Reducibility of Schr\"odinger equation on the sphere

TL;DR

This paper proves a reducibility result for the linear Schr"odinger equation on the sphere with quasi-periodic in time perturbations, including unbounded cases, without using pseudo-differential calculus.

Contribution

It provides one of the first reducibility results for multi-dimensional Schr"odinger equations with unbounded perturbations, under specific conditions.

Findings

Reduces the Schr"odinger equation to a simpler form

Handles unbounded perturbations of order less than 1/2

Does not require pseudo-differential calculus

Abstract

In this article we prove a reducibility result for the linear Schr\"odinger equation on the sphere $\mathbb{S}^{n}$ with quasi-periodic in time perturbation. Our result includes the case of unbounded perturbation that we assume to be of order strictly less than 1/2 and satisfying some parity condition. As far as we know, this is one of the few reducibility results for an equation in more than one dimension with unbounded perturbations. We notice that our result does not requires the use of the pseudo-differential calculus.

Equations

$${\rm i}\partial_{t}u=\Delta u+\varepsilon\big{(}W(\omega t,x)(-{\rm i}\partial_{\phi})^{\alpha}+V(\omega t,x)\big{)}u\,,\qquad u=u(t,x)\,,\quad t\in\mathbb{R}\,,\quad x\in\mathbb{S}^{2}\,,$$

$$\mathbb{T}^{d}_{\sigma}:=\big{\{}(a+{\rm i}b)\in\mathbb{C}^{d}\,:\,a\in\mathbb{T}^{d}\,,|b|<\sigma\big{\}}\,,$$

$$\mathrm{meas}({\mathcal{O}}_0\setminus {\mathcal{O}}_{\epsilon })\le {\epsilon }^{\delta }$$

$$\Vert \Psi (\phi )-\mathrm{Id}{\Vert }_{\mathcal{L}(H^{s^{\prime }}({\mathbb{S}}^2;\mathbb{C}))}+\Vert \Psi (\phi {)}^{-1}-\mathrm{Id}{\Vert }_{\mathcal{L}(H^{s^{\prime }}({\mathbb{S}}^2;\mathbb{C}))}\le {\epsilon }^{1-\delta },$$

$$\mathrm{i}{\partial }_{t}v=\Delta v+\epsilon Z(v).$$

$$(1-\epsilon C)\Vert u_0{\Vert }_{H^{s^{\prime }}}\le \Vert u(t){\Vert }_{H^{s^{\prime }}}\le (1+\epsilon C)\Vert u_0{\Vert }_{H^{s^{\prime }}},\forall t\in \mathbb{R},$$

$$|{\lambda }_k-{\lambda }_{k^{\prime }}|\ge k+k^{\prime },\forall k,k^{\prime }\in \mathbb{N},k\ne k^{\prime }.$$

$$\Delta +\epsilon M,M:H^s({\mathbb{S}}^n;\mathbb{C})\to H^{s+1-2\alpha }({\mathbb{S}}^n;\mathbb{C}),$$

$${\Lambda }_{k,j}\sim {\lambda }_k+O(\epsilon k^{-(1-2\alpha )}),j=1,\dots ,d_k.$$

$$|\omega \cdot l+{\Lambda }_{k,j}-{\Lambda }_{k^{\prime },j^{\prime }}|\ge \frac{\gamma }{|l{|}^{\tau }},l\in {\mathbb{Z}}^d,k,k^{\prime }\in \mathbb{N}$$

$${\lambda }_k:=k(k+n-1),k\in \mathbb{N}$$

$$\dim E_k:=d_k\le k^{n-1}.$$

$${\Phi }_{[k]}(x):=\{{\Phi }_{k,m}(x),m=1,\dots ,d_k\}$$

$$u(x)=\sum _{k\in \mathbb{N}}\sum _{m=1}^{d_k}{z}_{k,m}{\Phi }_{k,m}=\sum _{k\in \mathbb{N}}{z}_{[k]}\cdot {\Phi }_{[k]}(x),z_{[k]}=(z_{k,1},\cdots ,z_{k,d_k})\in {\mathbb{C}}^{d_k},$$

$$({\Pi }_{E_k}u)(x)=z_{[k]}\cdot {\Phi }_{[k]}(x)\Rightarrow (-\Delta ){\Pi }_{E_k}u={\lambda }_k{\Pi }_{E_k}u,k\in \mathbb{N}.$$

$$h^{s}:=\big{\{}z=\{z_{[k]}\}_{k\in\mathbb{N}}\,,z_{[k]}\in\mathbb{C}^{d_{k}}\,:\,\|z\|^{2}_{s}:=\sum_{k\in\mathbb{N}}\langle k\rangle^{2s}\|z_{[k]}\|^{2}<+\infty\big{\}}\,,$$

$$H^s=H^s({\mathbb{S}}^n,\mathbb{C}):=\{u(x)=\sum _{k\in \mathbb{N}}{z}_{[k]}\cdot {\Phi }_{[k]}(x)\mid z\in h^s\}$$

$$c|k|\le \sqrt{{\lambda }_k}\le C|k|$$

$$u(\phi ,x)=\sum _{\ell \in {\mathbb{Z}}^d,k\in \mathbb{N}}{z}_{[k]}(l)\cdot {\Phi }_{[k]}(x)e^{\mathrm{i}l\cdot \phi },z_{[k]}(l)\in {\mathbb{C}}^{d_k}$$

$${\mathbb{T}}_{\sigma }^d\ni \phi \mapsto H^{s_1}(\mathbb{S};\mathbb{C})\mathrm{analytic}\mathrm{for}|\mathrm{Im}(\phi )|<\sigma ,p_1-\mathrm{Sobolev}\mathrm{for}|\mathrm{Im}(\phi )|=\sigma .$$

$${\mathcal{A}}_{s,\sigma }:=\bigcap _{p_1+s_1=s}{H}^{p_1}({\mathbb{T}}_{\sigma }^d;H^{s_1}({\mathbb{S}}^n;\mathbb{C}))$$

$$\ell_{s,\sigma}:=\big{\{}z=\{z_{[k]}(l)\}_{l\in\mathbb{Z}^{d},k\in\mathbb{N}}\,,\,z_{[k]}\in\mathbb{C}^{d_{k}}:\,\|z\|_{s,\sigma}^{2}:=\sum_{l\in\mathbb{Z}^{d},k\in\mathbb{N}}\langle l,k\rangle^{2s}e^{2|l|\sigma}\|z_{[k]}(l)\|^{2}<+\infty\big{\}}$$

$$\Vert f{\Vert }_E^{sup}:=\Vert f{\Vert }_E^{sup,\mathcal{O}}:=\underset{\omega \in \mathcal{O}}{\sup }\Vert f(\omega ){\Vert }_E,\Vert f{\Vert }_E^{lip}:=\Vert f{\Vert }_E^{lip,\mathcal{O}}:=\underset{\begin{array}{c}{\omega }_1,{\omega }_2\in \mathcal{O}\\ {\omega }_1\ne {\omega }_2\end{array}}{\sup }\frac{\Vert f({\omega }_1)-f({\omega }_2){\Vert }_E}{|{\omega }_1-{\omega }_2|}.$$

$$\Vert f{\Vert }_E^{\gamma ,\mathcal{O}}:=\Vert f{\Vert }_E^{sup,\mathcal{O}}+\gamma \Vert f{\Vert }_E^{lip,\mathcal{O}}.$$

$$\Vert f{\Vert }_{{\ell }_{s,\sigma }}^{\gamma ,\mathcal{O}}=:\Vert f{\Vert }_{s,\sigma }^{\gamma ,\mathcal{O}}=\Vert f{\Vert }_{s,\sigma }^{sup,\mathcal{O}}+\gamma \Vert f{\Vert }_{s,\sigma }^{lip,\mathcal{O}}.$$

$$\ell_{s,\sigma}^{\gamma,\mathcal{O}}:=\big{\{}\mathcal{O}\ni\omega\mapsto z(\omega)\in\ell_{s,\sigma}\,:\,\|z\|_{s,\sigma}^{\gamma,\mathcal{O}}<+\infty\big{\}}\,.$$

$$\Vert (\mathrm{Id}-{\Pi }_N)z{\Vert }_{s,{\sigma }^{\prime }}\le \frac{C(s)e^{-(\sigma -{\sigma }^{\prime })N}}{(\sigma -{\sigma }^{\prime }{)}^d}\Vert z{\Vert }_{s,\sigma }.$$

$$L^2({\mathbb{S}}^n;\mathbb{C})=\underset{k\in \mathbb{N}}{⨁}{E}_k,$$

$$A_{[k]}^{[k^{\prime}]}:=\Big{(}A_{k,j}^{k^{\prime},j^{\prime}}\Big{)}_{\begin{subarray}{c}j=1,\ldots,d_{k},\\ j^{\prime}=1,\ldots,d_{k^{\prime}}\end{subarray}}\,.$$

$$(Au)(x)=\sum _{k\in \mathbb{N}}(Az{)}_{[k]}\cdot {\Phi }_{[k]}(x),z_{[k]}\in {\mathbb{C}}^{d_k},(Az{)}_{[k]}=\sum _{j\in \mathbb{N}}{A}_{[k]}^{[j]}{z}_{[j]}.$$

Full text

During the preparation of this work the two authors benefited from the support of the Centre Henri Lebesgue ANR-11-LABX- 0020-01 and of ANR -15-CE40-0001-02 “BEKAM” of the Agence Nationale de la Recherche. R. F. was also supported by ERC starting grant FAFArE of the European Commission and B.G. by ANR-16-CE40-0013 “ISDEEC” of the Agence Nationale de la Recherche.

