1 Introduction↩︎

This article is concerned with uncertainty principles at low energy for Schrödinger operators. A typical manifestation of such an uncertainty principle might be \[\label{eq:intro1} P_E V P_E \geq \kappa P_E ,\tag{1}\] where, in a Hilbert space \(\mathcal{H}\), \(P_E=\boldsymbol{1}_{[0,E]}(H)\) is the spectral projector associated to some selfadjoint operator \(H\ge 0\) onto energies below \(E \geq \min \sigma (H)\), \(V\) is a non-negative selfadjoint operator, and \(\kappa = \kappa (E)\) is some positive constant. The most prominent example is the case where \(H=-\Delta\) is the negative Laplacian in \(L^2 (\mathbb{R}^d)\) and \(V = \mathbf{1}_D\) with a so-called thick set \(D \subset \mathbb{R}^d\). In that case, the Logvinenko-Sereda theorem implies 1 for all \(E > 0\) with an explicit constant \(\kappa\) depending on \(E\) and the geometric properties of the set \(D\); see [1]–[3]. It is called an uncertainty principle since it says that \[\| f\mathbf{1}_D\|^2\ge \| f\|^2 for all f\in\mathrm{ran}(P_E) ,\] so that states of low energy are spread out in configuration space in the sense that they carry a reasonable amount of mass on the set \(D\). The proof of the Logvinenko-Sereda theorem strongly relies on harmonic analysis. The presence of a potential and of more general domains \(\Omega \subset \mathbb{R}^d\), i.e., considering \(H = -\Delta + W\) in \(L^2 (\Omega)\), places the problem beyond the reach of purely harmonic analysis methods. This setting has been investigated, e.g., in [4]–[8] under different assumptions on the set \(D\) and the potential \(W\).

From the viewpoint of applications, uncertainty relations of the type 1 have become an essential tool in spectral theory of random Schrödinger operators, control theory, or harmonic analysis, to mention a few. In the theory of random Schrödinger operators such uncertainty relations are typically used to prove Wegner estimates and in turn Anderson localization; see, for example, [4]–[6], [9], [10]. In control theory, uncertainty relations serve as an input to the Lebeau-Robbiano strategy [11], originally formulated for the Laplace-Beltrami operator on compact Riemannian manifolds \(M\). The general philosophy here is that an uncertainty relation implies an observability estimate, and by Douglas’ lemma [12], this is equivalent to null-controllability of a certain linear control problem. Subsequently, this strategy has been studied in various mathematical situations, including abstract Hilbert or Banach spaces; see, e.g., [13]–[20]. While the original Lebeau-Robbiano strategy requires 1 to hold for all \(E > 0\) with a constant \(\kappa = \kappa (E)\) not vanishing too fast as \(E\) tends to infinity, there exist also weak versions where the validity of 1 for some (eventually small) fixed \(E\) is sufficient; cf. the recent papers [21], [22].

Let us now turn to our contribution. Our key result is Theorem 5, which gives a computable lower bound for abstract Schrödinger operators. This theorem plays a crucial role in our proof of the equivalence of the uncertainty principle known from [9], [23], [24] and the strict positivity of Schrödinger operators. Moreover, this theorem allows us to pass from potentials of the form \(V=\mathbf{1}_D\) with relatively dense sets \(D\) to non-negative potentials \(V\) subject to what we call \((R,\varrho)\)-thickness.

It is evident that the technique employed here could equally well be applied in the discrete case of graph Laplacians; we refer to [25] for earlier results on graphs and to the upcoming [26], which features a metric measure space set-up that includes graphs and continuum configuration spaces.

In contrast to the equivalent quantitative uncertainty principles mentioned above, a qualitative uncertainty principle for \(H\) and \(V\) can be thought of as a unique-continuation statement: for suitable \(E\), it requires that if \(f\in\mathrm{ran}(P_E)\) and \(Vf=0\), then necessarily \(f=0\). It is a rather simple observation that such a unique-continuation is a consequence of the quantitative uncertainty principles mentioned earlier. For \(H=-\Delta\), the fact that functions in the range of the spectral projector are analytic shows that in this case the qualitative uncertainty principle holds for arbitrary \(E\) and every \(V\ge 0\) that is not \(0\) almost everywhere, whereas a quantitative uncertainty principle requires \(V\) to be \((R,\varrho)\)-thick for some \(R,\varrho>0\).

