ ) → U ( θ ) in norm and for all θ with U ( θ ) is an orthogonal matrix. In fact: Let U j ( θ ) = e X j ( θ ) we have ∏ j ≥ 1     U j ( θ ) converges if and only if ∑ j ≥ 1 ‖ U j ( θ ) − I ‖ converges, now since ‖ U j ( θ ) − I ‖ ≤ 2 ε j 1 − e − ρ j + 1 then we have the existence of U for all θ . Moreover from Lemma 4 and for ε 0 = 1 e 2 37 ( κ 2 τ 3 ) 4 α the matrix D ∞ ( θ ) is pure point with finite-dimensional eigenvectors for all θ and the measure of

σ ( D ) \ σ ( D + F ) goes to 0 as ε → 0 . The eigenvectors of D + F are formed by the columns of U.

2) Let ψ θ ( t , n ) = e − i t H θ ψ for ψ ∈ l 2 ( ℤ ) , so by the exponential decaying of eigenfunctions we can easily deduce that

sup t ∑ n ∈ ℤ ( 1 + n 2 ) | ψ θ ( t , n ) | 2 < ∞ .

The result in Theorem 1 improves the previous one by Eliasson, such that dynamical localization is proven with an appropriate potential. To my knowledge, there are no results on the spectral properties of Schrödinger operators with discontinuous potential. The ideas presented in this paper can be used to obtain new results for several models with discontinuities. For example, one can consider the quasi-periodic operator defined for θ in the torus T with a piecewise smooth potential or even functions in a piecewise Gevrey class, and see if similar localization results can be obtained.

The authors declare no conflicts of interest.

Refai, W. (2023) Dynamical Localization of the Quasi-Periodic Schrödinger Operators. Open Access Library Journal, 10: e10966. https://doi.org/10.4236/oalib.1110966