Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schr\"odinger operators

We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schr\"odinger operators. Let $\Lambda_L = (-L/2,L/2)^d$ and $H_L = -\Delta_L + V_L$ be a Schr\"odinger operator on $L^2 (\Lambda_L)$ with a bounded potential $V_L : \Lambda_L \to \mathbb{R}^d$ and Dirichlet, Neumann, or periodic boundary conditions. Our main result is of the type \[ \int_{\Lambda_L} \lvert \phi \rvert^2 \leq C_{\mathrm{sfuc}} \int_{W_\delta (L)} \lvert \phi \rvert^2, \] where $\phi$ is an infinite complex linear combination of eigenfunctions of $H_L$ with exponentially decaying coefficients, $W_\delta (L)$ is some union of equidistributed $\delta$-balls in $\Lambda_L$ and $C_{\mathrm{sfuc}}>0$ an $L$-independent constant. The exponential decay condition on $\phi$ can alternatively be formulated as an exponential decay condition of the map $\lambda \mapsto \lVert \chi_{[\lambda , \infty)} (H_L) \phi \rVert^2$. The novelty is that at the same time we allow the function $\phi$ to be from an infinite dimensional spectral subspace and keep an explicit control over the constant $C_{\mathrm{sfuc}}$ in terms of the parameters. Moreover, we show that a similar result cannot hold under a polynomial decay condition.


Introduction
Starting with the pioneering work [Car39], there has been plenty of research concerning unique continuation properties for elliptic operators L with non-analytic coefficients. That is, if the solution u of Lu = 0 in Ω ⊂ R d vanishes in a non-empty open set ω ⊂ Ω, then u will be identically zero, see e.g. [Hör89] and the references therein. More than this, there are several quantitative formulations of unique continuation which proved to be useful in a variety of applications, see e.g. [BK05,RL12,BK13,RMV13,NTTV16]. For instance, Bourgain and Kenig [BK05] showed that if ∆u = V u in R d , u(0) = 1 and u, V ∈ L ∞ (R d ) then for all x ∈ R d with |x| > 1 we have max |y−x|≤1 |u(y)| > c · exp −c ′ (log|x|)|x| 4/3 . (1) This quantitative formulation has been crucial for the proof of Anderson localization for the continuum Anderson model with Bernoulli-distributed coupling constants. An L 2 -variant of Ineq. (1) has been shown in [BK13] in order to study the density of states of Schrödinger operators. A similar quantitative formulation is an estimate of the type where u is in the range of some spectral projector of a Schrödinger operator with potential V , and C is some positive constant depending on the geometry of ω and the potential V . Such quantitative unique continuation principles have been applied to control theory for the heat equation and spectral theory of random Schrödinger operators, see e.g. the recent [TTV16] and the references therein. Let us emphasize that the dependence of C on the geometry of ω turned out to be important for some of these applications. To be more specific, let Ω ⊂ R d be a finite, open and non-empty connected set, W ∈ L ∞ (Ω) and H Ω = −∆ + W on L 2 (Ω) with Dirichlet boundary conditions Then Ineq.
(2) has been obtained in [RL12] in the case W ≡ 0, ω ⊂ Ω open and non-empty, u a (finite or infinite) linear combinations of eigenfunctions of H Ω . However, the dependence of C on the geometry of ω is not known.
(2) is proven for Ω = (−L/2, L/2) d , ω an equidistributed arrangement of δ-balls, u ∈ W 2,2 (Ω) satisfying |∆u| ≤ |W u|, and with where N > 0 depends only on the dimension. For the application to random Schrödinger operators it is crucial that the result is scale-free, i.e. C is independent of L. In [RMV13] the question was raised whether a similar estimate holds for finite linear combinations of eigenfunctions u ∈ Ran χ (−∞,b] (H Ω ). A partial answer to this question was given in [Kle13]. The full answer has been announced in [NTTV15], and full proofs have been given in [NTTV16]. There, the constant is derived. Let us emphasize that this was the missing step to study localization for random Schrödinger operators with non-linear dependence on the random parameters. The aim of this note is to extend the main result of [NTTV16] to the the natural setting of infinite dimensional spectral subspaces. For this purpose, we first extend the strategy of [NTTV16] to prove Ineq. (2) for infinite linear combinations of eigenfunctions with exponentially decaying coefficients, cf. Theorem 2.3. In a second step we show that Ineq. (2) holds if χ [λ,∞) (H Ω )φ 2 decays exponentially in λ, cf. Theorem 2.2. In order not to lose the explicit control over the constant C sfuc , in particular its L-independence, this step requires a detailed analysis using precise knowledge of the ∆-eigenvalues and eigenfunctions, cf. Lemma 3.5. While the proofs are given in Section 3, we will show in Section 4 that our results are optimal in the sense that they cannot hold under polynomial decay conditions.

