Quantitative uncertainty principles and the first non zero ... · An uncertainty principle for...

An uncertainty principle for graphsDFG-AIMS Workshop “EVOLUTIONARY PROCESSES ON NETWORKS” AIMS Rwanda

Quantitative uncertainty principles and the first non–zeroeigenvalue on graphs

Peter Stollmann, joint work with Daniel Lenz, Marcel Schmidt and Gunter Stolz

DFG-AIMS Workshop “EVOLUTIONARY PROCESSES ON NETWORKS” AIMS Rwanda

March 2018

· March 2018 · Peter Stollmann 1 / 22 http://www.tu-chemnitz.de/mathematik/analysis

http://www.tu-chemnitz.de/mathematik/analysis

Introduction and road map

I Uncertainty principles – unique continuation: classical topic . . .functions with low energy are spread out in space

I Uncertainty principles: important for random Schrodinger operators . . .I . . . many recent results: starting with Bourgain and Kenig, then Klein,

Rojas-Molina, Veselic . . .I . . . unique continuation NOT TRUE for graphs, but . . .I a spectral uncertainty principle gives nice uniform results for a large class of

graph laplacians.I KEY: [Spectral] geometry for graphs – lower bounds for Dirichlet Laplacians on

subsets



Introduction and road mapThe big picture



Introduction and road mapThe set up

Topology – Geometry:

A weighted graph (X, b) is given by

I a countable set X , finite orinfinite;

I a symmetricb : X ×X → [0,∞) calledthe weight function ( =conductance) satisfyingb(x, x) = 0 for all x ∈ X .

I edge stands for positive weight




Measuring length:

I path: γ = (x0, x1, ..., xk)where b(xj , xj+1) > 0 for allj = 0, ..., k − 1

I length of a path γ:

L(γ) :=

k−1∑j=0

1

b(xj , xj+1).

I distance between x and y:

d(x, y) := infL(γ) | γ a path from x to y; d(x, x) := 0.

Assume that (X, b) is connected⇒ d(x, y) <∞; d is pseudo–metric.




ExampleClassical undirected graphs: X at most countable, b : X ×X → 0, 1 indicateswhether there is an edge or not.In this case: d(x, y) is the combinatorial distance, counting the minimal number ofedges for a path connecting x and y.

Our results are interesting and new for combinatorial graphs. One last quantity: Theenergy of f ∈ F(X) := RX :

E(f) :=1

2

∑x,y∈X

b(x, y)|f(x)− f(y)|2 ∈ [0,∞].

Under appropriate conditions: E(f) <∞ for any

f ∈ Fc = f ∈ F(X) | supp(f) is finite .



Introduction and road mapRoad map

I A topological Poincare inequalityI Introducing a measureI Lower bounds in the finite volume caseI Lower bounds in the infinite caseI The uncertainty principleI Outlook



Topological Poincare inequality

Proposition (d satisfies a topological Poincare inequality)Let x, y ∈ X be arbitrary. Then for any path γ = (x0, . . . , xk) from x to y and f ∈ Dthe inequality

|f(x)− f(y)|2 ≤ L(γ)

k−1∑j=0

b(xj , xj+1)(f(xj)− f(xj+1))2

holds. In particular|f(x)− f(y)|2 ≤ d(x, y)E(f)

is valid.



Topological Poincare inequality

Proof:It suffices to show the first inequality. Take a path γ = (x0, . . . , xk) from x = x0 toy = xk:

|f(x)− f(y)| ≤k−1∑j=0

|f(xj)− f(xj+1)|

=

k−1∑j=0

1

b(xj , xj+1)12

· b(xj , xj+1)12 |f(xj)− f(xj+1)|

≤ L(γ)1/2

k−1∑j=0

b(xj , xj+1)(f(xj)− f(xj+1))2

12

,

using the triangle inequality and the Cauchy-Schwarz inequality.



Introducing a measure

Let m : X → (0,∞) and denote the corresponding measure by m as well.This gives a Hilbert space `2(X,m) and we can view E as a symmetric (bilinear) formon `2(X,m), giving an associated operator H ...Now we can introduce:

λN1 = λN1 (X, b,m) := infE(f) | ‖f‖2 = 1, f ⊥ 1

“first [non-zero] Neumann eigenvalue”

and for Ω X :

λD0 = λD0 (Ω;X, b,m) := infE(f) | ‖f‖2 = 1, f = 0 on D := X \ Ω.

“first [non-zero] Dirichlet eigenvalue” on Ω.

Both correspond to the graph Laplacian induced by E .



Bounds in the finite volume, finite diameter case

Now consider the case that X (Ω, respectively) has finite volume m(X) <∞ andfinite diameter diamd(X) := supd(x, y) | x, y ∈ X <∞.

