Improved energy estimates for a class of time-dependent perturbed Hamiltonians

We consider time-dependent perturbations which are relatively bounded with respect to the square root of an unperturbed Hamiltonian operator, and whose commutator with the latter is controlled by the full perturbed Hamiltonian. The perturbation is modulated by two auxiliary parameters, one regulates its intensity as a prefactor and the other one controls its time-scale via a regular function, whose derivative is compactly supported in a finite interval. We introduce a natural generalization of energy conservation in the case of time-dependent Hamiltonians: the boundedness of the two-parameter unitary propagator for the physical evolution with respect to the n/2-th power energy norm for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathbb {Z}$$\end{document}n∈Z. We provide bounds of the n/2-th power energy norms, uniformly in time and in the time-scale parameter, for the unitary propagators, generated by the time-dependent perturbed Hamiltonian and by the unperturbed Hamiltonian in the interaction picture. The physically interesting model of Landau-type Hamiltonians with an additional weak and time-slowly-varying electric potential of unit drop is included in this framework.


Introduction
We consider the physical evolution of a quantum system in a separable Hilbert space H generated by the time-dependent Hamiltonian operator H (ε, η, t) := H 0 + εg(ηt)H 1 for all t ∈ R, (1.1) where H 0 is the unperturbed Hamiltonian, H 1 is the perturbation switched on by a function g with supp g ⊂ (0, 1) and g(s) = 0 for s < 0, and ε ∈ (0, ε * ], η > 0 are parameters 1 regulating respectively the intensity and the time-scale of the perturbation. The variable t here stands for time and the positive parameter η is a convenient tool to control the rate at which the system changes. The function g regulates the switch-on time of the external Hamiltonian ε H 1 (notice that the perturbation is completely off for t ≤ 0). When the Hamiltonian H (ε, η, t) is t-independent, 2 namely H (ε, η, t) = H (ε), it is well known that, by an elementary consequence of Stone's theorem, one has that [U ε (t), H (ε)] = 0, where U ε (t) denotes the unitary propagator for the self-adjoint operator H (ε). In other words there is conservation of the energy and consequently one obtains that H −n/2 (ε)U ε (t)H n/2 (ε) has a bounded extension for every n ∈ Z. On the other hand, if there is a non-trivial t-dependence and the perturbation commutes with the unperturbed Hamiltonian, i. e. [H 1 , H 0 ] = 0, to establish that for all n ∈ Z the product H −n/2 (ε, η, t)U ε,η (t, r )H n/2 (ε, η, r ) extends to a bounded operator, one can use the representation formula for the unitary propagator U ε,η (t, r ) = e −i t r ds H(ε,η,s) (see [17,Proposition 2.5]) and rely on similar techniques developed in Proposition 2.9. In this paper, we deal with the more general case in which the commutator [H 1 , H 0 ] = 0 and "is controlled" by the full perturbed Hamiltonian H (ε, η, t), uniformly in (ε, η, t) (see Assumption (B(k))), beyond Assumption (A 2 ) on the perturbation H 1 to be selfadjoint and relatively bounded with respect to H 1/2 0 (see the hypotheses in the statement of Theorem 2.5).
Denoting by U ε,η (t, r ) the unitary propagator generated by H (ε, η, t), we will prove that for every n ∈ N one has that U ε,η (t, r ) is in L (n) ε,η (r , t) with the corresponding operator norm U ε,η (t, r ) L (n) ε,η (r ,t) uniformly bounded in the parameters (η, (t, r )) ∈ (0, ∞)×R 2 , which is equivalent to establish the following estimate 3 : For every n ∈ Z, for all ε ∈ (0, ε * ] and η > 0 we have that The precise assumptions and result are stated in Theorem 2.5. To the best knowledge of the author, in the standard results of well-posedness of non-autonomous linear evolution equations not even the statement U (t, r ) ∈ L (2) ε,η (r , t) is shown, the only exception being [8,Theorem 5.1]. Moreover, we are interested in working in the so-called interaction or intermediate picture 4 : First one computes the unitary propagator , generated by εg(ηt)H 1 (e. g. using again [17,Proposition 2.5]) and then one considers the time-dependent unitarily transformed 5 Hamiltonian Similarly to the previous case, we will prove the following inequality: For every n ∈ Z, for all ε ∈ (0, ε * ] and η > 0 we have that sup s,u∈R 3 We will prove this equivalent statement. 