Notation and main results
Let d ∈ N. For L, r > 0 we denote by Λ L = (−L/2, L/2) d ⊂ R d the d-dimensional cube with side length L and by B(x, r) the ball with center x and radius r with respect to the Euclidean norm. The Laplace operator on L 2 (Λ L ) with Dirichlet, Neumann or periodic boundary conditions is denoted by ∆ L . For Ω ⊂ R d open and ψ ∈ L 2 (Ω) we denote by ψ = ψ Ω = ψ L 2 (Ω) the usual L 2 -norm of ψ. If Γ ⊂ Ω we use the notation χ Γ ψ Ω = ψ Γ = ψ L 2 (Γ) . Moreover, for a measurable and bounded V : R d → R we denote by V L : Λ L → R its restriction to Λ L given by V L (x) = V (x) for x ∈ Λ L , and by the corresponding Schrödinger operator. We will also write V = V + − V − for the decomposition into positive and negative part and V ∞ for the L ∞ -norm of V . The operator H L is lower semibounded and self-adjoint with lower bound − V − ∞ and purely discrete spectrum.
Definition 2.1. Let G > 0 and δ > 0. We say that a sequence Corresponding to a (G, δ)-equidistributed sequence Z we define for L ∈ GN the set where we suppressed the dependence of W δ (L) on G and on the choice of Z.
we have .
For every measurable and bounded V : R d → R and every L ∈ N we denote the eigenvalues of the corresponding operator H L by E k , k ∈ N, enumerated in increasing order and counting multiplicities, and fix a corresponding sequence ψ k , k ∈ N, of normalized eigenfunctions. Note that we suppress the dependence of E k and ψ k on V and L. For φ ∈ L 2 (Λ L ) we set we have A special case of Theorem 2.2 and 2.
Let us assume G = 1 for convenience. In this case Inequality (5) holds with κ = 18e . Hence, the constants C B,A sfuc in Theorem 2.3 and 2.2 can be estimated as withÑ B andÑ A depending only on the dimension. This way we recover the original result of [NTTV16].
Remark 2.4 (Relation between κ and G). In Theorem 2.2 and 2.3, the parameters κ (decay of high energies) and G (grid size) are subject to the relation respectively. This is in accordance with the intuition of uncertainty principles: delocalization in momentum space (large κ) corresponds to localization in position space, i.e. a fine grid (small G) is required in order to obtain an estimate as in Ineq. (4). It also seems that the condition on G and κ appears naturally when using Carleman estimates to prove a scalefree quantitative unique continuation result as in Theorem 2.2 and 2.3. Indeed, a similar assumption is required in analogue results for solutions of variable coefficient second order elliptic operators with Lipschitz continuous coefficients, see [BTV15]. There, on a technical level the Lipschitz constant assumes the role of 1/κ from our setting and our condition turns into a smallness condition on the Lipschitz constant in the main result of [BTV15].
However, one could ask if a quantitative unique continuation principle as in Theorem 2.2 and Theorem 2.3 holds for every pair (κ, G). An indication for this is Proposition 5.6 in [RL12] where the following statement is proven in the special case V ≡ 0: Let ω ⊂ Λ L be open and κ > 0. Then for all functions u = k∈N α k φ k with |α k | ≤ exp(−κ √ E k ), k ∈ N, we have u ≡ 0 if u| ω ≡ 0. Even though it would be possible without much effort to turn this qualitative into a quantitative statement of the form the method in [RL12] does not provide any control over the constant C in terms of δ, L, and κ, which we study in this note. One possibility to treat arbitrary κ and G might be a so-called chaining argument, as used in [DF88,Kuk98,Bak13] in the context of quantitative uniqueness results and nodal sets for solutions of the Schrödinger equation. However, in order to obtain a strong dependence of C sfuc on the parameters δ and V ∞ as in Theorem 2.2 and 2.3, a direct adaptation of these chaining arguments to our setting might not be feasible.
Remark 2.5 (Optimality). As observed by Jerison and Lebeau in [JL99, Proposition 14.9], the square root in the exponent of Ineq.
(3) and (5) is optimal. The exponent 2/3 of V ∞ in C sfuc , is known from Meshkov's example [Mes92] to be optimal in the case of eigenfunctions  (3) and (5) are replaced by polynomials. We will show in Section 4 that this is not the case. For this purpose, we show that every φ ∈ C ∞ 0 (Λ L ) satisfies such a polynomial condition. Hence polynomial summability of the |α k | 2 does not imply such a quantitative unique continuation principle.