In the final section we sketch an application of our method to control theory: it provides \(\alpha\)-controllability with arbitrary \(\alpha>0\) for a new and fairly general class of control problems.

2 Main results↩︎

We start in a general setting: \((\mathcal{H}, (\cdot|\cdot))\) denotes a Hilbert space, \(H, V\) are selfadjoint operators in \(\mathcal{H}\), \(V\) is supposed to be bounded, and we require that \(H,V \ge0\). As usual, if \(A\) is selfadjoint and \(\varepsilon \in \mathbb{R}\), then \(A \ge \varepsilon\) means that \((Af|f) \ge \varepsilon (f|f)\) for all \(f \in {\rm dom}(A)\), which in turn is equivalent to the requirement that the spectrum \(\sigma(A)\) of \(A\) is contained in \([\varepsilon,\infty)\). We call \(A\) strictly positive and write \(A \gg 0\) if \(A \ge \varepsilon\) for some \(\varepsilon > 0\). Most of what follows is only interesting in the case where \(0 \in \sigma(H)\), that is, where \(H\) is not strictly positive.

For \(E > 0\), we denote by \(P_E := \boldsymbol{1}_{[0,E]}(H)\) the spectral projector of \(H\) associated with \([0,E]\). Following [9], we say that a quantitative uncertainty principle for spectral projectors holds for the pair \(H,V\) at \(E\) if there is a \(\kappa=\kappa(H,V,E) >0\) such that \[\label{sUP} P_E V P_E \ge \kappa P_E,\tag{2}\] where, of course, \(P_E V P_E \ge \kappa P_E\) means that \(P_E V P_E - \kappa P_E \ge 0\).

We say that a quantitative uncertainty principle at low energy holds for the pair \(H,V\) at \(E\) if there exists a \(\mu=\mu(H,V,E)>0\) such that \[\label{EUP}UP} f \in {\rm dom}(H) \;\: \land \;\: (Hf|f)\le E \lVert f \rVert^2 \;\: \Longrightarrow \,\: (Vf|f) \ge \mu \lVert f\rVert^2.\tag{3}\] Clearly, (3 ) implies (2 ) with \(\kappa=\mu\): since \((HP_Ef|P_Ef) \le E \lVert P_Ef\rVert^2\), we infer from (3 ) that \((P_EVP_Ef|P_Ef) =(VP_Ef|P_Ef) \ge \mu \lVert P_Ef\rVert^2\).

Here is our abstract main characterization of quantitative uncertainty principles.

We now turn to the special case where \(H=-\Delta: H^2(\mathbb{R}^d) \to L^2(\mathbb{R}^d)\) is the negative Laplacian and \(V\) is the operator of multiplication by a non-negative function in \(L^\infty(\mathbb{R}^d)\). We denote this function by \(V\), too. Given two numbers \(R,\varrho >0\), the function \(V\) is said to be \(\boldsymbol{(R,\varrho)}\)-thick if \[\frac{1}{R^d}\int_{C_R(y)} V(x) \mathrm{d}x \ge \varrho\] for all \(y \in \mathbb{R^d}\), where \(C_R(y)=y+(0,R)^d\), so that the left hand side is the mean of \(V\) over \(C_R(y)\). Using appropriate trial functions one can show that \[\label{Dia} \inf \sigma(-\Delta+V)\le\liminf_{R \to \infty} \left(\inf_{y \in \mathbb{R}^d}\frac{1}{R^d}\int_{C_R(y)} V(x) \mathrm{d}x\right),\tag{4}\] an estimate very reminiscent of results from [27]. We will give a proof of this inequality in Section 4. Herewith our second main result.

The only non-trivial implication of this equivalence is (ii)\(\Longrightarrow\)(i); the attentive reader will note that for \(V=\mathbf{1}_D\) this could be readily deduced from the above mentioned Logvinenko-Sereda Theorem combined with Theorem 1. However, we prefer to give a computable lower bound on \(\inf \sigma(-\Delta+V)\) in Theorem 6 and Corollary 7. These results are again based on Theorem 5. We want to stress the fact that the previous result gives uncertainty principles for large classes of operators, namely for \(H\ge \eta(-\Delta)\), \(\eta>0\) and \((R,\varrho)\)-thick \(V\). In Subsection 5.2 we discuss a concrete class of examples that can be treated in this way.