1. 1 Introduction
2. 2 Functional setting
3. 3 An abstract reducibility result and its application to ()
4. 4 The regularization step
5. 5 The iterative reducibility scheme
6. A Technical Lemmata

1. Introduction

In this article we are interested in the problem of reducibility for the linear Schrödinger equation on the sphere with quasi-periodic in time perturbation. In the introduction, to make our statement more readable, we state our results in the physical space $\mathbb{S}^{2}\subset\mathbb{R}^{3}$ and with an explicit linear perturbation. A more general statement, including the higher dimension case $\mathbb{S}^{n}$ for $n\geq 3$ (the case $n=1$ is much more simpler and already known, see [28]), is detailed in section 3.2. So we consider the following linear Schrödinger equation on the $\mathbb{S}^{2}$

[TABLE]

where $\Delta$ denotes the Laplace-Beltrami operator on $\mathbb{S}^{2}$, ${\rm i}\partial_{\phi}={\rm i}(x_{1}\partial_{x_{2}}-x_{2}\partial_{x_{1}})$ is the $x_{3}$ component of the orbital angular momentum (and the generator of rotations about the $x_{3}$ axis) and $0\leq\alpha<1/2$. The operator $(-{\rm i}\partial_{\phi})^{\alpha}$ is precisely defined in (3.14). The parameter $\varepsilon>0$ is small, the frequency vector $\omega$ belongs to $\mathcal{O}_{0}:=[1/2,3/2]^{d}\subset\mathbb{R}^{d}$, $d\geq 1$. The functions $W,\,V$ in (1) are real valued multiplicative potentials depending quasi periodically on time, i.e. $V$ is a function in $C^{0}(\mathbb{T}^{d}\times\mathbb{S}^{2};\mathbb{R})$, $\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}$. We assume that $W,\,V$ are real analytic functions with respect to the angle variable $\varphi\in\mathbb{T}^{d}$ with values in the Sobolev space $H^{s}(\mathbb{S}^{2};\mathbb{R})$ with $s>d/2+1$. In particular the map $\mathbb{T}^{d}\ni\varphi\mapsto V(\varphi,\cdot)\in H^{s}(\mathbb{S}^{2};\mathbb{R})$ analytically extends to

[TABLE]

for some $\sigma>0$.

We stress out that (1) doesn’t describe the most general case that we can consider, in particular $\partial_{\phi}$ could be replaced by some unbounded operator.

The purpose of reducibility is to construct a change of variables that transforms the non-autonomous equation (1) into an autonomous equation.

Our main result is the following.

Theorem 1.1**.**

Let $0<\delta<1$ and $0\leq\alpha<1/2$. Assume that $\varphi\mapsto V(\varphi,\cdot)\in H^{s+s_{0}}(\mathbb{S}^{2};\mathbb{R})$ and $\varphi\mapsto W(\varphi,\cdot)\in H^{s+s_{0}}(\mathbb{S}^{2};\mathbb{R})$ analytically extend to $\mathbb{T}^{d}_{\sigma}$ for some $\sigma>0$, $s>0$ and for some $s_{0}=s_{0}(d,\alpha)\geq(d+2)/2$ large enough. Assume furthermore that the potentials $V$ and $W$ are odd in the variable $x\in\mathbb{S}^{2}$. There exists $\varepsilon_{0}>0$ such that, for any $0<\varepsilon\leq\varepsilon_{0}$ there is a set $\mathcal{O}_{\varepsilon}\subset\mathcal{O}_{0}\subset\mathbb{R}^{d}$ with

[TABLE]

such that the following holds:

for any $\omega\in\mathcal{O}_{\varepsilon}$ there exists a family linear isomorphisms $\Psi(\varphi)\in\mathcal{L}(H^{s}(\mathbb{S}^{2};\mathbb{C}))$, analytically depending on $\varphi\in\mathbb{T}^{d}_{\sigma/2}$ and a Hermitian operator $Z\in\mathcal{L}(H^{s}(\mathbb{S}^{2};\mathbb{C});H^{s+1-\alpha}(\mathbb{S}^{2};\mathbb{C}))$ commuting with the Laplacian and satisfying

$\bullet$* $\Psi(\varphi)$ is unitary on $L^{2}(\mathbb{S}^{2};\mathbb{C})$;*

$\bullet$* for any $0\leq s^{\prime}\leq s$*

[TABLE]

$\bullet$* the function $t\mapsto u(t,\cdot)\in H^{s^{\prime}}(\mathbb{S}^{2};\mathbb{C})$ solves (1) if and only if the map $t\mapsto v(t,\cdot):=\Psi(\omega t)u(t,\cdot)$ solves the autonomous equation*

[TABLE]

As a consequence of our reducibility result, we get the following corollary concerning the solutions of (1).

Corollary 1.2**.**

Assume that $V$ and $W$ satisfy the same assumptions than in Theorem 1.1. Let $1\leq s^{\prime}\leq s$ and let $u_{0}\in H^{s^{\prime}}(\mathbb{S}^{2};\mathbb{C})$. Then there exists $\varepsilon_{0}>0$ such that for all $0<\varepsilon<\varepsilon_{0}$ and for all $\omega\in\mathcal{O}_{\varepsilon}$, there exists a unique solution u\in{C}^{0}\big{(}\mathbb{R}\,;\,H^{s^{\prime}}\big{)} of (1) such that $u(0)=u_{0}$. Moreover, $u$ is almost-periodic in time and satisfies

[TABLE]

for some $C=C(s^{\prime},s,d)$.

The study of the reducibility problem for Schrödinger equations with quasi-periodic in time perturbation has been very popular in recent years. The first results adapting the KAM technics were due to Kuksin [27, 28] (see also [34, 29, 32, 6, 30, 25]) and concerned only one dimensional case. More recently the technics were adapted to the higher dimensional case [17, 16, 23, 33]. To consider unbounded perturbations, a new strategy has been developed in [1, 2] using the pseudo-differential calculus. Without trying to be exhaustive we quote also [22, 13, 3, 21] regarding KAM theory for quasi-linear PDEs in one space dimension. This technics were successfully applied for reducibility problems in various case. For one dimensional linear equations with unbounded potential we quote [5, 4, 8, 20]. In higher space dimensions we refer to [18, 24] for bounded potential, and to [9, 31, 19, 7] for the unbounded cases.

In this paper we choose to present an intermediate result were pseudo-differential calculus is not required although the perturbation is unbounded. We believe that the simplicity of this paper justifies this choice.

Scheme of the proof. We now briefly describe the structure of the proof. Some key points concern

the matrix representation of the multiplication operator $u\mapsto bu$ by a function $b\in H^{s}(\mathbb{S}^{n};\mathbb{C})$;

• 2)

the properties of the Laplace-Beltrami operator on $\mathbb{S}^{n}$;

• 3)

a sufficiently accurate asymptotic expansion of the eigenvalues of the linear operator in the right hand side of (1).

Regarding item $1)$, the key property which is exploited is that the product of two eigenfunctions is a finite linear combinations of them. Hence the rule of multiplications of the eigenfunctions implies that the multiplication operator $u\mapsto bu$ can be represented, in the base of eigenfunctions, as a block matrix with off-diagonal decay. The block structure of this matrix is a consequence of the multiplicity of the eigenvalues of $\Delta$ on $\mathbb{S}^{n}$. For the analysis of these decay properties we refer to [14] and [12] in which it is considered the more general case of equations on Lie Group or on compact manifolds which are homogenenous with respect to a compact Lie Group. In [24] the use of these decay-norms was not possible since in the case of the quantum harmonic operator we need to use specific dispersive properties of the eigenfunctions.

Concerning item $2)$, we strongly use the fact that the eigenvalues $\lambda_{k}$, $k\in\mathbb{N}$ of $\Delta$ (see (2.1)) satisfy a very strong “separation property” i.e.

[TABLE]

These property holds for the Laplace-Beltrami operator on $\mathbb{S}^{n}$ and more in general holds for compact manifolds which are homogenenous with respect to a compact Lie Group of rank $1$. We remark that this property is not true for “any” homogeneous manifold. For instance, it is violated by the eigenvalues of $\Delta$ on the torus $\mathbb{T}^{n}$, $n\geq 2$, which have the form $|j|^{2}$ with $j\in\mathbb{Z}^{d}$. The separation property in (1.6) is deeply used in the preliminary regularization step in section 4. In this step we also require an oddness hypothesis on the multiplicative potential $W$, $V$.

To understand the use of item $3)$ we briefly discuss the difficulties related to reducibility in high space dimension. We first recall that the Laplace operator $\Delta$ diagonalizes on the basis of the spherical harmonics of the sphere $\mathbb{S}^{n}$. We denote by $E_{k}$ the eigenspace associated to the eigenvalues $\lambda_{k}$ (see (2.1)), $k\in\mathbb{N}$. It is also know that the dimension of $E_{k}$ grows to infinity as $k\to\infty$. We shall denote by $\Phi_{k,j}$, $j=1,\ldots,d_{k}:=\dim E_{k}$ an orthonormal basis of $E_{k}$. With this formalism, the matrix $A$, which represents the operator $W(\omega t,x)(-{\rm i}\partial_{\phi})^{\alpha}+V(\omega t,x)$ (see (1)) in the basis $\Phi_{k,j}$, has the form A:=A(\omega t):=\Big{(}A_{[k]}^{[k^{\prime}]}\Big{)}_{k,k^{\prime}\in\mathbb{N}} with blocks $A_{[k]}^{[k^{\prime}]}\in\mathcal{L}(E_{k^{\prime}};E_{k})$. The reducibility of (1) rely on the reducibility of the operator $\Delta+\varepsilon A$ which is divided into two steps.

The first one is to regularize the (1) equation to a Schrödinger equation with a smoothing quasi-periodic in time perturbation. This is the content of section 4. More precisely, using also the oddness assumption on the potentials, we are able to show that $i)$ the operator $\Delta+\varepsilon A$ can be conjugated to an operator of the form

[TABLE]

and $ii)$ the eigenvalues of $\Delta+\varepsilon M$ have the form

[TABLE]

We remark that, since $\alpha<1/2$, the matrix $M$ in (1.7) is a “regularizing” operator, and its eigenvalues in (1.8) are very “close” to the unperturbed eigenvalues $\lambda_{k}$.

