Eigenvalue Estimates for the Weighted Laplacian on Metric Trees

Abstract

The Laplacian on a metric tree is Δ u = u″ on its edges, with the appropriate compatibility conditions at the vertices. We study the eigenvalue problem on a rooted tree Γ: −λnormalΔu=Vu1emon normalΓ,2emufalse(ofalse)=0. Here V ⩾ 0 is a given ‘weight function’ on Γ, and o is the root of Γ. The eigenvalues for such a problem decay no faster than λn = O(n−2), this last case being typical for one‐dimensional problems. We obtain estimates for the eigenvalues in the classes lp, with p > ½, and their weak analogues lp, ∞ with p ⩾ ½. The results for p<1 and p>1 are of different character. The case p<1 is studied in more detail. To analyse this case, we use two methods; in both of them the problem is reduced to a family of one‐dimensional problems. One of the methods is based upon a useful orthogonal decomposition of the Dirichlet space on Γ. For a particular class of trees and weights, this method leads to the complete spectral analysis of the problem. We illustrate this in several examples, where we were able to obtain the asymptotics of the eigenvalues. 1991 Mathematics Subject Classification: primary 47E05, 05C05; secondary 34L15, 34L20.

Remarks about Hardy inequalities on metric trees

“…Orthogonal decomposition. In this subsection we recall the results of Carlson [C] and of Naimark and Solomyak [NS1,NS2]. For each integer k ≥ 1 we define the k-th branching functions g k : R + → N by…”

“…However, due to the existence of harmonic functions with finite Dirichlet integral, the Hardy weights have to decay rather fast. This led Naimark and Solomyak [NS1] to the study of (1.1) for functions in {u ∈ C ∞ 0 (Γ) : u(o) = 0} (and its closure with respect to the Dirichlet integral). For regular trees, see Subsection 2.1 for the definition, they gave a complete characterization of the validity of (1.1) on that class of functions.…”

“…Following the approach in [NS1] we can give a complete characterization of admissible weights and obtain two-sided estimates on the sharp constant C(ψ, Γ) in (1.1). As in the Euclidean case, Hardy weights typically decay like |x| −2 as |x| := dist(o, x) → ∞.…”

“…In this subsection we would like to recall some basic definitions about trees and fix our notation. We refer, e.g., to [NS2,NS1,S] for details. Let Γ be a rooted metric tree with root o.…”

Reconstruction of the Sturm-Liouville operators on a graph with δ′ s couplings

“…The question of physical meaning of such a coupling on graphs was addressed and a pair of simple nontrivial examples of the so-called δ ′ s couplings was presented in [12,13]. Recently, the spectral problems of quantum graphs have become a rapidly-developing field of mathematics and mathematical physics, and spectral properties of quantum graphs and different inverse problems have been studied in both forward [20,21,22,31,39] and inverse [4,23,32,38,40,41,42], etc. Nowadays there are only a number of papers devoted to inverse nodal problems for differential operators on graphs (for example, refer to [8,11,41,42]).…”