# Localization for a Multi-Particle Quantum Graph

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## Multi-particle quantum graphs

In the first part of this thesis, we study the N -particle analog of the model described in Section 1.1.3. We call it a multi-particle quantum graph to emphasize the fact that not only the Schrödinger operator H changes in this case, but also the geometric structure Γ, namely it is no longer a collection of vertices and edges.
To study the interaction between two particles, one lying on an edge e1 and the other lying on an edge e2, one intuitively has to consider the couple (e1, e2). So instead of studying a collection of edges E , one now has to study a collection of edge couples or rect-angles by identification. So one obtains a kind of two-dimensional network of rectangles. More generally, to study the interaction between N particles, one will have to consider an N -dimensional network of rectangular boxes. This is just the intuitive picture 1. More formally, if Γ(1) ⊂ Rd is the set introduced in Section 1.1.3, then the Hilbert space one considers for the study of 1-particle systems on Γ(1) is H1 = L2(Γ(1), dm(1)). Now quantum mechanics tells us that the Hilbert space corresponding to N distinguishable
particles, each living in Γ(1) is the tensor product HN = H1 ⊗ ⊗ H1. Taking the cartesian product
Γ(N ) := Γ(1) × × Γ(1) ⊂ (Rd)N and the product measure m := m(1) ⊗ ⊗m(1), HN may be identified 2 with L2(Γ(N ), dm). Each x = (x1, , x N ) ∈ Γ(N ) takes the form xk = mk + tk hjk , for some mk ∈ Zd, jk ∈ {1, , d } and tk ∈ [0, 1]. Hence, if for m = (m1, , m N ) ∈ (Zd)N and j = (j1, , j N ) ∈ {1, , d }N we put κm,j := [m1, m1 + hj1 ] × × [mN , mN + hjN ] .

### Ideas of the proof

The proof of these theorems is based on the multi-particle multi-scale analysis devel-oped by A. Boutet de Monvel, Chulaevsky and Suhov in  and . The multi-scale induction becomes significantly more involved in the multi-particle setting because one loses the independence at a distance. The passage from multiscale analysis bounds to localization is based on the approach of Germinet and Klein .
Before adapting these methods to multi-particle quantum graphs, there is a list of input to verify. Here are the main diﬃculties that lie in the passage from Γ(1) to Γ(N ).
First of all in the Combes-Thomas estimate, we need to have a good control on the decay exponent, because here the Combes-Thomas estimate is used not only to derive the initial length scale estimate, but also in the multiscale induction itself. Roughly speaking, the exponent should depend (as usual) on the distance of the energy to the spectrum, but not on the absolute value of the energy. We derived such a Combes-Thomas estimate using semigroups, via an improved Davies-Gaﬀney estimate. Next we had to derive some trace estimates which are necessary to obtain generalized eigenfunction expansions.
Next there is the eigenfunction decay inequality (EDI): in the usual derivation, one needs to be sure that the generalized eigenfunctions have some regularity, say they are locally of class W 1,2. However, to use the approach of Germinet and Klein, we needed the generalized eigenfunction expansion usually attributed to Berezansky (see ) whose functions are not a priori regular. Instead of proving they are indeed regular 3, we were able to derive EDI by means of approximations in the negative Hilbert space (that is, the Hilbert space in which the generalized eigenfunctions live). Finally we needed a stronger initial length scale estimate for H (1) than the one proved in  and this is why we proved some Lifshitz-type asymptotics. The perturbation argument used to derive them in the continuum also works here, but at some point one needs to know the width of the gap between the first two eigenvalues of H (1) restricted to a finite graph. It turns out that this problem had been studied in Riemannian geometry, and one way to estimate this gap is by means of Cheeger Inequality 4. This inequality was later proved for finite quantum graphs by Nicaise in  and it also appeared in an article by Post .
These are the main diﬃculties. There are also some details to settle to obtain the rest of the input, namely the geometric resolvent inequalities and the Wegner estimate. For the former, note that besides the usual resolvent inequality needed for single-particle models, one needs an additional one in the multi-particle setting, namely one that compares the Green functions of a certain type of N -cubes to those of n-cubes for n < N . A Combes-Thomas estimate with a good exponent enters again in the proof. For the latter, the main diﬀerence in the N -particle setting is that one needs to make sure that the random potential is well distributed in any 1-particle direction. More precisely, the Wegner bound one really needs for localization is not one that estimates the probability on the whole space Ω, but a stronger conditional one instead. This is needed to prove the two-volume bound later on.