We say that a qualitative uncertainty principle holds for the pair \(H,V\) at \(E\) if \[\label{QUP} f \in \mathrm{ran}(P_E) \;\: \land \;\: Vf=0 \;\: \Longrightarrow \,\: f=0.\tag{5}\] It is easily seen that (\(\sigma\)UP) and thus each of the three equivalent conditions in Theorem 1 imply (5 ). Indeed, if \(f \in P_E(\mathcal{H})\) and \(Vf=0\), then 2 gives \[0=(Vf|f)=(VP_Ef|P_Ef)=(P_EVP_Ef|P_Ef)\ge \kappa (P_Ef|P_Ef)=\kappa (f|f),\] implying that \(f=0\). The converse implication is totally wrong:

The following two sections are devoted to the proofs of the two theorems stated above. In Section 5 we deal with Observation 2.3, show that in the Schrödinger operator set-up, we get quantitative uncertainty principles for \(-\Delta\) replaced by rather general classes of operators and give some concrete examples, both of strictly positive operators \(-\Delta+V\) and of such operators that are not strictly positive. The last sections contains a short discussion of applications of the above results to control theory.

3 The abstract setting↩︎

In this section we prove Theorem 1.

The equivalence (i) \(\Longleftrightarrow\) (iii) is readily seen. Indeed, if (i) holds with \(H+V\ge E_0\), then (iii) holds with \(E\in (0,E_0)\) and \(\mu=E_0-E\). Conversely, suppose (iii) is true. Then \(H+V \ge \min(E,\mu)\), which is (i).

The obvious implication (iii) \(\Longrightarrow\) (ii) was already mentioned in the previous section. We are therefore left with the implication (ii) \(\Longrightarrow\) (i), whose proof will occupy the rest of this section. We start with a simple result for \(2 \times 2\) matrices.

Recall that a closed subspace \(\mathcal{K}\) of \(\mathcal{H}\) is called reducing for \(H\) if \(\mathcal{K}\subset\mathrm{dom}(H)\) and \(P_{\mathcal{K}} H \subset H P_{\mathcal{K}}\), where \(P_{\mathcal{K}}\) denotes the orthogonal projection onto \(\mathcal{K}\). This is equivalent to the split \(\mathrm{dom}(H) = (\mathcal{K}\cap \mathrm{dom}(H)) + (\mathcal{K}^{\perp} \cap \mathrm{dom}(H))\) together with the requirement that both \(\mathcal{K}\) and \(\mathcal{K}^\perp\) are invariant under \(H\), i.e., for \(\eta \in \mathcal{K}\cap \mathrm{dom}(H)\) we have \(H \eta \in \mathcal{K}\) and similarly on \(\mathcal{K}^\perp \cap \mathrm{dom}(H)\). In particular, this implies that \(P_{\mathcal{K}} \eta \in \mathrm{dom}(H)\) if \(\eta \in \mathrm{dom}(H)\).

The implication (ii) \(\Longrightarrow\) (i) of Theorem 1 follows from Theorem 5 applied to \(\mathcal{K}=P_E(\mathcal{H})\). In that case \(\mathcal{K}\) is a closed reducing subspace for \(H\) and we have \((H\xi|\xi) > E\lVert \xi\rVert^2\) for \(\xi\) in \(\mathrm{dom}(H) \cap \mathcal{K}^\perp\). The assumption \(P_EVP_E \ge \kappa P_E\) gives \((V\zeta|\zeta) \ge \kappa \lVert\zeta\rVert^2\) for \(\zeta=P_E g\) in \(\mathcal{K}\). From Theorem 5 we therefore obtain that \[H+V \ge \frac{\kappa E}{E+2\lVert V\rVert}.\]

4 The case \(\boldsymbol{H=-\Delta}\)↩︎

This section is devoted to the proof of Theorem 2 and inequality 4 . We start with the latter.

It remains to prove the implication (ii) \(\Longrightarrow\) (i). Our proof is based on Theorem 5, and it also delivers an explicitly computable lower bound; we state the corresponding estimates in the following theorem and its corollary.