4 Usually, the interaction picture is performed using the unitary propagator induced by the time-independent part of the time-dependent perturbed Hamiltonian (e. g. see [18, § X.12]). More generally, one can introduce the interaction picture via the two-parameter family of unitary operators generated by time-dependent part (see [15,§VIII.14]), fixing an initial time. In our framework, we choose the second kind of interaction picture with initial time t 0 = 0 . 5 In Sect. 5, where we deal with the physically interesting model of Landau-type Hamiltonians, this unitary transformation is the gauge transformation G(t, 0) = e −i ε η φ(ηt) 1 , where H 1 := 1 models an electric potential of negative unit drop for an electric field pointing in the negative 1-st direction (see Definition 5.1).
whereÛ ε,η (s, u) is the unitary propagator generated byĤ (ε, η, s) and D n is a finite constant independent of (ε, η). This result, formulated in Corollary 2.7, is obtained as a consequence of estimate (1.2), thanks to the following identitŷ and Proposition 2.9, which guarantees that for every integer number n, H n/2 Energy estimates in the form of (1.4) (or equivalently (1.2)) are relevant when one needs to keep track of localization in energy under the physical evolution, uniformly in the time-scale of the perturbation. More precisely, suppose that a family of operators O(s) with s ∈ R decays in energy with power m/2 with m ∈ N, in the sense that there exists a finite constant C O such that for every ε ∈ (0, ε * ], η > 0, s ∈ R and for all ψ ∈ D Ĥ m/2 (ε, η, s) . Then, by applying inequality (1.4) this energy localization is conserved by the evolved family of operatorsÛ ε,η (u, s)O(s)Û ε,η (s, u): for any s, u ∈ R and for every ψ ∈ D Ĥ m/2 (ε, η, u) . This work has been motivated in the first instance by the need to fill a gap in the proof of [2, Lemma 5.1], where Landau-type Hamiltonian operators with an additional weak and time-slowly-varying electric potential of unit drop are considered (see Sect. 5 for this application). While Theorem 2.5 implies [2, Lemma 5.1], Corollary 2.7 is relevant since it is explicitly used in the proof of [2, Theorem 2.2] (see [2,Remark (3), p. 599] for the case n = 0). The strategy proof of Theorem 2.5 is based on the one given in the aforementioned paper, with two essential differences: firstly we use H (ε, η, t) whose time derivative is compactly supported (while ∂ ∂sĤ (ε, η, s) is not compactly supported) and secondly in the proof of Theorem 2.5 we establish the induction step by computing the time derivative of the bounded operator H −1/2 (ε, η, t) (compare (3.8)) instead of the unbounded oneĤ 1/2 (ε, η, s). As it is briefly explained in Sect. 5, these kinds of energy estimates are used to prove the validity of the Kubo formula for the transverse conductance in the quantum Hall effect in a two-dimensional sample (e. g. see [1, 4-6, 11, 13, 14, 21]). But we are convinced that our results are of general conceptual interest, since we provide bounds on the growth of the n/2-th power energy norms for time-dependent Hamiltonian in a model-independent setting, and could be relevant for proving the linear response in quantum Hall systems for unbounded Hamiltonians (cf. Sect. 5). More specifically, we require mild properties: Beyond the technical hypotheses, i. e. Assumptions 2.1 and (C 2 (k)) for k = 2, which guarantee the self-adjointness of H (ε, η, t) andĤ (ε, η, s) on the same t-independent domain D(H 0 ) and spectrum condition (2.2), the operator H 1 associated with the perturbation must not be bounded but only H 1/2 0 -bounded (compare Assumption (A 2 )), and the two parameters ε, η, related to the perturbation, are independent. Furthermore, both estimates (1.2) and (1.4) are uniform in the time-scale parameter η > 0, while for fixed η > 0 these bounds are clearly expected, due to the hypothesis supp g ⊂ (0, 1), with η-dependent constants. Finally, the use of the symbols ε and η is not related to a smallness assumption, as far as this paper is concerned (however our results apply to the particular case considered in [2], where the limit ε = η = 1 τ → 0 + is considered).