Ghost dimension and interpolation inequalities
In this subsection we restate two interpolation inequalities from [NTTV16], on which the proof of Theorem 2.3 relies. For more details we refer to [NTTV16].
Given a measurable and bounded V : R d → R and L ∈ N we define extensions of V L and of the eigenfunctions ψ k (defined on Λ L ) to a larger cube Λ RL where R is the least odd integer larger than 18e √ d + 2. The type of the extension will depend on the boundary conditions, see [NTTV16]. In the case of • periodic boundary conditions we extend both V and ψ k periodically.
• Dirichlet boundary conditions we extend V iteratively by symmetric reflections with respect to the boundary of Λ L , and ψ k by antisymmetric reflections.
• Neumann boundary conditions we extend both V and ψ iteratively by symmetric reflections with respect to the boundary of Λ L .
We will use the same symbol for the extended V L and ψ k . Note that V L : , the extended ψ k are elements of W 2,2 (Λ RL ) with corresponding boundary conditions, they satisfy the eigenvalue equation ∆ψ k = (V L − E k )ψ k on Λ RL and their orthogonality relations remain valid. For a measurable and bounded V : R d → R, L ∈ N and φ ∈ L 2 (Λ L ) recall that α k = ψ k , φ whence φ = k∈N α k ψ k . We set ω k := |E k | and define for n ∈ N the function F n : Λ RL × R → C by where s k : R → R is given by Note that we suppress the dependence of F n on V , L and φ. The function F n fulfills the handy relations and ∂ d+1 F n (·, 0) = n k=1 α k ψ k =: φ n on Λ RL .
Proposition 3.3. For all T > 0, all measurable and bounded V : R d → R, all L ∈ N odd , all n ∈ N and all φ ∈ L 2 (Λ L ) we have Remark 3.4. The counterparts of Propositions 3.1, 3.2 and 3.3 in [NTTV16] are formulated with φ ∈ Ran χ (−∞,b] (H L ) instead of φ ∈ L 2 (Λ L ), and instead of F n . However, the proofs in [NTTV16] do not depend on the particular choice of the index set {k ∈ N : E k ≤ b} and apply to arbitrary finite index sets as well.

Proof of Theorem 2.3
First we consider the case G = 1 and L ∈ N odd . We note that Proposition 3.3 remains true if we replace R by 1, i.e. for all T > 0, n ∈ N and L ∈ N odd we have We haveX R 3 ⊂ Λ RL × [−R 3 , R 3 ]. By Ineq. (7) and Proposition 3.3 we have Since n → n k=1 θ k |α k | 2 n k=1 |α k | 2 is monotonously increasing, this implies for all n ∈ N : We use Propositions 3.1 and 3.2 and obtain Since U 3 (L) ⊂X R 3 we have By Ineq. (7), the square of the left hand side is bounded from below by Putting everything together we obtain by using and calculate where K i , i ∈ {1, 2, . . . , 4}, are constants depending only on the dimension. Hence, with some constantÑ =Ñ (d). Letting n tend to infinity and using φ n − φ L 2 (Λ L ) → 0 for n → ∞, we conclude the statement of the theorem in the case L ∈ N odd . Let now G > 0 be arbitrary and L/G ∈ N odd . We define the map g : Λ L/G → Λ L , g(y) = G · y. Then, on Λ L/G we have Hence, the functions ψ k • g are an orthonormal basis of eigenfunctions of the operator H L = −∆ L/G + G 2 V L • g with eigenvaluesẼ k = G 2 E k . We apply our theorem with G = 1 to the functionφ = φ • g and obtain The general case L/G ∈ N follows by a similar scaling argument and the explicit dependence ofC B sfuc on the parameters, see [NTTV16] for details.

Proof of Theorem 2.2
Recall that for a measurable and bounded V : R d → R we denote by V L : Λ L → R its restriction to Λ L , by ∆ L the Laplace operator on L 2 (Λ L ) subject to either Dirichlet, Neumann or periodic boundary conditions, and by the corresponding Schrödinger operator. Moreover, we denote the eigenvalues of H L by E k , k ∈ N, enumerated in increasing order and counting multiplicities, and fix a corresponding sequence ψ k , k ∈ N, of normalized eigenfunctions.
For the proof of Lemma 3.5 and 4.1 we shall need explicit formulas for the eigenvalues and eigenfunctions of the negative Laplacian −∆ L on L 2 (Λ L ). Depending on the boundary conditions we choose the index set I = N in the case of Dirichlet boundary conditions, I = N 0 in the case of Neumann boundary conditions, and I = 2Z in the case of periodic boundary conditions. Then, the eigenvalues of −∆ L are given by with corresponding normalized eigenfunctions The normalization constants e y −1 can be easily calculated, though we will not need them. Moreover, there exists a bijection p : N → I d such that is the k-th eigenvalue of −∆ L enumerated in increasing order counting multiplicities. This bijection is unique up to permutations of sites y ∈ I d with the same Euclidean norm.
Hence there is a constantÑ depending only on the dimension such that This shows the statement of the theorem in the case G = 1. The general case follows by scaling, analogously to the end of the proof of Theorem 2.3.

Discussion on optimality
In Remark 2.5 we discussed whether the class of functions φ satisfying k∈N max{0, E k } κ |α k | 2 ≤ D B k∈N |α k | 2 (12) for some D B , κ > 0 can still exhibit a unique continuation principle as in Theorem 2.3. The following lemma leads to a counterexample in the case V ≡ 0.
Proof of Lemma 4.1. Since the eigenfunctions and eigenvalues of −∆ on Λ L are explicitly known, cf. Section 3.3, we can replace the sum on the left hand side by where N ∈ 2N is the least even integer larger than κ. For the eigenfunctions, see Eq. (9), we have ∂ N i e y = −(π/L) N |y i | N e y for i ∈ {1, · · · , d}. We calculate using integration by parts