Corollary 7 gives the implication (ii)\(\Rightarrow\)(i) of Theorem 2 and the proof of Theorem 2 is complete.0◻

5 Miscellanea↩︎

5.1 (5 ) does not imply \(\boldsymbol{H + V \gg 0}\)↩︎

Here we derive Observation 3, which in particular implies that the title of this subsection is correct. Actually, this is an understatement: while (5 ) holds for any nontrivial \(V\), we already know that \({H + V \gg 0}\) requires \(V\) to be \((R,\rho)\)-thick. For the Fourier transform of \(f \in L^2(\mathbb{R}^d)\), \[\hat{f}(\xi)=(2\pi)^{-d/2}\int f(x)e^{-i\xi x}\mathrm{d}x,\] it follows that \[(-\Delta f)\hat{~}(\xi)=\lvert \xi \rvert^2\hat{f}(\xi)\] and hence \(P_E\) is the orthogonal projection onto the subspace of all functions \(f\in L^2(\mathbb{R}^d)\) whose Fourier transform \(\hat{f}(\xi)\) is supported in the ball \(\lvert \xi \rvert^2 \le E\).

5.2 Uncertainty principles for operators with irregular coefficients↩︎

As mentioned earlier, the equivalence in Theorem 2 in combination with Theorem 1 gives that (3 ) holds for all selfadjoint operators satisfying \(H\ge \eta(-\Delta)\) with some \(\eta>0\) and all \((R,\rho)\)-thick potentials. Note that for the comparison of the operators, it is no problem to consider non-densely defined forms, so \(H\) could be a selfadjoint operator in some closed subspace of \(L^2(\mathbb{R}^d)\).

Here is a concrete class of examples that arise in this way. Assume that \(\Omega\subset \mathbb{R}^d\) is open, \(a(x)\ge \eta\) is a \(d\times d\) matrix for \(x\in \Omega\) so that \(\Omega\ni x\mapsto a_{i,j}(x)\) is locally integrable for all \(i,j= 1, \ldots, d\). Moreover, let \(W:\Omega\to[0,\infty]\) be locally integrable.

We would like to compare our uncertainty estimates to those obtained in [10]. The method in this latter paper is restricted to the case \(V=\mathbf{1}_D\), where \(D\) satisfies a strong relative density condition, strictly more incisive than what we need here. On the other hand, in the more specialized situation, Theorem 1.1 of the latter paper gives a slightly better energy range and 2 than our result above. At the same time, our proof here is much simpler.

5.3 Examples of thick and non-thick potentials↩︎

We consider \(A=-\Delta+V\) on \(L^2(\mathbb{R}^d)\) with \(V \ge 0\) in \(L^\infty(\mathbb{R}^d) \setminus\{0\}\). In the following we permanently employ Theorem 3.

Let first \(d=1\). To ensure that \(A\gg 0\), we must find \(R>0\) and \(\varrho>0\) such that \((1/R)\int_y^{y+R}V(x) \mathrm{d}x \ge \varrho\) for all \(y \in \mathbb{R}\). This is impossible if the essential supremum of \(V\) over \((y,\infty)\) goes to zero as \(y\) goes to infinity, which in particular happens if \(V\) is piecewise continuous and monotonically decreases to zero. Thus, in this case \(A\) is not strictly positive. Situations in which \(V\) does not decrease to zero are more interesting. The function \(V(x)=\lvert \sin(x) \rvert\) is \((\pi,2/\pi)\)-thick and hence induces a strictly positive operator \(A\). Clearly, \(A \gg 0\) even for every periodic potential \(V\). Choosing natural numbers \(k_1 < k_2 < k_3 < \ldots\) and letting \(V(x)=1\) on \([k_j,k_j+1] \cup [-k_j,-k_j-1]\) and \(V(x)=0\) otherwise, we obtain that \(T \gg 0\) if the distances \(k_{j+1}-k_j\) remain bounded whereas \(A\) is not strictly positive if \(k_{j+1}-k_j \to \infty\) as \(j \to\infty\). To have a last example, let \(V(x)=1\) for \(x\) in \(\bigcup_{k\in \mathbb{Z}\setminus \{0\}}[k,k+1/\lvert k\rvert^\beta]\) with some \(\beta >0\) and \(V(x)=0\) elsewhere. Then \((1/n)\int_1^{n+1} V(x) \mathrm{d}x \to 0\) as \(n \to\infty\), implying that \(A\) is not strictly positive.