The second part of the proof consists in a quite standard KAM step following [24] or [18]. We note that in this second step we use the decay-norms introduced in [14] (see also [12]) which provides a simpler algebraic framework. A key point of a reducibility scheme is the resolution of the so called “homological equation”, which relies on the invertibility of an infinite dimensional matrix which is block diagonal with respect to the orthogonal splitting $L^{2}=\oplus_{k\in\mathbb{N}}E_{k}$ (see (2.16)). The fact that $\dim E_{k}\sim k^{n-1}$ makes hard the control of the inverse of such matrix, and could, in principle, creates loss of regularity at each step of the iteration. To overcome this problem we take advantages of the regularizing effect of the matrix $M$ to solve the homological equation using a trick previously used in literature, see for instance [23, 24]. We refer the reader to Lemma 5.3 where the properties (1.7), (1.8) are used to prove suitable estimates on the solution of the homological equation (see the bound (5.24)). We remark that, in [24], the regularizing effect of the perturbations is proved by using special dispersive properties of the eigenfunctions which do not hold in our context.

It is also know that reducibility of a matrix $M$ (even in finite dimension) requires some non-degeneracy conditions on differences of two eigenvalues, the so called “second order Melnikov conditions”. More precisely we shall prove that, for “most” parameters $\omega$, one has lower bounds of the form

[TABLE]

and $j=1,\ldots,d_{k}$, $j^{\prime}=1,\ldots,d_{k^{\prime}}$ (see (5.8) for more details). In order to prove that the set of “good” parameters has large Lebesgue measure it is fundamental to show that for any fixed $l\in\mathbb{Z}^{d}$, there are only finitely many indexes $k,k^{\prime}$ such that the conditions (1.9) are violated. Since the asymptotic of the eigenvalues in (1.8) is superlinear, i.e. $\sim k^{\mathtt{d}}$ with $\mathtt{d}>1$, it is quite easy to show that the (1.9) are violated only if $k+k^{\prime}\leq|l|$. The case $k=k^{\prime}$ is more delicate and the asymptotic (1.8) play a fundamental role. For more details we refer the reader to Lemma 5.1.

We note that the regularization of section 4 could be obtained by using a pseudo-differential calculus in the spirit of [1]. Actually in a subsequent paper we will extend our result using the regularization procedure developed in [10]. We expect to generalized Theorem 1.1 to the case of a quasi-periodic in time perturbation of order less or equal than $1/2$.

Acknowledgments. The authors wish to thank M. Procesi for many useful discussions.

2. Functional setting

In this section we introduce the space of functions, sequences and linear operators we shall use along the paper. We shall write $a\leq_{s}b$ to denote $a\leq Cb$ for some constant $C=C(s,d,n)$ depending only on $s,d,n$ (which are fixed parameters of the problem).

2.1. Space of functions and sequences

We denote by ${\mathcal{E}}:=\{\lambda_{k}\,,k\in\mathbb{N}\}$ with

[TABLE]

the spectrum of $-\Delta$ where $\Delta$ is the Laplace-Beltrami operator on the sphere $\mathbb{S}^{n}$ and let $E_{k}$ be the eigenspace associated to $\lambda_{k}$. We have

[TABLE]

We denote by

[TABLE]

an orthonormal basis of $E_{k}$ so that any function $u\in L^{2}(\mathbb{S}^{n};\mathbb{C})$ can be written as

[TABLE]

where ${}^{\prime\prime}\cdot^{\prime\prime}$ denotes the usual scalar product in $\mathbb{R}^{d_{k}}$. We denote by $\Pi_{E_{k}}$ the $L^{2}$-projector on the eigenspace $E_{k}$, i.e.

[TABLE]

For $s\geq 0$, we define the (Sobolev) scale of Hilbert sequence spaces

[TABLE]

where $\langle k\rangle:=\max\{1,|k|\}$ and $\|\cdot\|$ denotes the $L^{2}(\mathbb{C}^{d_{k}})$-norm. By a slight abuse of notation we define the operator $\Pi_{E_{k}}$ on sequences as $\Pi_{E_{k}}z=z_{[k]}$ for any $z\in h^{s}$ and $k\in\mathbb{N}$.

We note that

[TABLE]

is the standard Sobolev space and $\|u\|_{s}:=\|z\|_{s}$ is equivalent to the standard Sobolev norm.

Remark 2.1**.**

First of all notice that the weight $\langle k\rangle$ we use in the norm in (2.6) is related to the eigenvalues of the Laplace-Beltrami operator, indeed

[TABLE]

for some suitable constants $0<c\leq C$.

In the paper we shall also deal with functions of the space-time $u(\varphi,x)$ which can be expanded, using the standard Fourier theory, as

[TABLE]

where $e^{{\rm i}l\cdot\varphi}\Phi_{k,m}(x)$, $l\in\mathbb{Z}^{d}$, $k\in\mathbb{N}$, $m=1,\ldots,d_{k}$ is an orthogonal basis of $L^{2}(\mathbb{T}^{d}\times\mathbb{S}^{n};\mathbb{C})$. For $p_{1},s_{1}\geq 0$ we define the space $H^{p_{1}}(\mathbb{T}_{\sigma}^{d};H^{s_{1}}(\mathbb{S}^{n};\mathbb{C}))$ as the space of functions

[TABLE]

We shall work with functions $u(\varphi,x)$ in the space $\mathcal{A}_{s,\sigma}$, $s\geq 0$, $\sigma>0$,

[TABLE]

which we identify (using (2.8)) with the space of sequence

[TABLE]

and we endow the space $\mathcal{A}_{s,\sigma}$ with the norm $\|u\|_{\mathcal{A}_{s,\sigma}}:=\|z\|_{s,\sigma}$.

Lipschitz norm. Consider a compact subset $\mathcal{O}$ of $\mathbb{R}^{d}$, $d\geq 1$. For functions $f:\mathcal{O}\to E$, with $(E,\|\cdot\|_{E})$ some Banach space, we define the sup norm and the lipschitz semi-norm as

[TABLE]

For any $\gamma>0$ we introduce the weighted Lipschitz norms

[TABLE]

In order to simplify the notation, if $E=\ell_{s,\sigma}$ in (2.10), we shall write

[TABLE]

We finally define the space of sequences

[TABLE]

We have the following Lemma.

Lemma 2.2**.**

For $s>(d+n)/2$, for any $z,v\in\ell_{s,\sigma}$ there is $C(s)>0$ such that

(1) Sobolev embedding: $\|z\|_{L^{\infty}}\leq C(s)\|z\|_{s,\sigma}\,$;

(2) algebra: $\|zv\|_{s,\sigma}\leq C(s)\|z\|_{s,\sigma}\|v\|_{s,\sigma}$.

(3) Setting, for $N>0$, $\Pi_{N}z=\{z_{a}(l)\}_{|l|\leq N,a\in\mathcal{E}}$, one has

[TABLE]

Similar bounds holds also replacing $\|\cdot\|_{s,\sigma}$ with the norm $\|\cdot\|_{s,\sigma}^{\gamma,\mathcal{O}}$.

Proof.

Items $(1)$ and $(2)$ are classical estimates for Sobolev spaces, see for instance Lemma $2.13$ in [14]. Item $(3)$ follows by the definition of the norm. ∎

2.2. Linear operators

According to the orthogonal splitting

[TABLE]

we identify a linear operator acting on $L^{2}(\mathbb{S}^{n};\mathbb{C})$ with its matrix representation A:=\Big{(}A_{[k]}^{[k^{\prime}]}\Big{)}_{k,k^{\prime}\in\mathbb{N}} in $\mathcal{L}(h^{0})$ (recall (2.6)) with blocks $A_{[k]}^{[k^{\prime}]}\in\mathcal{L}(E_{k^{\prime}};E_{k})$. Notice that each block $A_{[k]}^{[k^{\prime}]}$ is a $d_{k}\times d_{k^{\prime}}$ matrix.

Notation. We shall write

[TABLE]

The action of the operator $A$ on functions $u(x)$ as in (2.4) of the space variable in $L^{2}(\mathbb{S}^{n};\mathbb{C})$ is given by

[TABLE]

Time-dependent matrices. In this paper we also consider $\varphi$-dependent families of linear operators

[TABLE]

where $A(l)\in\mathcal{L}(h^{0})$, for any $l\in\mathbb{Z}^{d}$ . We also regard $A$ as an operator acting on functions $u(\varphi,x)$ of space-time (see (2.9)) as

[TABLE]

More precisely, expanding $u$ as in (2.8), we have

[TABLE]

On operators as in (2.18) we define the following norm.

Definition 2.3**.**

($(s,\sigma)$-decay norm)* We define the $(s,\sigma)$-decay norm of a matrix $A$ in (2.18) as*

[TABLE]

where $\|\cdot\|_{\mathcal{L}(L^{2})}$ is the $L^{2}$-operator norm in $\mathcal{L}(E_{k^{\prime}},E_{k})$. We denote by $\mathcal{M}_{s,\sigma}$ the space of matrices of the form (2.18) with finite $(s,\sigma)$-decay norm.

Consider a family $\mathcal{O}\ni\omega\mapsto A(\omega)\in\mathcal{M}_{s,\sigma}$ where $\mathcal{O}$ is a compact subset of $\mathbb{R}^{d}$, $d\geq 1$. For $\gamma>0$ we define the Lipschitz decay norm as

[TABLE]

We denote by $\mathcal{M}^{\gamma,\mathcal{O}}_{s,\sigma}$ the space of families of matrices $A(\omega)$ with finite $|\cdot|_{s,\sigma}^{\gamma,\mathcal{O}}$-norm.

For the properties of the $(s,\sigma)$-decay norm we refer the reader to Lemma A.1 in Appendix A.

Remark 2.4**.**

Notice that, if the $(s,\sigma)$-decay norm of a matrix $A$ is finite, then

[TABLE]

We deal with a larger class of linear operators.

