Spectral inequality for the Schr odinger equation g + V ( x ) in R - - PowerPoint PPT Presentation

Spectral inequality for the Schr¨

• dinger equation

−△g + V (x) in Rd

Gilles Lebeau †, IvanMoyano‡

†D ´ epartementdeMath´ ematiques, Universit´ e Nice Sophia Antipolis, lebeau@unice.fr ‡CenterforMathematicalSciences, im449@dpmms.cam.ac.uk

Marrakech, April 19, 2018 Premier (?) Congr` es Franco Marocain de Math´ ematiques Appliqu´ ees

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Outline

1

Spectral inequality on a compact manifold

2

Spectral inequality for −△ in Rd, a short review

3

Main result

4

Sketch of proof Interpolation for holomorphic functions Holomorphic extensions

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Outline

1

Spectral inequality on a compact manifold

2

Spectral inequality for −△ in Rd, a short review

3

Main result

4

Sketch of proof Interpolation for holomorphic functions Holomorphic extensions

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Spectral inequality on a compact manifold

Let (M, g) be a smooth connected compact manifold. Let △g be the (negative) Laplace operator on M, and let −△gej = λjej, 0 = λ0 < λ1 ≤ λ2 ≤ ... be its spectral decomposition. The following spectral inequality was proved by Jerison-Lebeau and Lebeau-Zuazua in 1998-1999.

Theorem

Let ω ⊂ M be a non void open subset of M. There exists constants A = A(ω) > 0, C = C(ω) > 0 such that for all λ > 0 and all sequence {zj}j∈N of complex numbers, one has

• λj<λ

|zj|2 ≤ AeCλ1/2

ω

|

• λj<λ

zjej(x)|2dgx . (1.1)

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Outline

1

Spectral inequality on a compact manifold

2

Spectral inequality for −△ in Rd, a short review

3

Main result

4

Sketch of proof Interpolation for holomorphic functions Holomorphic extensions

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

In the rest of the talk, we will work in Rd, and ω ⊂ Rd will denote a measurable set satisfying the density assumption: ∃R, δ > 0, such that inf

x∈Rd mes {t ∈ ω, |x − t| < R} ≥ δ.

(2.1)

Theorem

Let ω ⊂ Rd be a measurable set satisfying the geometric condition (2.1). There exists constants A = C(ω), C = C(ω) > 0 such that the following hods true. For all µ > 0 and all f ∈ L2(Rd), such that support (ˆ f ) ⊂ {ξ ∈ Rd, |ξ| ≤ µ}, where ˆ f is the Fourier transform of f , one has f 2

L2(Rd) ≤ AeCµ

• ω

|f (x)|2dx. (2.2)

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Using techniques from Harmonic Analysis, Logvinenko and Sereda proved (Tero. Funk. Anal. i Prilozen, 1974) that in 1-d, the condition E ⊂ R measurable s.t. ∃γ > 0, a > 0 s.t. mes(E ∩ I) mes(I) ≥ γ (2.3) whenever I is an interval of length a, is sufficient to ensure that, when support(ˆ f ) ⊂ [−b, b]: ∃C = C(γ, a, b) > 0 s.t.

• E

|f (x)|2dx ≥ Cf 2

L2(R).

(2.4) On the other hand, the authors were not able to quantify the dependence

• f C with respect to the parameters a, b, γ. This was achieved in the
• ne-dimensional case by Kovrojkine (The Uncertainty principle for

relatively dense sets and lacunary spectra, 2002) where the author proves that ∃K > γ such that C(γ, a, b) = γ K ab+1 ,

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Outline

1

Spectral inequality on a compact manifold

2

Spectral inequality for −△ in Rd, a short review

3

Main result

4

Sketch of proof Interpolation for holomorphic functions Holomorphic extensions

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Let g be a given Riemannian metric on Rd. Let ∆g be the (negative) Laplace operator defined by the metric g. Let V = V (x) a real potential function such that limx→∞ V (x) = 0. One defines the Schr¨

• dinger operator associated to (g, V ) by

Hg,V := −1 2∆g + V (x) (3.1) With reasonable hypothesis on (g, V ), Hg,V is a (bounded from below) unbounded self adjoint operator in L2(Rd), and its spectrum σ(g, V ) ⊂ [E0, ∞[ satisfies the following. σ(g, V )∩] − ∞, 0[ is purely discrete with eigenvalues of finite multiplicity, and its only possible accumulation point is 0. σ(g, V )∩]0, ∞[ is absolutely continuous. For E ∈ R, we denote by ΠE the spectral projector on ] − ∞, E[ associated to Hg,V . .

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

We will assume that (g, V ) satisfy the following hypothesis (H) The metric g and the potential V are real analytic and there exists a > 0 such that they extend holomorphicaly in the complex domain Ua = {|Im(z)| < a}. One has g = Id + ˜ g, where ˜ g is a symbol of degree < 0 in Ua. V is a symbol of degree < 0 in Ua. – Observe that even in the case g = Id, the assumption on the potential V allows long range perturbation. Short range perturbations are associated to potentials V which are symbols of degree < −1 in Ua. For the analysis of scattering theory for long range perturbation, we refer to H¨

• rmander, in The analysis of linear pde’s vol 4, ch. XXX.

– Observe also that the metric g may have trapped trajectories.

