Sharp spectral stability for a class of singularly perturbed pseudo-differential operators

Let $a(x,\xi)$ be a real H\"ormander symbol of the type $S_{0,0}^0(\mathbb{R}^{d}\times \mathbb{R}^d)$, let $F$ be a smooth function with all its derivatives globally bounded, and let $K_\delta$ be the self-adjoint Weyl quantization of the perturbed symbols $a(x+F(\delta\, x),\xi)$, where $|\delta|\leq 1$. First, we prove that the Hausdorff distance between the spectra of $K_\delta$ and $K_{0}$ is bounded by $\sqrt{|\delta|}$, and we give examples where spectral gaps of this magnitude can open when $\delta\neq 0$. Second, we show that the distance between the spectral edges of $K_\delta$ and $K_0$ (and also the edges of the inner spectral gaps, as long as they remain open at $\delta=0$) are of order $|\delta|$, and give a precise dependence on the width of the spectral gaps.


Introduction and main results
Let ( , ) be a real Hörmander symbol [14] of class 0 0,0 (R × R ), i.e. a smooth function on R 2 satisfying the estimate: It belongs to the same class.We denote by = ( ) the self-adjoint operator generated by the Weyl quantization, which means that: where , ∈ S (R ) and • , • denotes the usual scalar product in 2 (R ) (considered to be anti-linear in the first variable).The operator has a distribution kernel that can be written as the oscillatory integral: In [11], Gröchenig et.al.proved that the spectral edges of the spectrum ( ) are Lipschitz at = 0.The problem is non-trivial because the map ↦ → is not necessarily differentiable in the operator norm topology, which can already be seen at the level of the symbol: should have some extra linear decay in both and in order to make sure that − 0 has a norm of order , which would imply that the Hausdorff distance between the spectra of and 0 is Lipschitz continuous at zero.Nevertheless, the authors of [11] show that such a strong decay is far from necessary if one is only interested in the spectral edges.Actually they even consider more general operators corresponding to symbols of Sjöstrand type, operators which belong to certain weighted modulation spaces; see [10] and references therein for an introduction to the subject.
A similar phenomenon appears in the case of long range magnetic perturbations [2,4,5,8,9].In fact, the two problems are very much related, see Section 3.2 of the current manuscript for more details.

A more general perturbation
In this manuscript we are interested in a more general perturbation of the symbol, where the dilation treated in [11] becomes just a particular case.In order to achieve that, we have to "rotate" the operators in the following way: Lemma 1.1.Denote by the unitary transformation in 2 (R ) given by Then * equals the Weyl quantization of the symbol ( + , ), and is isospectral with .
The proof of this lemma, rather straightforward, will be given in the next section.The advantage of working with symbols shifted only in is that we can identify a larger class of perturbations, where the same spectral results as proved in [11] hold true.More precisely, instead of + we will consider + ( ) where satisfies the following assumptions: Hypothesis 1.2.Let ∈ [ ∞ (R )] be a smooth real vector-valued function with all its derivatives of all order uniformly bounded (thus can grow linearly at infinity).
We will only work with symbols of class 0 0,0 (R × R ) (included in the Sjöstrand class of symbols considered in [11]) since they are more suitable for the less symmetric perturbation which we consider.

The main results
We start by recalling the definition of the Hausdorff distance between any two compact sets , ⊂ R: Our first main result gives a sharp upper bound on how much the spectra can "move" as sets: The next straightforward corollary spells out in a detailed way how the interior non-trivial gaps in the spectrum of 0 may vary with .Corollary 1.4.Assume that 0 has an open spectral gap ( 0 , 0 ) with 0 , 0 ∈ ( 0 ).Then there exists a constant > 0 (the same as in Theorem 1.3), independent on the spectral gap, such that for all

non-empty and belongs to the resolvent set of . Moreover, both intervals
The next main result states that the spectral edges of have a Lipschitz variation at = 0: The next corollary describes the variation of the edges of those interior gaps which remain open at = 0, and gives a precise control with respect to the width of the spectral gap: Corollary 1.6.Consider the same setting and the same notation as in Corollary 1.4.
, and both sets ( ) ∩ (−∞, ( 0 + 0 )/2) and ( ) ∩ (( 0 + 0 )/2, ∞) are non-empty, we may define Then there exists a constant ˜ > 0, independent of 0 − 0 , and some Remark 1.7.Corollary 1.6 is stronger than Corollary 1.4 only when | | is much smaller than the width of the gap 0 − 0 .An important point is that the constant in Corollary 1.4 is independent of the gap, while the Lipschitz constant in Corollary 1.6 is inverse proportional with the width of the gap at = 0.This is compatible with Theorem 1.3: when | | increases and becomes of order ( 0 − 0 ) 2 , the gap might even close.
Remark 1.8.When ( ) = , the results of Theorem 1.5 and Corollary 1.6 are also obtained in [11].On the other hand, the results of Theorem 1.3 and Corollary 1.4 are new.We note that if one is only interested in proving Lipschitz behavior of the inner gap edges and , one does not need the explicit estimate in our Theorem 1.3, but only some a-priori knowledge of their continuity, as in [11].

Technical preliminaries
We consider that Lemma 1.1 is a rather well-known fact and omit its proof.

Known facts about the Hausdorff distance between spectra
The following lemma is well-known but also very important, hence we prove it for completeness, see also [9].Lemma 2.1.Let and be self-adjoint and bounded.Let + ( ) = sup ( ), − ( ) = inf ( ), and ± ( ) denotes the same for .Then Proof.Let us prove the first inequality but only for " + ".Let us assume, without loss of generality, that + ( ) ≤ + ( ).Then Now let us prove the second inequality.Let ∉ ( ).We have and ∉ ( ).This means that the spectrum of is located within a neighborhood of width − of the spectrum of .The same conclusion holds for replaced with .
Another useful inequality is the following.Lemma 2.2.Let , , , be bounded self-adjoint operators.Then Proof.Direct application of the triangle inequality and of Lemma 2.1.