Definition 2.5**.**

Define the diagonal $\varphi$-independent operator $\mathcal{D}$, acting on sequences $z\in\ell_{0,\sigma}$ (see (2.10)), as (recall (2.1))

[TABLE]

For $\beta\in\mathbb{R}$ we define the norm $[\![\cdot]\!]_{\beta,s,\sigma}$ of a matrix $A$ in (2.18) as

[TABLE]

We denote by $\mathcal{M}_{\beta,s,\sigma}$ the space of matrices of the form (2.18) with finite $[\![\cdot]\!]_{\beta,s,\sigma}$-norm.

Consider a family $\mathcal{O}\ni\omega\mapsto A(\omega)\in\mathcal{M}_{\beta,s,\sigma}$ where $\mathcal{O}$ is a compact subset of $\mathbb{R}^{d}$, $d\geq 1$. For $\gamma>0$ we define the Lipschitz norm as

[TABLE]

We denote by $\mathcal{M}^{\gamma,\mathcal{O}}_{\beta,s,\sigma}$ the space of families of matrices $A(\omega)$ with finite $[\![\cdot]\!]_{\beta,s,\sigma}^{\gamma,\mathcal{O}}$-norm. If $\beta<0$ we say that $A\in\mathcal{M}^{\gamma,\mathcal{O}}_{\beta,s,\sigma}$ is a $\beta$-smoothing operator. If $A\in\mathcal{M}^{\gamma,\mathcal{O}}_{\beta,s,\sigma}$ does not depend on $\varphi$ we simply write $A\in\mathcal{M}^{\gamma,\mathcal{O}}_{\beta,s}$.

Remark 2.6**.**

We have the following simple inclusions for $\beta^{\prime}>\beta$ and $\nu_{1},\nu_{2}\geq 0$:

[TABLE]

The inclusions are continuous. For further properties of the operators of Def. 2.5 we refer to Appendix A.

2.3. Hamiltonian structure

In this subsection we introduce a special class of linear operators.

Definition 2.7**.**

Consider a linear operator $M\in\mathcal{L}(h^{0})$ and a family of maps $\varphi\mapsto A(\varphi)$ in $\mathcal{M}_{0,\sigma}$.

$\bullet$* (Hermitian operators). We say that $M$ is Hermitian if*

[TABLE]

for any $k,k^{\prime}\in\mathbb{N}$. To lighten the notation we shall also write that $M_{[k]}^{[k^{\prime}]}=\overline{M_{[k^{\prime}]}^{[k]}}$ instead of the (2.25). We say that $A(\varphi)$ is Hermitian if and only if

[TABLE]

$\bullet$* (Hamiltonian operators). We say that $M$ is Hamiltonian if  ${\rm i}M$ is Hermitian. We say that $A(\varphi)$ is Hamiltonian if and only if*

[TABLE]

$\bullet$* (Block-diagonal operators). We say that $A(\varphi)$ is block-diagonal if and only if $A_{[k]}^{[k^{\prime}]}(\varphi)=0$ for any $k\neq k^{\prime}$ and any $\varphi\in\mathbb{T}^{d}_{\sigma}$.*

Definition 2.8**.**

(Normal form)* We say that a matrix $M$ is in normal form if it is $\varphi$-independent, Hermitian and block-diagonal according to Definition 2.7. Given a Hermitian family of maps $\varphi\mapsto A(\varphi)$ in $\mathcal{M}_{0,\sigma}$ we define its normal form {\rm Diag}A=\big{(}({\rm Diag}A)_{[k]}^{[k^{\prime}]}(l)\big{)}_{l\in\mathbb{Z}^{d},k,k^{\prime}\in\mathbb{N}} as*

[TABLE]

Let $\omega\cdot\partial_{\varphi}$ be the diagonal operator acting on sequences $z\in\ell_{0,\sigma}$ (see (2.10)) defined by

[TABLE]

This operator is Hamiltonian and thus an operator of the form $\omega\cdot\partial_{\varphi}+M(\varphi)$ is Hamiltonian if and only if $M(\varphi)$ is Hamiltonian.

Conjugation under Hamiltonian flows. Consider the operator

[TABLE]

where $\omega\cdot\partial_{\varphi}$ is defined in (2.28), the operator $M=M(\varphi)\in\mathcal{M}_{0,\sigma}$ is Hamiltonian (see Def. 2.7). We shall study how the operator $L(\varphi)$ changes under the map

[TABLE]

for some $A\in\mathcal{M}_{0,\sigma}$ Hermitian. For the well-posedness of a map of the form (2.30) we refer to Lemma A.5 in Appendix A. By using Lie expansion the conjugate operator $M^{+}=M^{+}(\varphi):=e^{{\rm i}A}M(\varphi)e^{-{\rm i}A}$ has the form

[TABLE]

where

[TABLE]

Using the (2.31) we also deduce that (recall (2.28))

[TABLE]

Lemma 2.9**.**

If $M$ and ${\rm i}A$ are Hamiltonian linear operators then $M^{+}$ and $\widetilde{M}^{+}$ in (2.31) and (2.33) are Hamiltonian.

Proof.

To prove the lemma it is sufficient to check that $[{\rm i}A,M]$ and ${\rm i}\omega\cdot\partial_{\varphi}A$ are Hamiltonian. We have that, for any $k,k^{\prime}\in\mathbb{N},l\in\mathbb{Z}^{d}$

[TABLE]

Hence the claim follows using that ${\rm i}A$ and $M$ are Hamiltonian, i.e. their coefficients satisfy (2.26). Reasoning similarly one deduces the claim for ${\rm i}\omega\cdot\partial_{\varphi}A$. ∎

Notice that in view of Lemma 2.9 the map of the form (2.30) with $A$ Hermitian is symplectic.

Remark 2.10**.**

Lemma 2.9 provides only a formal rule of conjugation of matrices. It does not guarantees that such conjugate is a bounded operator on the spaces $\ell_{s,\sigma}$ with $s\gg 1$. The key information is that (at least formally) the flow $\Phi$ of a Hamiltonian operator (see (2.30)) preserves the Hamiltonian structure, i.e. the map $\Phi$ is symplectic.

3. An abstract reducibility result and its application to

(1)

In this section we state our main abstract result and we give some applications for the Schrödinger equation on spheres.

3.1. Abstract reducibility result

Fix the parameters $s>(d+n)/2$, $\sigma>0$, $\gamma>0$ as in the previous sections and let us add three new parameters

[TABLE]

Consider (recall Def. 2.3, 2.5) an operator of the form

[TABLE]

where $\omega\cdot\partial_{\varphi}$ and $\mathcal{D}$ are defined respectively in (2.28) and (2.22) and $\mathcal{O}$ is a compact subset of $\mathbb{R}^{d}$. Assume also that $R$ and $R^{\prime}$ are Hamiltonian according to Definition 2.7 and that $R$ is diagonal free i.e.

[TABLE]

We notice that $R$ is unbounded while $R^{\prime}$ is smoothing.

Theorem 3.1**.**

(Reducibility)* Let $\gamma>0$. There exist $\epsilon_{0}>0$ and $C>0$ depending only on $s,d,n,\alpha$ such that, if*

[TABLE]

then the following holds. There exist:

(i) (Cantor set) A cantor set $\mathcal{O}_{\infty}\subset\mathcal{O}$ such that

[TABLE]

(ii) (Normal form) an operator $\mathcal{Z}\in\mathcal{M}^{\gamma,\mathcal{O}_{\infty}}_{-\beta,s}$ in normal form (see Def. 2.8) satisfying

[TABLE]

and the eigenvalues of the block $\mathcal{Z}_{[k]}^{[k]}$, denoted $\mu^{(\infty)}_{k,j}$, $j=1,\ldots,d_{k}$, are Lipschitz functions from $\mathcal{O}$ into $\mathbb{R}$, and satisfy

[TABLE]

(iii) (Conjugacy) A Lipschitz family of invertible and symplectic maps $\Phi=\Phi(\omega):\ell_{s,\sigma/4}^{\gamma,\mathcal{O}_{\infty}}\to\ell_{s,\sigma/4}^{\gamma,\mathcal{O}_{\infty}}$, of the form $\Phi={\rm Id}+\Psi$ satisfying

[TABLE]

such that, for any $\omega\in\mathcal{O}_{\infty}$,

[TABLE]

Theorem 1.1 will be proved in sections 4, 5

3.2. Application to (1) on the sphere

In this section we consider a more general setting than in introduction. In fact we consider the Schrödinger equation

[TABLE]

where $\Delta$ denotes the Laplace-Beltrami operator on $\mathbb{S}^{n}$ and $\mathcal{R}$ and $\mathcal{R}^{\prime}$ are time-dependent families of linear operators corresponding, in their matrix representation with respect to the spherical harmonics basis, to Hamiltonian matrices $R\in\mathcal{M}^{\gamma,\mathcal{O}_{0}}_{\alpha,s+\nu,\sigma}$, $R^{\prime}\in\mathcal{M}^{\gamma,\mathcal{O}_{0}}_{-\beta,s,\sigma}$ with $R$ diagonal free as in (3.1), (3.2), (3.3). Let us choose $\gamma=\varepsilon^{\delta}$ for some $0<\delta<1$. The assumption (3.4) reads $\varepsilon<\varepsilon_{0}$ with $\varepsilon_{0}^{1-\delta}=([\![R]\!]_{\alpha,s+\nu,\sigma}^{\gamma,\mathcal{O}_{0}}+[\![R^{\prime}]\!]_{-\beta,s,\sigma}^{\gamma,\mathcal{O}_{0}})^{-1}\epsilon_{0}$. So we have the following.

Theorem 3.2**.**

Let $0<\delta<1$, $0\leq\alpha<1/2$ and $s>(d+n)/2$. There exists $\varepsilon_{0}>0$ such that, for any $0<\varepsilon\leq\varepsilon_{0}$ there is a set $\mathcal{O}_{\varepsilon}\subset\mathcal{O}_{0}\subset\mathbb{R}^{d}$ with

[TABLE]

such that the following holds.