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Main result

For E ∈ R, we define E 1/2

±

by E 1/2

±

= √ E for E ≥ 0, E 1/2

±

= ±i

• |E|

for E < 0.

Theorem

Let (g, V ) satisfying hypothesis (H). There exists constants A = A(ω, g, V ), C = C(ω, g, V ) > 0 such that for all E ∈ R and for all f ∈ L2(Rd), one has ΠEf L2(Rd) ≤ A|eCE 1/2

± | ΠEf L2(ω).

(3.2)

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Main result

Observe that under the hypothesis (H), which allows long range perturbation, we may have dim(range(Π0)) = ∞. In particular, inequality 3.2 implies that any function f ∈ range(Π0) satisfies: f (x) = 0 for all x ∈ ω ⇒ f = 0. In fact, in the course of the proof, we will show that any f ∈ range(Π0) extends holomorphicaly in Ua for a > 0 small enough. Thus uniqueness holds true for any measurable set ω of positive measure.

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Outline

1

Spectral inequality on a compact manifold

2

Spectral inequality for −△ in Rd, a short review

3

Main result

4

Sketch of proof Interpolation for holomorphic functions Holomorphic extensions

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

The first main ingredient is to use the following ” classical” interpolation inequality for holomorphic functions. Let ω ⊂ Rd satisfying the hypothesis (H). There exists constants Cint = Cint(ω, a) > 0 and δ = δ(ω, a) ∈ (0, 1) such that

• Rd |f |2dx ≤ Cint
• ω

|f |2dx δ

Ua

|f |2|dz| 1−δ , (4.1) for any f ∈ L2(Ua) ∩ H(Ua) satisfying sup

0≤b<a

• |y|=b

|f (x + iy)|2dx < ∞ (4.2)

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Poisson kernel

We denote by dπE(x, y) the kernel of the spectral measure of Hg,V , i.e dπE(f , g) =

• Rd×Rd f (x)g(y)dπE(x, y),

f , g ∈ L2(Rd). Recall that dπE(f , f ) is the positive measure on the line E ∈ R equal to the derivative of the left continuous and non decreasing function E → ΠE(f )2

L2.

The Poisson kernel Ps,±(x, y) is the smooth function on ]0, ∞[×Rd × Rd given by the formula Ps,±(x, y) =

• R

e−sE 1/2

± dπE(x, y).

(4.3) For any f ∈ L2(Rd), the smooth function on ]0, ∞[×Rd defined by u(s, x) =

• Rd Ps,±(x, y)f (y)dy satisfies the elliptic boundary problem

(−∂2

s + Hg,V )u = 0,

lim

s→0+ u(s, x) = f (x) in L2(Rd).

(4.4)

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

An analog of the Boutet de Monvel theorem

The second main ingredient is the following Lemma which is an extension

• f the analytic form of the famous result of Louis Boutet de Monvel on the

extension in the complex domain of eigenfunctions of an elliptic operator

• n a compact riemannian manifold.

Lemma

Let u(s, x) be a solution of the elliptic equation (−∂2

s + Hg,V )u = 0 on

]0, ∞[×Rd. Then for a > 0 and δ > 0 small enough, u extends holomorphically in the open set Ba = {(s, z) ∈ C × Cd Re(s) > 0, |Im(z)| ≤ min(a, δ|Re(s)|)},

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Zerner Lemma

The proof of the above Lemma uses the following classical Zerner Lemma.

Lemma

(Zerner) Let Q(z, ∂z) =

α,|α|≤m qα(z)∂α z be a linear differential

• perator with holomorphic coefficients defined near 0 in CN and let

q(z, ζ) =

|α|=m qα(z)ζα be its principal symbol. Let f : CN → R be a

C 1 function such that f (0) = 0 and such that, with ζ0 = 2i∂f (0), one has q(0, ζ0) = 0. Then, if u(z) is an holomorphic function defined in a half-neighborhood of 0 in f < 0, such that Q(u) extends holomorphically near 0, then u extends holomorphically near 0. Recall that the Zerner Lemma was the starting block for the introduction by M. Kashiwara of micro-hyperbolic tools in the analysis of PDE’s, and it leads further to the construction by M. Kashiwara and P. Schapira of the so called Microlocal Sheaf Theory.

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Simple pseudo-differential calculus

In order to prove our main result, we will also use the following simple ” pseudo-differential calculus” defined by the symbolic calculus of the self adjoint unbounded operator Hg,V . For a measurable and bounded function χ(E) on R, the operator A(χ) = χ(Hg,V ) has the following kernel: A(χ)(x, y) =

• R

χ(E)dπE(x, y). In particular, one has Ps,±(x, y) = A(e−sE 1/2

± ).

Then, we use the following elementary fact: For any given λ, and any f such that Πλ(f ) = f , one has Ps,±f = Ps−δ,±A(χ)(f ), χ(E) = eδE 1/2

± 1E<λ

and we use Lemma 4.1.

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Fr` eres humains, qui apr` es nous vivez, N’ayez les coeurs contre nous endurcis, Car, si piti´ e de nous pauvres avez, Dieu en aura plus tˆ

• t de vous mercis.

Fran¸ cois Villon La ballade des pendus, 1462.

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20

Hier Encore

Puisqu’en France, tout finit par des chansons: H´ elas, mille fois H´ elas ....

Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality Marrakech, April 19, 2018 Premier (?) Congr` es / 20