TheoremAssume diamd(X) <∞. Then, for any finite measurem onX of full support,

λN1 ≥4

diamd(X) ·m(X).

... follows by a clever decomposition of f ⊥ 1

‖f‖22 ≤1

4supx,y∈X (f(x)− f(y))

2m(X) ≤ 1

4diamd(X)m(X)E(f)

and the pointwise bound from the Proposition.



Bounds in the finite volume, finite diameter case

TheoremAssume Ω X , Inr(Ω) <∞ andm(Ω) <∞. Then

λD0 ≥1

Inr(Ω)m(Ω).

I Let R > Inr(Ω), x ∈ Ω.I ∃x0 ∈ X \ Ω, path γ fromx to x0 s.t. L(γ) < R.

I Proposition gives

|f(x)|2 ≤ L(γ)E(f).

I Summing over Ω gives theestimate.



The infinite volume case

The estimateλD0 ≥

1

Inr(Ω)m(Ω)

does not help, if m(Ω) =∞! Of course, we assume Inr(Ω) <∞. Way out: Voronoidecomposition.

I Let the geometry be nice andR > Inr(Ω):

I ∃ decomposition Vα, α ∈ Aof X .

I such that Vα ∩D 6= ∅I and diam(Vα) < R

I apply finite estimate on each ofthe cells,

‖f Vα ‖22 ≤ Rm(Vα)E(f Vα).



The infinite volume case

TheoremAssume Ω X , Inr(Ω) <∞ and

vol][Inr(Ω)] := infs>Inr(Ω)

supm(Us(x)) | x ∈ X <∞

ThenλD0 ≥

1

Inr(Ω)vol][Inr(Ω)].

We need that Ω is relatively dense, Inr(Ω) <∞ and that the corresponding relativevolume vol][Inr(Ω)] is bounded. New result, even for very special cases like thecombinatorial lattice Zd, for which there had been earlier estimates by Rojas–Molinaand Elgart/Klein.



The uncertainty principle




For what follows we assume that (X, b,m) is such that the corresponding energy formE and hence the associated graph Laplacian H is bounded, and that the geometry isnice enough, namely(B) Assume supx∈X

1m(x)

∑y∈X b(x, y) =: δ <∞,

so that (Hf)(x) = 1m(x)

∑y∈X b(x, y)(f(x)− f(y)) is bounded by 2δ.

(M) Moreover, assume mmax := supx∈X m(x) <∞

Corollary (Qualitative version)Let (X, b,m) be as above,D ⊂ X relatively dense, i.e. Ω := X \D has finite inradius.Let f have “small energy” in the sense that f ∈ Range(PI(H)), wheremax I < 1

Inr(Ω)·vol][Inr(Ω)]. Then

‖f1D‖2 ≥ κ‖f‖2.




Corollary (Quantitative version)Let (X, b,m) be as above,D ⊂ X relatively dense, i.e. Ω := X \D has finite inradius.Let I ⊂ R such that max I < 1

Inr(Ω)·vol][Inr(Ω)]and f ∈ Range(PI(H)). Then

‖f1D‖2 ≥( 1

Inr(Ω)·vol][Inr(Ω)]−max I)2

16‖H + 1‖4‖f‖2.



Outlook

I A suitable distance d(x, y) provides insight into spectral properties of graphs.I This leads to lower bounds on eigenfunctions of low energy.I Possible extensions to spectral geometry on quantum/metric graphs.I Applications of lower bounds for functions of low energy to more complicated

networks.



Outlook

M. Barlow, T. Coulhon and A. Grigor’yan, Manifolds and graphs with slow heatkernel decay, Invent. Math. 144 (2001), 609–649

A. Boutet de Monvel, D. Lenz and P. Stollmann, An uncertainty principle, Wegnerestimates and localization near fluctuation boundaries, Math. Z. 269 (2011),663–670.A. Elgart and A. Klein, Ground state energy of trimmed discrete Schrodingeroperators and localization for trimmed Anderson models, J. Spectr. Theory 4(2014), 391–413.

D. Lenz, M. Schmidt, and P. Stollmann, Topological Poincare type inequalities andlower bounds on the infimum of the spectrum for graphs, arXiv preprint, arXiv:1801.09279 (2018)

D. Lenz, P. Stollmann and Gunter Stolz: An uncertainty principle and lower boundsfor the Dirichlet Laplacian on graphs. arXiv: 1606.07476, to appear in J. Spectr.Theory

C. Rojas-Molina, The Anderson model with missing sites, Oper. Matrices 8 (2014),287–299.



Quantitative uncertainty principles and the first non zero ... · An uncertainty principle for...

Documents

Transcript of Quantitative uncertainty principles and the first non zero ... · An uncertainty principle for...