For any $\omega\in\mathcal{O}_{\varepsilon}$ there exist a family of linear isomorphisms $\Psi(\varphi)\in\mathcal{L}(H^{s}(\mathbb{S}^{n};\mathbb{C}))$, analytically depending on $\varphi\in\mathbb{T}^{d}_{\sigma/2}$ and a block diagonal Hermitian operator $Z\in\mathcal{L}(H^{s}(\mathbb{S}^{n};\mathbb{C});H^{s+\beta}(\mathbb{S}^{n};\mathbb{C}))$ satisfying

$\bullet$* $\Psi(\varphi)$ is unitary on $L^{2}(\mathbb{S}^{n};\mathbb{C})$;*

$\bullet$* for any $0\leq s^{\prime}\leq s$*

[TABLE]

$\bullet$* the function $t\mapsto u(t,\cdot)\in H^{s^{\prime}}(\mathbb{S}^{n};\mathbb{C})$ solves (3.2) if and only if the map $t\mapsto v(t,\cdot):=\Psi(\omega t)u(t,\cdot)$ solves the autonomous equation*

[TABLE]

Now it remains to give examples of $\mathcal{R}$ and $\mathcal{R}^{\prime}$ that satisfy the right hypothesis. In particular, we need to make sure that (1) is in the right framework in such a way Theorem 1.1 holds true.

First we verify that a multiplicative potential is an admissible perturbation.

Lemma 3.3**.**

Assume that $\varphi\mapsto V(\varphi,\cdot)\in H^{s+s_{0}}(\mathbb{S}^{n};\mathbb{R})$ analytically extends to $\mathbb{T}^{d}_{\sigma}$ for some $\sigma>0$ and with $s_{0}=s_{0}(n)$. Then the matrix that represents the multiplication operator by $V$ and still denoted by $V$ belongs to $\mathcal{M}_{s,\sigma^{\prime}}$ for any $0<\sigma^{\prime}<\sigma$. Furthermore if $V$ is an odd function in the space variable then $V$ is diagonal free:

[TABLE]

Proof.

The fact that $V\in\mathcal{M}_{s,\sigma^{\prime}}$ is a consequence of Proposition $2.19$ in [14] (see also Lemma $3.1$ in [12]). Actually this is the reason why we use the $s$-decay norm (see Definition 2.3). So we only have to verify the second statement. By definition we have

[TABLE]

Now the spherical harmonic $\Phi_{k,j}$ has the same parity than $k$: $\Phi_{k,j}(-x)=(-1)^{k}\Phi_{k,j}(x)$. Therefore, if $V$ is odd, we conclude

[TABLE]

which implies the (3.12). ∎

Now we consider the perturbation term $W(\omega t,x)(-i\partial_{\phi})^{\alpha}$ appearing in (1). We know that $L_{x_{3}}=-i\partial_{\phi}$ also diagonalizes in spherical harmonic basis111 Recall that in (1) we are in $\mathbb{S}^{2}$ and the spherical harmonic basis is given by $\Phi_{k,j}=Ce^{ij\phi}P^{j}_{k}(\cos\theta)$ for $k\in\mathbb{N}$ and $-k\leq j\leq k$ and where $P_{k}^{j}$ are the Legendre polynomials (see for instance wikipedia.org/wiki/Spherical-harmonics). :

[TABLE]

and we define $(-i\partial_{\phi})^{\alpha}$ by

[TABLE]

Lemma 3.4**.**

Assume that $\varphi\mapsto W(\varphi,\cdot)\in H^{s+s_{0}}(\mathbb{S}^{n};\mathbb{R})$ analytically extends to $\mathbb{T}^{d}_{\sigma}$ for some $\sigma>0$ and with $s_{0}=s_{0}(n,d,\alpha)\geq(d+n)/2$. Then the matrix that represents the unbounded operator $R=W(\omega t,x)(-i\partial_{\phi})^{\alpha}$ belongs to $\mathcal{M}_{\alpha,s+\nu,\sigma^{\prime}}$ with $\nu$ as in (3.1) and $0<\sigma^{\prime}<\sigma$. Furthermore if $W$ is an odd function in the space variable then $R$ is diagonal free:

[TABLE]

Proof.

Since $|j|\leq k$ we have (see (2.22)) $-\mathcal{D}\leq-i\partial_{\phi}\leq\mathcal{D}$ in the sense of operators on $\ell_{0,0}$ (see (2.10)). Thus, in view of Definition 2.5 and Lemma 3.3, we get the first part of the Lemma. It remains we only have to verify the second part. By definition we have

[TABLE]

So we use again that the spherical harmonic $\Phi_{k,j}$ has the same parity than $k$ to conclude that if $W$ is odd then $R_{[k]}^{[k^{\prime}]}$ satisfies (3.13) and hence (3.15) holds. ∎

Proof of Theorem 1.1.

The result follows by Lemmata 3.3, 3.4 and by Theorem 3.2. ∎

We conclude this section with examples of regularizing perturbations $R^{\prime}\in\mathcal{M}^{\gamma,\mathcal{O}}_{-\beta,s,\sigma}$. The natural framework is that of pseudo-differential operators.

We denote by $S_{\rm cl}^{m}(\mathbb{S}^{n})$ the space of classical real valued symbols of order $m\in\mathbb{R}$ on the cotangent $T^{*}(S^{n})$ of $\mathbb{S}^{n}$ (see Hörmander [26] for more details).

Definition 3.5**.**

We say that $A\in\mathcal{P}_{m}$ if it is a pseudodifferential operator (in the sense of Hörmander [26], see also [10] ) with symbol of class $S^{m}_{\rm cl}(M)$.

We have

Lemma 3.6**.**

Let $\beta>0$ and assume that $\varphi\mapsto R(\varphi,\cdot)\in\mathcal{P}_{\beta}$ analytically extends to $\mathbb{T}^{d}_{\sigma}$ for some $\sigma>0$. Then the matrix that represents the operator $R$ belongs to $\mathcal{M}_{-\beta,s,\sigma}$ for all $s>(n+d)/2$.

Proof.

We use the so called commutator Lemma: Let A be a linear operator which maps $H^{s}(\mathbb{S}^{n})$ into itself and define the sequence of operators

[TABLE]

we have for any $\Phi_{k}\in E_{k}$, $\Phi_{k^{\prime}}\in E_{k^{\prime}}$

[TABLE]

Consider the operator $A:=D^{\beta}R$, by hypothesis $A\in\mathcal{A}_{0}$ and $(-\Delta)^{1/2}\in\mathcal{A}_{1}$ so by the fundamental property of pseudo-differential operators we deduce that for all $N\geq 1$, $A_{N}\in\mathcal{A}_{0}$. As a consequence $\|A_{N}\Phi_{k}\|\leq C_{N}\|\Phi_{k}\|$ and thus by (2.32)

[TABLE]

Taking $N=N(s)$ large enough we deduce that $A\in\mathcal{M}_{s,\sigma^{\prime}}$ and thus $A\in\mathcal{M}_{-\beta,s,\sigma^{\prime}}$. ∎

4. The regularization step

In this section we show that Theorem 1.1 (where $R$ is unbounded) can be reduced to a reducibility problem with a smoothing perturbation. To do this, we use the properties of the eigenvalues of the Laplacian operator on the spheres to show that the operator $G$ in (3.2) can be conjugated to a diagonal operator plus a smoothing remainder. More precisely we have the following (We use the same set of constants as in the section 3.1).

Proposition 4.1**.**

There exists $\overline{\varepsilon}>0$ and $C>0$ (depending only on $s,n,d$) such that for any $0<\varepsilon\leq\overline{\varepsilon}$, if $R$ as in (3.2) satisfies

[TABLE]

then the following holds. There exists a Lipschitz family of invertible and symplectic maps map $\mathcal{T}:=\mathcal{T}(\omega):={\rm Id}+\mathcal{F}$, with $\mathcal{F}\in\mathcal{M}_{\alpha-1,s+\nu,\sigma}^{\gamma,\mathcal{O}}$ and (see Def. 2.5)

[TABLE]

such that the conjugate of the operator $G$ in (3.2) has the form

[TABLE]

where $Z\in\mathcal{M}_{-\beta,s}^{\gamma,\mathcal{O}}$ is in normal form (see Def. 2.8) and $M\in\mathcal{M}_{-\beta,s,\sigma/2}^{\gamma,\mathcal{O}}$ is Hamiltonian (see Def. 2.7) and satisfy

[TABLE]

Finally $M$ is such that $M_{[k]}^{[k]}(0)=0$ for any $k\in\mathbb{N}$.

Proof.

Consider the matrix

[TABLE]

with $\lambda_{k}$ defined in (2.1). Since $R$ is Hamiltonian one verifies that $\mathcal{A}$ is Hamiltonian. Moreover, using that, for $k\neq k^{\prime}$, one has $|\lambda_{k}-\lambda_{k^{\prime}}|\geq k+k^{\prime}$, we deduce that (recall (2.22))

[TABLE]

Reasoning in a similar way for $\mathcal{A}\mathcal{D}^{1-\alpha}$ one obtain

[TABLE]

for some $C=C(s,n)>0$. We set $\mathcal{T}:={\rm Id}+\mathcal{F}:=e^{\mathcal{A}}$ which has the form (2.30) with ${\rm i}A\rightsquigarrow\mathcal{A}$. Estimates (4.6), (4.1) implies (A.10) for $\varepsilon$ small enough. Hence the bound (4.2) follows by Lemma A.5. By (4.5) and the hypothesis (3.3) we have that

[TABLE]

Thus formulæ  (2.31), (2.33) and (4.7) imply that that $\mathcal{T}\circ G\circ\mathcal{T}^{-1}$ has the form (4.3) with

[TABLE]

We define $-{\rm i}Z$ as the normal form (see (2.27) in Def. 2.8) of the previous expression while $M$ is defined by difference. Let $0<\sigma_{+}<\sigma$. Then we have

[TABLE]

With a similar reasoning one concludes

[TABLE]

By estimate (A.4) in Lemma A.3 and (3.1) we also obtain

[TABLE]

The (4.4) follows by using the smallness condition (4.1), the estimates (4.6), (4.9), (4.10), and reasoning as in Lemma A.5. Finally the operator $M$ is Hamiltonian by Lemma 2.9. ∎

5. The iterative reducibility scheme

In this section we prove Theorem 3.1 taking into account the regularization step given in Proposition 4.1. This mean that we show how to block-diagonalize the operator

[TABLE]

with $Z\in\mathcal{M}_{-\beta,s}^{\gamma,\mathcal{O}}$ in normal form and $M\in\mathcal{M}_{-\beta,s,\sigma}^{\gamma,\mathcal{O}}$ Hamiltonian satisfying that

[TABLE]

are small enough. Actually at the beginning of our iterative process we can take $\Theta_{0}=\varepsilon_{0}$ (see (4.4)) but during the process it will be important to distinguish between the size of the normal form (which essentially will not change) and the size of the remainder term (which will converge rapidly to zero). Consider the diophantine set

[TABLE]

We remark that it is know that ${\rm meas}(\mathcal{G}_{0})\lesssim\gamma$. In the following we shall assume that the set of parameters $\mathcal{O}$ satisfies $\mathcal{O}\subseteq\mathcal{G}_{0}$.

