Resolvent Estimates Near the Boundary of the Range of the Symbol

Abstract

The purpose of this chapter is to give quite explicit bounds on the resolvent near the boundary of Σ(p) (or more generally, near certain “generic boundary-like” points.) The result is due (up to a small generalization) to Montrieux (Estimation de résolvante et construction de quasimode pres du bord du pseudospectre, 2013) and improves earlier results by Martinet (Sur les propriétés spectrales d’opérateurs nonautoadjoints provenant de la mécanique des fluides, 2009) about upper and lower bounds for the norm of the resolvent of the complex Airy operator, which has empty spectrum (Almog, SIAM J Math Anal 40:824–850, 2008). There are more results about upper bounds, and some of them will be recalled in Chap. 10 when dealing with such bounds in arbitrary dimension.

6.A Appendix: Quick Review of Almost Holomorphic Functions

Almost analytic functions have been introduced by Hörmander [81], Nirenberg [108] and Dynkin [42]. We prefer the terminology “almost holomorphic functions” to emphasize that these functions are defined in the complex domain. The work [81] has been very important for the author’s works and we refer to the more recent commentary by the author (JS) at the end of the online version of [81], with pointers to more recent works.

If ω ⊂R ⁿ and Ω ⊂C ⁿ are open with Ω ∩R ⁿ = ω, and if u ∈ C ^∞(ω), there exists a function $\widetilde {u}\in C^\infty (\Omega )$ such that

$$\displaystyle \begin{aligned} \widetilde{u}{\vert}_{\omega }=u, \end{aligned} $$

(6.A.1)

$$\displaystyle \begin{aligned} \overline{\partial }_z\widetilde{u}(z)={\mathcal{O}}(|\Im z |{}^\infty ),\mbox{ locally uniformly on }\Omega . \end{aligned} $$

(6.A.2)

Here in the last equation, we write

$$\displaystyle \begin{aligned}\overline{\partial }u=\overline{\partial }_z\widetilde{u}=\partial _{\overline{z}_1}\widetilde{u}d\overline{z}_1+\cdots+\partial _{\overline{z}_n}\widetilde{u}d\overline{z}_n\simeq (\partial _{\overline{z}_1}\widetilde{u},\ldots,\partial _{\overline{z}_n}\widetilde{u})\end{aligned}$$

with $\partial _{\overline {z}_j}=(1/2)(\partial _{\Re z_j}+i\partial _{\Im z_j})$ and the meaning of the estimate is that $\overline {\partial }\widetilde {u}(z)={\mathcal {O}}(|\Im z|{ }^N)$ locally uniformly on Ω for every N ∈N. We call $\widetilde {u}$ an almost holomorphic (rather than almost analytic) extension of u. The Taylor expansion of $\widetilde {u}$ is uniquely determined by that of u at every real point, so if $\widehat {u}$ is another almost holomorphic extension of the same function u, then $\widetilde {u}-\widehat {u}={\mathcal {O}}(|\Im z|{ }^\infty )$ locally uniformly on Ω (or on the intersection of the domains of the two extensions, when these domains are different). The extension $\widetilde {u}$ can be constructed by means of truncated Taylor expansions and if 0 < r ∈ C(ω), we can choose $\widetilde {u}$ with support in $\{ z\in \Omega ;\, |\Im z|<r(\Re z) \}$. See [81, 102] for further details.

In the case n = 1, let γ : [a, b] →C be a smooth injective curve with injective differential, −∞ < a < b < +∞, and let u be a smooth function on γ(]a, b[). Then we can find a smooth function $\widetilde {u}$ on neigh (γ(]a, b[), C) such that

$$\displaystyle \begin{aligned} \begin{aligned} \widetilde{u}\vert_{\gamma (]a,b[)}&=u,\\ \overline{\partial }\widetilde{u}&={\mathcal{O}}(\mathrm{dist}\,(\cdot ,\gamma (]a,b[)) ^\infty) , \end{aligned} \end{aligned} $$

(6.A.3)

locally uniformly, on the neighborhood of definition. Again, $\widetilde {u}$ is unique mod ${\mathcal {O}}(\mathrm {dist}\,(\cdot , \gamma (]a,b[))^\infty )$ locally uniformly, and we call $\widetilde {u}$ an almost holomorphic extension.

Indeed, if $\widetilde {\gamma }$ is an almost holomorphic extension of γ, then by the implicit function theorem, $\widetilde {\gamma }$ is a diffeomorphism: neigh (]a, b[ , C) →neigh (γ(]a, b[), C) and letting $\widetilde {\gamma }^{-1}$ denote the inverse, we can take

$$\displaystyle \begin{aligned} \widetilde{u}=\widetilde{ u\circ \gamma }\circ \widetilde{\gamma }^{-1}, \end{aligned} $$

(6.A.4)

where $\widetilde {u\circ \gamma }$ denotes an almost holomorphic extension of u ∘ γ. This can be generalized to higher dimensions, n ≥ 2, by replacing the range of a curve by a totally real manifold of maximal dimension. (See [102].)

Notes

The main result of this chapter, Theorem 6.1.4, is a small generalization of a result of Montrieux [18]. In Chap. 10 we will review more general results in all dimensions, which are less precise however.

Metaplectic (Fourier integral) operators (in the semi-classical limit) are Fourier integral operators L ²(R ⁿ) → L ²(R ⁿ) with quadratic phase functions and constant amplitudes. The phase function is assumed to satisfy standard nondegeneracy assumptions which imply that the operator can be associated to an affine canonical transformation κ : T ^∗ R ⁿ → T ^∗ R ⁿ. Such an operator U with non-zero amplitude, associated to κ has a bounded inverse U ⁻¹ associated to κ ⁻¹. If A = a ^w(x, hD;h) is the h-Weyl quantization of the symbol a and U is a metaplectic operator as above, then B := U ⁻¹ AU is the h-Weyl quantization of the symbol b = a ∘ κ. This is an exact form of Egorov’s theorem. It suffices that $a\in {\mathcal {S}}'(T^*{\mathbf {R}}^n)$. See e.g., [41, Chapter 7, Appendix].