#### Variations

In Chapter 3, we give a variation of these results. Namely, we are able to prove exponential localization and strong dynamical localization of any order without relying on generalized eigenfunction expansions. However, using this approach, we cannot achieve it in the Hilbert-Schmidt norm. We also explain how far our results can be generalized if the distribution of the random variables is only log-Hölder continuous. Finally, we give a Combes-Thomas estimate which is valid for energies in arbitrary spectral gaps, but the price to pay is that the proof becomes quite complicated.

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Abstract Wegner estimates

The second part of this thesis is concerned with the formulation of some abstract Wegner estimates and their application to concrete models.

Wegner estimates in general

The objective of Wegner estimates is to derive good bounds on the average number of eigenvalues of a random Schrödinger operator HΛ(ω) restricted to a cube Λ in a fixed interval I. More precisely, if χI (HΛ(ω)) is the spectral projection of HΛ(ω) onto the interval I, then one seeks estimates of the type E{tr[χI (HΛ(ω))]} ≤ CW · |Λ|α · |I|β for some CW ≥ 0, α ≥ 1 and β > 0. Such estimates can be used in a proof of localization via multiscale analysis, or in the study of continuity properties of the integrated density of states (IDS) if α = 1.

Motivation

The motivation for this work came again from quantum graphs. A very interesting feature of quantum graphs is that one can study randomness not only in the potential, but also in the graph itself. This gives rise to two new models: quantum graphs with random vertex couplings (RCM, i.e. random boundary conditions) and quantum graphs with random edge lengths (RLM); see ,  and  for localization results for these models.
I was first interested in the RLM, so I studied the articles  and . These works establish many results including localization near the spectral edge and continuity of the IDS. I focused on the Wegner bound, which I found to be particularly interesting as the dependence on the random parameter becomes quite unusual for such models.
The precise description of these models would be too long, so I will only discuss two major tricks that were applied in these works to obtain the Wegner bound. In , the idea was to establish a relationship between the Schrödinger operator H (lω ) on the quantum graph ⊕e E L2(0, leω ) and a discrete counterpart M (lω , I) which acts on the set of vertices, i.e. in the Hilbert space ℓ2(V ). Here I is the interval where the Wegner bound is to be established. This simplified the task a little bit, because for finite subgraphs (EΛ, VΛ), the space ℓ2(VΛ) is finite-dimensional. However, the dependence of M (l ω , I) on lω = (leω ) becomes diﬃcult (terms like (sin leω )−1 arise), so new arguments have to be conceived.
In , the first idea was to work with the random variables (ln leω )e E instead of the random variables (leω )e E . This makes the dependence on the random parameter simpler, and some precise relationships between the eigenvalues can be derived. Of course the probability space needs to be modified in a corresponding way. The conclusion is not immediate because even with this, the dependence on the random parameter is unusual. Both works assumed the random variables (leω ) are i.i.d. with an absolutely continuous distribution. So I asked myself how we could relax this condition and work with a Hölder continuous distribution.
A second motivation came from the usual Schrödinger operators on ℓ2(Zd) and L2(Rd), but which have no covering condition (i.e. the single site potentials do not cover all points in Zd or all regions in Rd respectively). For these operators the available proofs of the Wegner bound are complicated if one does not assume the probability measure has a density, see e.g. .

Introduction (version Française)
0.1 Généralités
0.2 Graphe quantique multi-particule
0.3 Estimées de Wegner abstraites
0.4 Perspectives
1 Introduction
1.1 Background
1.2 Multi-particle quantum graphs
1.3 Abstract Wegner estimates
1.4 Perspectives
2 Localization for a Multi-Particle Quantum Graph
2.1 Multi-particle Quantum Graphs
2.2 Finite-volume operators and geometry of cubes
2.3 Combes-Thomas estimate
2.4 Geometric Resolvent Inequalities
2.5 Wegner Estimates
2.6 Initial Length Scale Estimate
2.7 Multi-Particle Multiscale Analysis
2.8 Generalized Eigenfunctions
2.9 Exponential Localization
2.10 Dynamical Localization
2.11 Appendix
3 Additional properties and alternative proofs
3.1 Ergodicity and almost sure spectrum
3.2 Combes-Thomas estimate for general energies
3.3 Regularity of generalized eigenfunctions
3.4 Localization without generalized eigenfunctions: prelude
3.5 Localization
3.6 Log-Hölder continuous distributions
4 Abstract Wegner estimates with applications
4.1 Introduction
4.2 Outline
4.3 Abstract Theorems
4.4 Applications
4.5 Proofs of the general theorems
4.6 Appendix
5 More results on Wegner bounds
5.1 A refinement of the discrete bounds
5.2 Discrete bounds without UP
5.3 More applications
5.4 Some difficulties
5.5 A deterministic variant of the UP
A Generalized Eigenfunction Expansions
A.1 Introduction
A.2 Main results