5.1. KAM strategy

We begin with $L$ given by (5.1), we seach for $\Phi=e^{S}$ a canonical change of variable such that

[TABLE]

where $Z_{+}$ is block-diagonal and $\varphi$-independent, $N_{+}=\mathcal{D}^{2}+Z_{+}$ is the new normal form, $\varepsilon$ close to $N_{0}=\mathcal{D}^{2}$ and the new perturbation $M_{+}$ is expected of size $O(\varepsilon^{2})$.

Using the expansion (2.31), (2.33) with ${\rm i}A\rightsquigarrow S$ we have that

[TABLE]

Formally, if we are able to construct $S=O(\varepsilon)$ satisfying the the so-called homological equation222In fact the homological equation that we will solve contains a small remainder in the right hand side (see (5.15)) because we cannot solve all the Fourier modes at the same time.

[TABLE]

where ${\rm Diag}M$ is defined as in (2.27), then $L^{+}$ is of the form (5.4) with $Z_{+}=Z+{\rm Diag}M$ and where $M_{+}$ is a sum of terms containing at least two operators of size $\varepsilon$ and thus is formally of size $\varepsilon^{2}$.

Repeating infinitely many times the same procedure we will construct a change of variable $\Phi$ such that

[TABLE]

with $Z_{\infty}$ in normal form according to Definition 2.8 which is our final goal.

5.2. The homological equation

5.2.1. Control of the small divisors

Let $Z\in\mathcal{M}_{-\beta,s}^{\gamma,\mathcal{O}}$ be in normal form and denote by $\mu_{k,j}$, $k\in\mathbb{N}$ and $j=1,\ldots,d_{k}$ (see (2.2)), the eigenvalues of the block $Z_{[k]}^{[k]}$.

We define the set $\mathcal{O}_{+}\subseteq\mathcal{O}\subseteq\mathcal{G}_{0}\subset[1/2,3/2]^{d}$ of parameters $\omega$ for which we have a good control of the small divisors. Let us fix once for all

[TABLE]

with $\tau_{0}$ in (5.3). We set

[TABLE]

We have the following.

Lemma 5.1**.**

Assume that $[\![{Z}]\!]_{-\beta,s}^{\gamma,\mathcal{O}}\leq\gamma/4$ for some $0<\gamma\leq 1/8$ then for any $K\geq 1$ we have

[TABLE]

for some $C=C(s,d,n)>0$.

Before giving the proof of Lemma 5.1 we recall the following classical result regarding the measure of sublevels of Lipschitz functions.

Lemma 5.2**.**

Let $m\geq 1$, $\eta>0$ and let $\mathcal{O}$ be a subset of $\mathbb{R}^{m}$, $m\geq 1$ such that ${\rm meas}(\mathcal{O})<+\infty$. Consider a Lipschitz function $f:\mathcal{O}\to\mathbb{R}$ such that

[TABLE]

Then, setting ${\mathcal{O}}_{\eta}:=\{x\in\mathcal{O}\;:\;|f(x)|\leq\eta\}$ we have

[TABLE]

Proof.

Let us set ${\rm diam}({\mathcal{O}_{\eta}}):={\rm sup}_{x_{1},x_{2}\in{\mathcal{O}}}|x_{1}-x_{2}|$. Notice that ${\rm meas}({\mathcal{O}_{\eta}})\leq{\rm diam}({\mathcal{O}_{\eta}})$. For any $x_{1},x_{2}\in{\mathcal{O}_{\eta}}$ such that $x_{1}\neq x_{2}$, we have that

[TABLE]

This implies the thesis. ∎

Proof of Lemma 5.1.

We write

[TABLE]

where

[TABLE]

We claim that, for $k\neq k^{\prime}$, $l\neq 0$,

[TABLE]

for some constant $C>0$ depending only on $n,d$ and $|\omega|$. Indeed, by hypothesis, there is $\omega\in\mathcal{O}$ such that

[TABLE]

On the other hand, since $[\![{Z}]\!]_{-\beta,s}^{\gamma,\mathcal{O}}\leq\gamma/4$ , by Lemma A.6 and Corollary A.7, we have that

[TABLE]

Then using (2.1) and the first in (5.12), we conclude for $k\neq k^{\prime}$

[TABLE]

Hence, by (5.11), we have

[TABLE]

which implies (5.10).

We also notice that when $l=0$ and $k\neq k^{\prime}$ then $R_{l,k,k^{\prime}}^{j,j^{\prime}}=\emptyset$ for all $j,j^{\prime}$. Indeed in such case, using again (5.13), we get $|\omega\cdot l+\lambda_{k}+\mu_{k,j}-\lambda_{k^{\prime}}+\mu_{k^{\prime},j^{\prime}}|\geq\frac{1}{2}|k+k^{\prime}|\geq\frac{1}{2}>\frac{2\gamma}{K^{\tau}}$.

Let us now consider the case $l\neq 0$ and $k=k^{\prime}$. We claim that

[TABLE]

We recall that, by assumption, the set $\mathcal{O}$ is contained in the set $\mathcal{G}_{0}$ in (5.3). Hence, for $\omega\in{\mathcal{O}}$, we deduce by (5.12)

[TABLE]

using that $|k|^{\beta}\geq|l|^{\tau_{0}}$ which implies claim (5.14) since $\tau_{0}<\tau$.

Now it remains to estimate the measure of

[TABLE]

In order to estimate the measure of a single bad set $R_{l,k,k^{\prime}}^{j,j^{\prime}}$ we compute the Lipschitz norm of the function

[TABLE]

The second condition in (5.12) implies that (recall that $l\neq 0$)

[TABLE]

Then Lemma 5.2 implies that ${\rm meas}(R_{l,k,k^{\prime}}^{j,j^{\prime}})\leq 2\frac{\gamma}{K^{\tau}}$. Finally, we recall that, by (2.2), (5.10) and (5.14), we have that

[TABLE]

Hence

[TABLE]

which is the (5.9). ∎

5.2.2. Resolution of the Homological equation

In this section we solve the following homological equation equation

[TABLE]

where ${\rm Diag}M$ is defined as in (2.27) and $R$ is some remainder to be determined.

Lemma 5.3**.**

(Homological equation)* Let $Z\in\mathcal{M}^{\gamma,\mathcal{O}}_{-\beta,s}$ in normal form and $M\in\mathcal{M}^{\gamma,\mathcal{O}}_{-\beta,s,\sigma}$. Assume that $[\![{Z}]\!]_{-\beta,s}^{\gamma,\mathcal{O}}\leq\gamma/4$ and let $0<\sigma_{+}<\sigma$ such that*

[TABLE]

For any $\omega\in\mathcal{O}_{+}\equiv\mathcal{O}_{+}(\gamma,K)$ (defined in (5.8)) there exist Hamiltonian operators $S,R\in\mathcal{M}^{\gamma,\mathcal{O_{+}}}_{-\beta,s,\sigma_{+}}$ satisfying

[TABLE]

such that equation (5.15) is satisfied.

Proof.

The proof is an adaptation of (for instance) Lemma $4.3$ in [24]. We set

[TABLE]

and $R_{[k]}^{[k^{\prime}]}(l)=0$ for $|l|\leq K$. By Lemma A.3 and (5.16) one deduces the (5.18). Moreover, recalling (2.27), we have that equation (5.15) is equivalent to

[TABLE]

for any $l\in\mathbb{Z}^{d}$, $k,k^{\prime}\in\mathbb{N}$ with $(l,k,k^{\prime})\neq(0,k,k)$ where the operator $\mathcal{G}(l,k,k^{\prime},\omega)$ is the linear operator acting on complex $d_{k}\times d_{k^{\prime}}$-matrices as

[TABLE]

Now, since $Z_{[k]}^{[k]}$ is Hermitian, there is a orthogonal $d_{k}\times d_{k}$-matrix $U_{[k]}$ such that

[TABLE]

where $\mu_{k,j}$ are the eigenvalues of $Z_{[k]}^{[k]}$. By setting

[TABLE]

equation (5.20) reads

[TABLE]

For $\omega\in\mathcal{O}_{+}$ (see (5.8)) the solution of (5.22) is given by (recalling the notation (2.17))

[TABLE]

Since $M$ is Hamiltonian (see Def. 2.7 and (2.26)) it is easy to check that also $S$ is Hamiltonian. We claim that

[TABLE]

Proof of the claim (5.24).

To prove the claim we follows the strategy used in the proof of Lemma 4.3 in [24] (see also Proposition 2.2.4 in [15]) and we prove (5.24) considering three different regimes of the indexes $k,k^{\prime}$.