Mathematical setting and main results
In this section we set up the mathematical framework and state our main results, under different assumptions. Let H denote a separable Hilbert space. Firstly, we write hypotheses on each summand of the perturbed Hamiltonian H (ε, η, t). 1] |g(s)| and M := max Here ε ∈ (0, ε * ], where ε * is chosen so that condition (2.3) is fulfilled, and η > 0. Furthermore, the Hamiltonian operator H (ε, η, t) satisfies the following properties: and is H 1/2 0 -bounded, namely there exists a finite constant a > 0 such that As it is explained respectively in Remark 2.4.(i) and Remark 2.4.(ii), the above assump- Secondly, we write hypotheses on "how the perturbed Hamiltonian H (ε, η, t) behaves with respect to the unperturbed one H 0 ".

Remark 2.4
Here we explain some useful consequences of the hypotheses above.
(i) Under Assumptions (A 1 ) and (A 2 ), we have that H 1 is H 0 -bounded, with relative boundã < 1. Indeed, notice that for every C > 0 where a is defined in Assumption (A 2 ). Hence, for every ψ ∈ D(H 0 ) we obtain that In view of hypothesis (A 1 ) and the previous remark, we get that and thus there exists ε * > 0 such that Before stating the main results, namely Theorem 2.5 and Corollary 2.7, it is convenient to recall the problem of well-posedness of non-autonomous linear evolution equations. As it is emphasized in [18, Notes of Section X.12], the Cauchy problem for linear evolution equations under general suitable conditions, was solved first by Kato [7] and then by Yosida [23] (for the comparison of these works see [19]). For more general results, considering that A(t) has domain which does depend on time, see e. g. [8,20,22] , η, t). This means that U ε,η (t, r ) is the two-parameter family of unitary operators, jointly strongly continuous in t ∈ R and r ∈ R, such that for every t, r , u ∈ R In order to keep the reader's attention on the main results, i. e. Theorem with α, β and γ n finite constants defined as and C −n (ε) := C n (ε) for all n ∈ N.

Remark 2.6
Both in the Gell-Mann and Low [3] and the Kubo [10] formula the standard choice for the switch-on procedure in time is to make use of the exponential function for the non-positive time-axis R − := (−∞, 0]. More specifically, in our setting of reference, we replace the function g with the exponential and restrict the whole real time-axis to the non-positive one, i. e. one considers the time-dependent Hamiltonian operator Clearly, the main difference between H exp (ε, η, t) and H (ε, η, t) ≡ H g (ε, η, t), defined in (1.1), is that the the switch-on process acts respectively on the infinite time-interval R − and on a finite time-interval (precisely, under Assumption 2.1: supp g ⊂ (0, 1)). Under the assumptions of Theorem 2.5 except for the substitution of g with the exponential and the restriction to R − (applying the these two replacements everywhere in the nested hypotheses), a type of inequality similar to (2.4) still holds true. Precisely, denoting by U exp,ε,η (t, r ) the unitary propagator generated by for all η > 0, where C n (ε) is defined iteratively as with α, β and γ n finite constants, and C −n (ε) := C n (ε) for all n ∈ N.
Here, for completeness we sketch a proof of the above statement. We follow the argument of the proof of Theorem 2.5 in Sect. 3 for −∞ < t ≤ r ≤ 0. By using Grönwall's inequality and that 0 −∞ dτ ηe ητ = 1, we conclude that Therefore, it emerges that the crucial properties of the switch-on procedure modeled by a generic function f : I → R, f ∈ C k (I ) with k ≥ 1 on a subset I ⊆ R to deduce a type of inequality in the form of (2.4), which is uniform in the time-scale parameter η, is to have that both f L ∞ (I ) and f L 1 (I ) are finite.

Lemma 2.8 Let
On the other hand, the next proposition turns out to be useful to deduce the energy estimates for the unperturbed Hamiltonian in the interaction pictureĤ (ε, η, s) from the ones for the perturbed Hamiltonian H (ε, η, t).

Proof of Corollary 2.7
Notice that identity (1.5) holds true since for every ϕ ∈ D(H 0 ) one has that due to strong differentiability of U ε,η (t, r ) on D(H 0 ), Assumption (C 2 (k)) for k = 2 and D(H 0 ) ⊂ D(H 1 ) by Assumption (A 2 ), and similarly one verifies the other properties in (2.9). Therefore, fixed any n ∈ N, in view of Assumption (C 2 (k)) for k = n, for every ψ ∈ D(H n/2 0 ) we have that Thus, we deduce that by using Theorem 2.5 and Proposition 2.9. Finally, the Riesz Lemma implies the thesis for all n = − |n| ∈ Z.