Reduction to compact support in the second variable of the distribution kernel.
We refer to Hypothesis 1.2 for the notation involving and .
Lemma 2.3.Let 0 ≤ ≤ 1 be smooth and compactly supported, with ( ) = 1 in a neighborhood of 0. Let ˜ be the operator with the integral kernel ˜ ( , Proof.We may assume ≥ 0. Denote by ˆ the Fourier transform of .The symbol of ˜ is a convolution: where we used that (0) = 1.Using the integral Taylor formula we have for every ≥ 1.Using that all the partial derivatives of at zero equal zero, we may write the symbol of − ˜ as ′ , for all ≥ 1.After a change of variables, this symbol reads as for all ≥ 1.This symbol obeys (1.1) where the supremum is bounded by /2 for all .An application of the Calderón-Vaillancourt Theorem [3] finishes the proof.

A localization result.
Let 0 ≤ ≤ 1 be smooth with compact support such that  Then there exists a constant independent of | | ≤ 1 such that Proof.Given ∈ Z we denote by the set of all ′ ∈ Z with the property that the support of ˜ , ′ has a non-empty overlap with the support of ˜ , , including ′ = .Denote by ∈ N \ {0} the cardinal of ; it is clearly independent of and .For ∈ 2 (R ): where in the last equality we used (2.2).

Proof of Theorem 1.3
For simplicity, let 0 ≤ ≤ 1.From Lemma 2.3 and Lemma 2.1 we infer that the Hausdorff distance between the spectra of and ˜ is of order ∞ .Let us define the operator ˚ through its integral kernel given by ˚ ( , ) = ( √ ) 0 ( , ).Then with the same proof as in Lemma 2.3 one can show that ˚ − 0 = O( ∞ ), and the same is true for the Hausdorff distance between their spectra.Therefore, according to the second inequality in (2.1), it is enough to prove that the Hausdorff distance between the spectra of ˜ and ˚ is of order √ .Let be the unitary operator induced by the translation with − , i.e. ( ) ( ) = ( − ); we use the notations introduced in Lemma 2.4 and work with Γ , i.e. with ˜ = .We shall prove the following statement.Proposition 3.1.Let ∈ C be in the resolvent set of ˚ defined above.Let us define: Proof.Let us consider the following distribution kernel: .
Denoting by ∇ 1 the partial gradient with respect to the " ∈ R " variables, we can write the above distribution kernel as: From our Hypothesis 1.2 we have | ( ) − ( )| ≤ | − |, hence both above kernels correspond to 0 0,0 symbols due to the fact that the growth in | − | is controlled by the Gaussian factor.
Moreover, in the second kernel we can couple one power of with the quadratic term | − | 2 and thus we can bound this second kernel by a constant times and conclude that all the seminorms of its associated symbol will be of order .
For the first kernel we use the Taylor expansion: The remarkable fact is that the linear term vanishes identically after integration in .
The quadratic term can be dealt with as we did with the second kernel concluding that it will generate an operator with norm of order .
Using the notation from Lemmas 4.1 and 4.2, the inequality from Lemma 4.3 reads as: Another crucial observation is that the operator with the integral kernel given by ( √ ) appearing in (4.3), is unitarily equivalent, by a conjugation with ( ) , with the operator denoted from now on by given by the distribution kernel 2 16 0 ( , ).
These operators have the same spectrum, for all ∈ R , thus from (4.3) and Lemma 4.1 we have Finally, we see that the operator − ˚ has the integral kernel The factor 2 multiplied by 0 will generate (by the usual integration by parts procedure for oscillatory integrals) some second order derivatives in of the symbol ( , ), while the Gaussian is just a smooth function depending on √ , which on the support of remains bounded.Hence the operator − ˚ has a norm bounded by , which together with (4.4) implies: i.e. + ( ˜ ) ≤ + ( ˚ ) + .The inequality where ˜ and ˚ exchange places can be proved in a similar way.

Proof of Corollaries 1.4 and 1.6
Corollary 1.4 is just a direct consequence of the definition of the Hausdorff distance.For Corollary 1.6 we use a similar trick with the one used in [11].Let 0 = (2 0 + 0 )/3 and let us define Let us assume for the moment that ± ( ) are Lipschitz at = 0, a fact which we will prove later.If ≥ 0 is small enough, then by using the spectral theorem, the fact that 0 is closer to 0 than to 0 , and the a-priori estimate from Theorem 1.3 which says that The Lipschitzianity of − ( ) implies the existence of a constant 1 > 0 such that: we have some small enough 1 < 0 such that and we are done.
The only thing which remains to be proved is that ± ( ) are Lipschitz at = 0. We start with a lemma.
Using a Taylor expansion for together with the strong localization of ℌ 0 in the variable, one may also show that for every ≥ 1 there exists > 0 such that Up to a use of the Schur test in Thus up to an error of order in operator norm, we have that 2 is given by , ′ ∈Z ′′ ∈Z (• + ) ˜ , ′′ ( ) ˜ ′′ , ′ ( ) (• + ′ ).
Finally, by again using a Taylor expansion and a Schur test, one shows that this operator and M ( 2 0 ) differ from each other by something of order in the operator topology, and the proof is finished.

Theorem 1 . 3 .
Consider the notation introduced in Hypothesis 1.2.Then there exists > 0 such that ℎ ( ), ( 0 ) ≤ | | for all | | ≤ 1.This bound is sharp, in the sense that one can construct a 0 such that 0 ∈ ( 0 ) while the spectrum of develops gaps of order | | near zero.