Case 1. Assume that

[TABLE]

for some $K_{1}>0$ large to be determined. Without loss of generality we can assume ${k}>K_{1}{k^{\prime}}$. We note that

[TABLE]

if $\lambda_{k}\geq K_{1}^{2}\geq|\omega|K\geq|\omega\cdot l|$ and using that, by hypothesis on $Z$, $|\mu_{k,j}|\leq 1/4$. We choose $K_{1}:=8K$. Equation (5.22) can be written

[TABLE]

where \mathcal{B}_{k,k^{\prime}}(l)\widehat{S}_{[k]}^{[k^{\prime}]}(l):=\big{(}-{\rm i}\omega\cdot l+{\rm i}D_{[k]}\big{)}^{-1}\widehat{S}_{[k]}^{[k^{\prime}]}(l){\rm i}D_{[k^{\prime}]}. Since

[TABLE]

thanks to the fact that $K_{1}\geq 8$, we have that the operator $({\rm Id}+\mathcal{B}_{k,k^{\prime}}(l))$ is invertible using Neumann series. Therefore we have

[TABLE]

Case 2. Assume that

[TABLE]

for some $K_{2}>0$ to be determined. The (5.28) implies that

[TABLE]

Using Corollary A.7 we also note that for all $k$

[TABLE]

and thus

[TABLE]

Equation (5.22) is equivalent to

[TABLE]

where the operator $\mathcal{B}_{k,k^{\prime}}^{+}(l)$ acts on $d_{k}\times d_{k^{\prime}}$-matrices as

[TABLE]

We need to estimate the operator norm of $\mathcal{B}_{k,k^{\prime}}^{+}(l)$. First notice that, for any $\omega\in\mathcal{O}_{+}$ (see 5.8),

[TABLE]

providing

[TABLE]

Combining (5.31) and (5.33) we get that, in operator norm,

[TABLE]

providing (5.34). Recalling $K_{1}=8K$ we choose

[TABLE]

Now, by (5.35), the operator \big{(}{\rm Id}+\mathcal{B}_{k,k^{\prime}}^{+}(l)\big{)} is invertible, and hence by (5.32) and (5.33) we get

[TABLE]

Case 3. Assume that

[TABLE]

In that case the size of the blocks are less than $K_{2}^{n}$ and we have, for any $j=1,\ldots,d_{k}$, $j^{\prime}=1,\ldots,d_{k^{\prime}}$,

[TABLE]

and hence

[TABLE]

By collecting the bounds (5.27), (5.37) and (5.39) we get (5.24). ∎

Estimate (5.24) allows us to conclude that

[TABLE]

Indeed (recall (2.20))

[TABLE]

To obtain (5.17), it remains to estimate the Lipschitz variation of the matrix $S$. For any family of operators $\omega\mapsto A=A(\omega)$ and any $\omega_{1},\omega_{2}\in\mathcal{O}_{+}$ with $\omega_{1}\neq\omega_{2}$ we set

[TABLE]

Hence, by (5.20), we obtain

[TABLE]

which is an equation of the same form of (5.20) with different non-homogeneous term. Using that $|Z|^{lip,\mathcal{O}}\leq 1/4$ we deduce from (5.21)

[TABLE]

Then, reasoning as in the proof of (5.24), we deduce

[TABLE]

which, following the proof of (5.40) and using (5.16) and recalling the choice (3.1), implies (5.17). ∎

5.3. The KAM step

Now we compute the new $L_{+}$ (see (5.4)) generated by the change of variable $\Phi=e^{S}$ where $S$ satisfies the homological equation (5.15).

We first prove the following.

Lemma 5.4**.**

There is $C(s)>0$ (depending only on $s$) such that, if

[TABLE]

then the map $\Phi=e^{S}={\rm Id}+\Psi$, with $S$ given by Lemma 5.3, satisfies

[TABLE]

Proof.

By (5.17) and (5.42) we have that

[TABLE]

This implies te smallness condition (A.10). Hence the (5.43) follows by Lemma A.5. ∎

5.3.1. The new normal form

As said in section 5.1 we define the new normal form $Z_{+}$ as

[TABLE]

We have the following.

Lemma 5.5**.**

(New normal form)* We have that $Z_{+}$ in (5.45) is in normal form (see Def. 2.8) and satisfies*

[TABLE]

There is a sequence of Lipschitz function

[TABLE]

such that, for $\omega\in\mathcal{O}_{+}$, the functions $\mu_{k,j}^{+}$, for $j=1,\ldots,d_{k}$, are the eigenvalues of the block $(Z_{+})_{[k]}^{[k]}$ satisfying

[TABLE]

Proof.

The matrix $Z_{+}$ is $\varphi$-independent, block-diagonal and Hermitian by construction. Estimate (5.47) is a consequence of Corollary A.7. ∎

5.3.2. The new remainder

Now we compute and estimate $M_{+}$ given by (5.4).

Lemma 5.6**.**

(The new remainder)* Assume that the smallness condition (5.42) holds true. The new remainder $M_{+}\in\mathcal{M}^{\gamma,\delta,\mathcal{O}_{+}}_{-\beta,s,\sigma_{+}}$ is Hamiltonian and satisfies*

[TABLE]

Proof.

Equations (5.15), (5.45) and (5.1) lead to the following formula for $M_{+}$

[TABLE]

with $R$ satisfying (5.18). Thus, in order to prove (5.48) we need to estimate $\widetilde{M}_{+}$. By (5.18) and (A.6), we have

[TABLE]

for some $C(s)>0$. The term $[S,M]$ can be estimated in the same way. Hence

[TABLE]

Using formula (5.49) we have that the estimates (5.18), (5.50) and (5.17) imply the (5.48). By Lemma 2.9 we have that $M_{+}$ is Hamiltonian. ∎

5.4. The iterative Lemma

We fix $1<\chi<2$, $K_{0}\geq 1$, $\sigma_{0}:=\sigma/2$ (see (4.4)) and we recall that $\Theta_{0}=\varepsilon_{0}>0$ (see (5.2)). For $k\in\mathbb{N}$ we introduce the following parameters:

[TABLE]

Consider an operator $L_{0}$ of the form (5.1) with $\mathcal{O}\rightsquigarrow\mathcal{G}_{0}\cap\mathcal{O}$, $\sigma\rightsquigarrow\sigma_{0}$ where $\mathcal{G}_{0}$ is in (5.3). We prove the following.

Proposition 5.7**.**

(Iterative Lemma)* There are $K_{\star},\Theta_{\star}>0$ depending on $n,s,d$, $\chi$, with $1<\chi<2$, such that if*

[TABLE]

then for all $k\geq 0$ we can construct:

$\bullet$* sets $\mathcal{O}_{k+1}\subset\mathcal{O}_{k}\subset\mathcal{G}_{0}$ satisfying*

[TABLE]

$\bullet$* Lipschitz family of canonical change of variables $\Phi_{k}\equiv\Phi_{k}(\omega):={\rm Id}+\Psi_{k}$ with $\Psi_{k}\in\mathcal{M}_{-\beta,s,\sigma_{k}}^{\gamma,\mathcal{O}_{k}}$ and*

[TABLE]

where $C_{\star}=C_{\star}(n,d,s,K_{0})$.

$\bullet$* Lipschitz family of operators*

[TABLE]

with $Z_{k}\in\mathcal{M}_{-\beta,s}^{\gamma,\mathcal{O}_{k}}$ in normal form and $M_{k}\in\mathcal{M}_{-\beta,s,\sigma_{k}}^{\gamma,\mathcal{O}_{k}}$ Hamiltonian (see Def. 2.8, 2.7) satisfying

[TABLE]

[TABLE]

such that for any $k\geq 1$

[TABLE]

Proof of Proposition 5.7.

We proceed by induction. At step $k=0$ the operator $L_{0}$ is defined on $\mathcal{O}_{0}$ by (5.1) which is of the form (5.54) and satisfies (5.56). Now assume that we have construct the sets $\mathcal{O}_{p}$, the operators $L_{p}$ and the changes of variables $\Phi_{p}$ for $p=1,\cdots,k$ and let us construct them at step $k+1$.

Since (5.56) implies $[\![Z_{k}]\!]_{-\beta,s}^{\gamma,\mathcal{O}_{k}}\leq\gamma/4$ for $\Theta_{0}$ small enough333Notice that $\sigma_{k+1}\searrow\sigma_{0}/2$. Hence, by (5.51), we have

$\sigma_{k}-\sigma_{k+1}=\sigma_{k}\frac{1}{2^{k+3}}\geq\sigma_{0}\frac{1}{2^{k+4}}\geq\frac{1}{K_{k}}=\frac{1}{4^{k}{K_{0}}}$

for ${K_{0}}>0$ large enough., we use Lemmata 5.1 and 5.3 to construct $\mathcal{O}_{k+1}$, $S_{k+1}$ and $R_{k+1}$. The set $\mathcal{O}_{k+1}$ is defined as in (5.8) with $\mathcal{O}\rightsquigarrow\mathcal{O}_{k}$, $K\rightsquigarrow K_{k}$ and satisfies the (5.52) by Lemma 5.1. By the induction hypothesis (5.56) we have

[TABLE]

provided that

[TABLE]

The (5.58) implies the smallness condition (5.42) with $K\rightsquigarrow K_{k}$, $M\rightsquigarrow M_{k}$. Then Lemma 5.4 provides a map $\Phi_{k+1}={\rm Id}+\Psi_{k+1}$ such that

[TABLE]

which, by (5.58), implies the (5.53). By Lemmata 5.5 and 5.6 we construct

[TABLE]

with $M_{k+1}=M_{+}$ Hamiltonian and

[TABLE]

is in normal form (see the (2.27) in Def. 2.8). Moreover, by the estimate (5.46), we deduce that

[TABLE]

On the other hand we note that Lemma 5.6 implies

[TABLE]

where we used that $(\sigma_{k}-\sigma_{k+1})K_{k}\geq K_{0}2^{k-3}\geq 2^{k}+\sqrt{K_{0}}$ for $K_{0}$ large enough. Hence

[TABLE]

provided $\varepsilon_{0}$ small enough and $K_{0}$ large enough. The (5.61) and (5.62) yields (5.56) with $k\rightsquigarrow k+1$. ∎

5.5. Convergence and Proof of Theorem 3.1

Proof of Theorem 3.1.