Let now \(d \ge 2\). Again \(A\) is not strictly positive if the essential supremum of \(V\) over \(\{y \in \mathbb{R}^d: \lvert y \rvert \ge R\}\) decays to zero as \(R \to\infty\). On the other hand, \(A\gg 0\) if \(V\) is periodic, for example, if \(V(x)=1\) on the black cubes and \(V(x)=0\) on the white cubes of an infinite \(d\)-dimensional chessboard. Finally, take spherical shells \(S_k=\{x\in \mathbb{R}^d: k \le \lvert x \rvert \le k+\delta_k\}\) (\(k=1,2, \ldots\)) with \(\delta_k\le 1\). For \(\delta_k=1\), the volume \(\lvert S_k \rvert\) of \(S_k\) is \(B_d[(k+1)^d-k^d] > B_ddk^{d-1}\), where \(B_d\) is the volume of the unit ball in \(\mathbb{R}^d\). Thus, for each \(\beta >0\), we can choose \(\delta_k \in (0,1)\) so that the shells do not intersect and \(\lvert S_k \rvert=B_ddk^{d-1}/k^\beta\). A simple computation reveals that \((1/n^d)\sum_{k=1}^{n}\lvert S_k\rvert\to 0\) as \(n \to \infty\). Consequently, if \(V=1\) on the union of the shells \(S_k\) and \(V=0\) otherwise, then \(A\) is not strictly positive. However, if we take \(\delta_k=\mu\in(0,1)\) for all \(k\), then \(\lvert S_k\rvert =B_d [(k+\mu)^d-k^\mu] >B_dd\mu k^{d-1}\) and hence \((1/n^d)\sum_{k=1}^{n}\lvert S_k\rvert\) remains bounded away from zero as \(n\to \infty\). Thus, in this case the characteristic function of the union of the shells generates a strictly positive operator \(A\).

6 Application to control theory↩︎

Let us discuss an application of our results to control theory. We assume that \(H\) is a selfadjoint operator in \(\mathcal{H}\subset L^2(\mathbb{R}^d)\) as in Subsection 5.2 above, i.e. \(H\ge \eta(-\Delta)\) for some \(\eta>0\). For \(T > 0\) we consider the linear control problem \[\label{eq:control1} \dot{f}(t) +Hf(t) = V u(t), \quad t \in (0,T], \quad f(0) = f_0 \in \mathcal{H},\tag{8}\] where \(V\) is the operator of multiplication by a non-negative function \(V \in L^\infty (\mathbb{R}^d)\), and \(u \in L^r ((0,T) ; \mathcal{H})\) with \(r \in [1,\infty]\) is the so-called control function. The solution of 8 is given by Duhamel’s formula \[f (t) = \mathrm{e}^{-tH} f_0 + \int_0^t \mathrm{e}^{-(t-s)H} V u(s) \mathrm{d}s, \quad t\in [0,T] .\] Given four numbers \(\alpha,K\geq 0\), \(T > 0\) and \(r \in [1,\infty]\), the linear control problem 8 is called cost-uniformly \(\boldsymbol{\alpha}\)-controllable in time \(T\) with respect to \(L^r((0,T);\mathcal{H})\) and cost \(K\) if for every \(f_0\in \mathcal{H}\) there is a control function \(u \in L^r((0,T);\mathcal{H})\) such that \[\lVert f(T) \rVert \leq \alpha \lVert f_0 \rVert \quad \text{and}\quad \lVert u \rVert_{L^r((0,T);\mathcal{H})} \le K \lVert f_0 \rVert .\]

Based on earlier works [30]–[32], it has recently been observed in [21] that an uncertainty relation at low energy of the form 2 is sufficient to guarantee \(\alpha\)-controllability for all \(\alpha > 0\) and sufficiently large \(T > 0\). This is in contrast to earlier results, where even for \(\alpha\)-controllability an uncertainty relation at all energies was required. Indeed, as a consequence of our Corollary 8 in combination with [21] we obtain a so-called weak observability estimate. Note that the first assumption of [21] is satisfied by Corollary 8, while its second assumption follows easily from the spectral theorem. This weak observability estimate is in turn equivalent to \(\alpha\)-controllability, see [30] for the case of Hilbert spaces (i.e.\(r=2\)), and [21] for the general framework of Banach spaces. Thus, we obtain the following corollary.

Note that in the present general context, we do not obtain \(\alpha = 0\), a property called null-controllability. This would require an uncertainty relation at all energies. We want to stress the fact that we can treat a general class of operators \(H\), merely satisfying \(H \geq \eta(-\Delta)\) for some \(\eta > 0\); see Section 5.2 for examples. As our examples cover operators with irregular coefficients, it seems (currently) beyond reach to show an uncertainty relation at all energies by standard methods to the best of our knowledge. Note, moreover, that in our previous result, the control operator \(V\) is merely supposed to be a \((R,\varrho)\)-thick potential, generalizing the case of indicator functions of thick sets treated previously in the literature.

References↩︎