Proof of Lemma 2.8
In view of D(H 1/2 (ε, η, t)) = D(H 1/2 0 ) by Remark 2.4.(iii), equality (3.5) and the second resolvent identity, we have that (4.1) In the last expression, for the second summand we observe that where we have used the hypothesis . Using the last inequality in (4.1) the thesis is obtained.

Proof of Proposition 2.9
First of all, notice that for any k ∈ N if one supposes Assumption (C 1 (k)) then Remark 2.4.(iii) ensures that the products of operators H k/2 0 H −k/2 (ε, η, t) and H (ε, η, t) k/2 H −k/2 0 are well defined on H. We are going to prove inequality (2.10) for every n ∈ N 0 , proceeding by induction. The induction step will be proved by using the base cases for 0 ≤ n ≤ 3 and estimate (2.11) for n = 1. For n = 0 it is trivial. For n = 1, in view of equality (3.5) and the second resolvent identity we obtain that where we have used the hypothesis H 1 H −1/2 0 = a < ∞ and condition (2.2).
Analogously, by virtue of Lemma 2.8 and condition (2.2), one obtains (2.11) for n = 1. For n = 2 rewriting thus by applying Lemma 2.8 and condition (2.2), inequality (2.10) is obtained. For n = 3 notice that where on the right-hand side the first summand is bounded 11  and A(x 1 , x 2 ) := B/2(−x 2 , x 1 ) with B > 0, λ ∈ R and the potential V is such that V ∞ is finite. 13 The perturbed Hamiltonian is defined as 14 where 0 < ε 1, 1 is a l 1 -switch function in the 1-st direction and g fulfills the hypotheses in Assumption 2.1. The multiplication operator 1 models an electric potential of negative unit drop for an electric field pointing in the negative 1-st direction. One is interested in computing the Hall conductance G Hall , defined as a ratio between the (excess of) induced current intensity when the perturbation is fully switched on and the voltage difference. More precisely, one introduces the operator i[H 0 , 2 ] standing for the current intensity in the 2-nd direction and ρ ε (t) the density operator, representing the state of the system at time t evolving from the Fermi projection P 0 of the unperturbed Hamiltonian H 0 with associated Fermi energy in a spectral gap of H 0 . Thus, one is in shape to define the Hall conductance as for any t ≥ 1/ε (when the perturbation is fully switched on). In [2] first, by exploiting the invariance of the trace under unitary conjugation, one rewrites 15 where s := εt is the scaled time,Ĥ (s) := e iφ(s) 1 H 0 e −iφ(s) 1 ,ρ ε (s) := e iφ(s) 1 ρ ε (s/ε)e −iφ(s) 1 andP 0 (s) = e iφ(s) 1 P 0 e −iφ(s) 1 . Then, in order to derive an explicit formula for the Hall conductance G Hall , they use an asymptotic expansion in ε powers ofρ ε (s)  (3), p. 599] for the case n = 0). Now we are going to verify that the general assumptions of Sect. 2 are fulfilled by this specific model. Clearly, H (ε, t) satisfies Assumptions (A 1 ) and (A 2 ). Assumptions (B(k)), (C 1 (k)) and (C 2 (k)) hold true under certain regularity conditions on V . Fix any k ∈ Z, assume that the Sobolev norm 16 V |k|+1,∞ is finite then hypothesis (B(k)) holds true. Indeed, since [ 1 , H (ε, t)] = i 2 p A,1 1 + 1 p A,1 , applying [2, Proposition 3.1.(i)] we deduce that there exists a finite constant e k : H (ε, t)]H (k−2)/2 (ε, t) ≤ e k 1 |k|+2,∞ , for all ε ∈ (0, 1) and t ∈ R. Now let k ∈ N with k ≥ 2, assume that V 2(k−1),∞ is finite then it follows that for all ε ∈ (0, 1) and t ∈ R namely the hypothesis (C 1 (k)) is fulfilled. Indeed, observe that  17 As in the previous sections, up to a shift of a constant, we can assume that H 0 ≥ 1.
here the product k−1 m=1 is ordered in the sense that a factor with larger index m stands to the left of ones with smaller m and, hence [2, Proposition 3.