By the smallness condition (3.4) we have that hypothesis (4.1) holds for $\epsilon$ sufficiently small. Hence Proposition 4.1 applies to the operator $G$ in (3.2). The operator (4.3) has the form (5.1) with $M_{0}\rightsquigarrow R_{+}$, $\mathcal{O}\rightsquigarrow\mathcal{G}_{0}\cap\mathcal{O}$, with $\mathcal{G}_{0}$ in (5.3) and $\sigma_{0}:=\sigma_{+}=\sigma/2$ and $\varepsilon_{0}$ in (5.2) satisfies $\varepsilon_{0}\leq_{s}\epsilon$. So taking again $\epsilon$ small enough we can apply Proposition 5.7 for some $K_{0}\geq K_{\star}$.

Let us define the set

[TABLE]

By (5.52) we deduce (3.5). We also notice that $\sigma_{k}\geq\sigma_{0}/2$ for all $k\geq 0$. Then, by (5.55) and (5.56), we deduce that

[TABLE]

and thus, since $\sum\varepsilon_{k}<+\infty$, $Z_{k}$ is a Cauchy sequence in $\mathcal{M}^{\gamma,\mathcal{O}_{\infty}}_{-\beta,s}$ and we can define the block diagonal hermitian operator

[TABLE]

As a consequence of Corollary A.7 we deduce (3.6).

Then definig $\widetilde{\Phi}_{\nu}:=\Phi_{1}\circ\Phi_{2}\circ\cdots\Phi_{k}={\rm Id}+\widetilde{\Psi}_{k}$ we have by (5.53)

[TABLE]

Thus $\widetilde{\Psi}_{k}$ is a Cauchy sequence in $\mathcal{M}^{\gamma,\mathcal{O}_{\infty}}_{-\beta,s,\sigma_{0}/2}$ and we can define its limit $\widetilde{\Psi}_{\infty}\in\mathcal{M}^{\gamma,\delta,\mathcal{O}_{\infty}}_{-\beta,s,\sigma_{0}/2}$ which satisfies

[TABLE]

Then the map $\Phi_{\infty}:={\rm Id}+\widetilde{\Psi}_{\infty}$ satisfies $\Phi_{\infty}:=\lim_{k\to+\infty}\tilde{\Phi}_{k}$. Finally for $\omega\in\mathcal{O}_{\infty}$ we set

[TABLE]

where $\mathcal{T}$ is the map given by Proposition 4.1. Since $\alpha-1\leq-\beta$ (see (3.1)), by Remark 2.6 we have that $\mathcal{F}$ belongs to $\mathcal{M}_{-\beta,s,\sigma_{0}/2}^{\gamma,\mathcal{O}_{\infty}}$. The (3.7) follows by composition using (4.2), (5.63) and (5.64). The (3.8) follows by Lemma A.4. The (3.9) follows by the construction. ∎

Appendix A Technical Lemmata

In this appendix we assume $s>(d+n)/2$ and $\sigma>0$.

Lemma A.1**.**

Let $A,B\in\mathcal{M}_{s,\sigma}$. Then the following holds:

• $(i)$

for any $z\in\ell_{s,\sigma}$ one has $\|Az\|_{s,\sigma}\leq C(s)|A|_{s,\sigma}\|z\|_{s,\sigma}$;

• $(ii)$

one has $|AB|_{s,\sigma}\leq C(s)|A|_{s,\sigma}|B|_{s,\sigma}$;

• $(iii)$

by setting (recall (2.18)) $\Pi_{N}A:=\sum_{|l|<N}A(l)e^{{\rm i}l\cdot\varphi}$ one has

[TABLE]

Similar bounds holds also replacing $|\cdot|_{s,\sigma}$, $\|\cdot\|_{s,\sigma}$ with the norms $|\cdot|_{s,\sigma}^{\gamma,\mathcal{O}}$, $\|\cdot\|_{s,\sigma}^{\gamma,\mathcal{O}}$ respectively.

• $(iv)$

Let $\beta>0$ and $A\in\mathcal{M}_{s+\beta,\sigma}^{\gamma,\mathcal{O}}$ then (recall (2.22)) $\mathcal{D}^{\beta}A\mathcal{D}^{-\beta},\mathcal{D}^{-\beta}A\mathcal{D}^{\beta}\in\mathcal{M}^{\gamma,\mathcal{O}}_{s,\sigma}$ and

[TABLE]

Proof.

Items $(i)$ and $(ii)$ follows by lemmata $2.6$, $2.7$ in [12]. Item $(iii)$ follows by the definition of the norm in (2.20). To prove item $(iv)$ we reason as follows. We study the operator $\mathcal{D}^{\beta}A\mathcal{D}^{-\beta}$. The bound for $\mathcal{D}^{-\beta}A\mathcal{D}^{\beta}$ can be deduced in the same way. First we note that

[TABLE]

If $k^{\prime}\geq 1/2k$ then, recalling (2.7), we deduce

[TABLE]

If on the contrary $k^{\prime}\leq 1/2k$, then $|k-k^{\prime}|\geq 1/2k$. Hence we have

[TABLE]

Bounds (A.2) and (A.3) imply (A.1) for the norm $|\cdot|_{s,\sigma}$. The bound for the Lipschitz norm in (2.21) follows in the same way. ∎

Lemma A.2**.**

Let $A$ be a matrix as in (2.18) with finite $|\cdot|_{s,\sigma}$ norm. Then

[TABLE]

Proof.

For any $\varphi\in\mathbb{T}_{\sigma^{\prime}}$ we have (using Cauchy-Schwarz inequality and taking $s_{0}:=(d+1)/2$)

[TABLE]

where we used that the function $e^{-(\sigma-\sigma^{\prime})x}x^{s_{0}+d}$ has a maximum in $x=s_{0}/(\sigma-\sigma^{\prime})$. ∎

Lemma A.3**.**

Let $\alpha,\beta\in\mathbb{R}$ and consider $A\in\mathcal{M}^{\gamma,\mathcal{O}}_{\alpha,s+\beta,\sigma}$ and $B\in\mathcal{M}^{\gamma,\mathcal{O}}_{\beta,s+\alpha,\sigma}$. There is $C(s)>0$ such that

[TABLE]

Moreover, if $\alpha\leq\beta<0$ then

[TABLE]

Proof.

To prove (A.4) we need to bound the decay norms of the operators $\mathcal{D}^{-\alpha-\beta}AM$ and $AM\mathcal{D}^{-\alpha-\beta}$. We have that

[TABLE]

Hence, by item $(iv)$ in Lemma A.1, $|AM\mathcal{D}^{-\alpha-\beta}|_{s,\sigma}{\leq_{s}}[\![A]\!]_{\alpha,s,\sigma}[\![M]\!]_{\beta,s+|\alpha|,\sigma}\,.$ Reasoning similarly one obtains the (A.4) for the Lipschitz norm $[\![\cdot]\!]_{\alpha+\beta,s,\sigma}^{\gamma,\mathcal{O}}$ . The (A.6) and (A.5) follow by Lemma A.1. ∎

Lemma A.4**.**

Let $\beta\in\mathbb{R}$ and consider $A\in\mathcal{M}^{\gamma,\mathcal{O}}_{-\beta,s,\sigma}$. Then

[TABLE]

In particular (recall (2.6)) $A(\varphi):h^{s}\mapsto h^{s+\beta}\,,\;\;\forall\varphi\in\mathbb{T}_{\sigma^{\prime}}^{d}\,,\sigma^{\prime}<\sigma\,,$ and

[TABLE]

for any $v\in h^{s}$.

Proof.

The (A.7) follows by Lemma A.1 and (2.23). To prove (A.8) we reason as follows. We have

[TABLE]

where $C(s):=\sum_{j\in\mathbb{N}}\frac{\langle k\rangle^{2s}}{\langle k-j\rangle^{2s}\langle j\rangle^{2s}}\,.$ It is easy to check that $C(s)<+\infty$. By Lemma A.2 and (A.9) we have

[TABLE]

which implies the thesis. ∎

Lemma A.5**.**

Let $\beta<0$, consider $A\in\mathcal{M}_{\beta,s,\sigma}^{\gamma,\mathcal{O}}$ and assume

[TABLE]

for some large $C(s)>0$. Then the map $\Phi:={\rm Id}+\Psi$ defined in (2.30) satisfies

[TABLE]

Proof.

By (A.6) we have

[TABLE]

for some (large) $C(s)>0$. By the smallness condition (A.10) one deduces the bounds (A.11). ∎

We end with two results on the eigenvalues of Hermitian matrix.

Lemma A.6**.**

Let $\omega\mapsto A(\omega)$ be a Lipschitz mapping from $\mathcal{O}$ a compact set of $\mathbb{R}^{d}$ into the set Hermitian matrix of finite dimension $p$. Then the eigenvalues of $A(\omega)$ can be ordered $\mu_{1}(\omega)\leq\mu_{2}(\omega)\leq\cdots\leq\mu_{p}(\omega)$ in such a way each eigenvalue $\mu_{j}$ is Lipschitz and

[TABLE]

where $\|\cdot\|$ denotes the operator norm.

Proof.

This is a consequence of the Courant Fischer formula:

[TABLE]

∎

As a consequence we get the following.

Corollary A.7**.**

If $Z\in\mathcal{M}^{\gamma,\mathcal{O}}_{-\beta,s,\sigma}$ is block diagonal then the eigenvalues of the block $\mathcal{Z}_{[k]}^{[k]}$, denoted $\mu_{k,j}$, $j=1,\ldots,d_{k}$, are Lipschitz functions from $\mathcal{O}$ into $\mathbb{R}$, and satisfy

[TABLE]