[Paper Review] An uncertainty principle and lower bounds for the Dirichlet Laplacian on graphs

Daniel Lenz, Peter Stollmann+1 more|arXiv (Cornell University)|Jun 23, 2016

Spectral Theory in Mathematical Physics30 references3 citations

TL;DR

This paper establishes a quantitative uncertainty principle for low-energy states of the Dirichlet Laplacian on weighted graphs, proving explicit lower bounds for Dirichlet eigenvalues in terms of geometric invariants such as inradius and volume growth. By leveraging spectral theory and a Voronoi-type decomposition to reduce infinite graphs to finite components, the authors derive uniform estimates that hold even in the presence of compactly supported eigenfunctions, extending known results to general graphs with explicit control over constants via geometric data.

ABSTRACT

We prove a quantitative uncertainty principle at low energies for the Laplacian on fairly general weighted graphs with a uniform explicit control of the constants in terms of geometric quantities. A major step consists in establishing lower bounds for Dirichlet eigenvalues in terms of the geometry.

Motivation & Objective

• To establish a quantitative uncertainty principle for low-energy states of the graph Laplacian, even in settings where compactly supported eigenfunctions exist.
• To derive explicit lower bounds for Dirichlet eigenvalues in terms of geometric quantities like inradius and volume growth.
• To provide uniform estimates with explicit constants depending only on geometric data, applicable to broad classes of weighted graphs.
• To extend these results to Schrödinger operators with bounded potentials by using ground state transforms and metric equivalence.

Proposed method

• Utilizes a spectral theoretic uncertainty principle from [11], relating spectral projections to a weight operator W=1_D on a relatively dense set D.
• Applies the variational principle and functional calculus to derive a lower bound on the spectral projection norm via the energy shift λ_t = minσ(H + tW).
• Reduces the infinite graph to a disjoint union of finite graphs via a Voronoi-type decomposition, enabling finite-volume estimates.
• Establishes a lower bound for the Dirichlet Laplacian on Ω = X\D via the inradius R and volume growth: H_Ω ≥ 1/(R · vol[B_R]), where vol[B_R] is the volume of a ball of radius R.
• Uses the ground state transform to relate the Schrödinger operator with potential V to a transformed graph (X, b_φ, m_φ), preserving geometric structure up to a constant factor c.
• Applies the main bound to the transformed graph and translates the result back to the original setting, yielding a lower bound involving c^4 and rescaled volume growth.

Experimental results

Research questions

• RQ1 Can a quantitative uncertainty principle be established for low-energy states of the graph Laplacian despite the existence of compactly supported eigenfunctions?
• RQ2 What geometric quantities control the lower bounds for Dirichlet eigenvalues on weighted graphs?
• RQ3 How can spectral projections of the Laplacian be uniformly bounded below in terms of geometric invariants like inradius and volume growth?
• RQ4 To what extent can such bounds be extended to Schrödinger operators with bounded potentials on graphs?
• RQ5 How does the ground state transform preserve geometric estimates under metric and measure equivalence?

Key findings

• A lower bound for the Dirichlet Laplacian on Ω = X\D is established as H_Ω ≥ 1/(R · vol[B_R]), where R is the inradius of Ω and vol[B_R] is the volume of a ball of radius R.
• The bound is stronger than known results when volume growth exceeds linear, and holds under mild assumptions on the weighted graph.
• For Schrödinger operators with bounded potential V, if a regular ground state φ exists with bound c, then the Dirichlet restriction satisfies E_V,Ω ≥ λ_V + 1/(c^4 · R · vol[c^2 R]).
• Under the volume doubling condition with exponent N, the bound improves to E_V,Ω ≥ λ_V + 1/(c^{4+2N} · R · vol[R]).
• The method applies uniformly across large classes of graphs, including Z^d and other lattices, with explicit constants in terms of geometric parameters.
• The approach avoids reliance on unique continuation in the classical sense, instead using spectral and geometric control to derive uncertainty principles for low-energy states.