Remarks on counting negative eigenvalues of Schr\"odinger operator on regular metric trees

Abstract

We discuss estimates on the number $N_-(\alpha)$ of negative eigenvalues of the Schr\"odinger operator $-\Delta-\alpha V$ on regular metric trees, as depending on the properties of the potential $V\ge 0$ and on the value of the large parameter $\alpha$. We obtain conditions on $V$ guaranteeing the behavior $N_-(\alpha)=O(\alpha^p)$ for any given $p\ge 1/2$. For a special class of trees we show that these conditions are not only sufficient but also necessary. For $p>1/2$ the order-sharp estimates involve a (quasi-)norm of $V$ in some `weak' $L_p$- or $\ell_p(L_1)$-space. We show that the results can be easily derived from the ones of an earlier paper by Naimark and the author, Proc. London Math. Soc. (3) 80 (2000), 690-724. The results considerably improve the estimates found in the recent paper by Ekholm, Frank, and Kova\v{r}\'{i}k, arXive:0710.5500.

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arXiv:0809.0752v1 [math.SP] 4 Sep 2008
REMARKS ON COUNTING NEGATIVE
EIGENVALUES OF SCHR¨ODINGER OPERATOR ON
REGULAR METRIC TREES
MICHAEL SOLOMYAK
To Victor Petrovich Khavin, a friend and colleague, on his 75-th birthday
Abstract. We discuss estimates on the number N−(HαV) of neg-
ative eigenvalues of the Schr¨ odinger operator HαV = −∆−αV on
regular metric trees, as depending on the properties of the potential
V ≥ 0 and on the value of the large parameter α. We obtain con-
ditions on V guaranteeing the behavior N−(HαV) = O(αp) for any
given p ≥ 1/2. For a special class of trees we show that these con-
ditions are not only sufficient but also necessary. For p > 1/2 the
order-sharp estimates involve a (quasi-)norm of V in some ‘weak’
Lp- or ℓp(L1)-space. We show that the results obtained can be
easily derived from the ones of [8].
The results considerably improve the estimates found in the
recent paper [5].
1. Introduction
intro
1.1. Preliminaries. In this note we discuss estimates for the num-
ber N−(HV) of the negative eigenvalues of the Schr¨ odinger opera-
tor HV = −∆ − V on regular metric trees. The classical Birman –
Schwinger principle allows one to reduce such estimates to the study
of the spectrum of a certain compact, self-adjoint operator acting in
an appropriate Hilbert space; see Section 5 below for an explanation
of this reduction. The detailed study of this spectrum was initiated by
Naimark and the author [8], and some estimates for N−(HV) immedi-
ately follow from there.
Further results on the behavior of N−(HV) were obtained recently
by Ekholm, Frank, and Kovaˇ r´ ık [5]. The main novelty in [5] is an
analysis of the connection between the estimates for N−(HV) and the
global dimension of a regular tree. This is an important step in the
understanding of spectral properties of the Schr¨ odinger operator on
trees.
Date: September 4, 2008.
1

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2MICHAEL SOLOMYAK
The proofs in [5] are based upon a scheme developed by Naimark
and Solomyak [8, 9], and independently by Carlson [4].
goal in this note is to show that the estimates obtained in [5] can be
considerably improved by the direct reduction to the results of [8]. We
also show that the estimates in terms of the ‘weak Lp-spaces’ obtained
by this approach, are order-sharp in the strong coupling limit. This
means that when applied to the operator HαV, where α > 0 is a large
parameter, these estimates are of the type
Our main
estim
(1.1) N−(HαV) = O(αp),α → ∞,
where ‘O’ cannot be replaced by ‘o’. On the contrary, in the estimates
obtained in [5] in terms of the standard Lp, such replacement is always
possible.
We also discuss (in Theorem 4.3) estimates of a different nature,
where the function N−(HαV) is evaluated in terms of a certain number
sequence associated with the potential. In Theorem 4.4 we single out
a sub-class of regular trees, for which the conditions guaranteeing the
order N−(HαV) = O(αp) with p > 1/2 are not only sufficient, but also
necessary.
Any metric tree has local dimension one. As a consequence, for the
rapidly decaying potentials V one always has N−(HαV) = O(α1/2).
For the slowly decaying potentials, such that (1.1) is satisfied with
some p > 1/2, the estimates in Lp-terms are never order-sharp: any
order-sharp estimate must involve a norm, or a quasi-norm, of some
non-separable function space. In this connection, see a discussion in
[11]. The results of the present paper once more substantiate this claim.
prel
1.2. Geometry of a tree. Let Γ be a rooted metric tree, with the
root o. We suppose that the number of vertices and the number of
edges are infinite, and that each edge has finite length. The distance
ρ(x,y) and the partial ordering x ? y on Γ are introduced in a natural
way, and we write x ≺ y if and only if x ? y and x ?= y. We denote
|x| = ρ(o,x). On each edge e ⊂ Γ the partial ordering turns into the
standard ordering defined by the inequality |x| ≤ |y|. With each vertex
v ∈ Γ we associate its generation Gen(v), i.e. the number of vertices
w such that o ? w ≺ v. In particular, Gen(o) = 0. The branching
number b(v) of a vertex v is the number of edges emanating from v.
We suppose that b(o) = 1 and that 1 < b(v) < ∞ for any vertex v ?= o.
In this paper we consider the regular trees, see [8, 9] for more detail.
A tree Γ is said to be regular (or symmetric, or radial), if all vertices of
the same generation have equal branching numbers and lie on the equal
distance from the root. Each regular tree is completely determined by

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NEGATIVE EIGENVALUES OF SCHR¨ODINGER OPERATOR ON GRAPHS3
specifying two number sequences, {bn} and {tn}, where bn= b(v) and
tn= |v| for any vertex v of generation n, n = 0,1,.... We suppose that
the sequence {tn} is unbounded. The branching function of Γ, defined
as
g
(1.2)g0(t) = b0b1...bnfor tn< t ≤ tn+1,n = 1,2,...
is an important characteristic of a regular tree.
The natural measure dx on Γ is induced by the Lebesgue measure
on the edges. The spaces Lp(Γ) are understood as the Lebesgue spaces
with respect to this measure. The norm in Lp(Γ) is denoted by ? · ?p;
for p = 2, we write simply ? · ?.
sob
1.3. Sobolev spaces on a tree. Differentiation, in the direction com-
patible with the ordering on Γ, is well-defined at any point x ∈ Γ except
for the vertices. The Sobolev space H1(Γ) consists of all continuous
functions u on Γ, such that u↾e ∈ H1(e) on each edge e ⊂ Γ, and the
condition
?u?2
Γ
is satisfied. It is easy to show that any function u ∈ H1(Γ) vanishes as
|x| → ∞, see, e.g., Lemma 3.1 in [12]. It immediately follows that the
functions u ∈ H1(Γ) with bounded support are dense in H1(Γ). The
boundary condition u(o) = 0 selects the subspace
H1(Γ):=
?
(|u′(x)|2+ |u(x)|2)dx < ∞
H1,0(Γ) ⊂ H1(Γ).
The functions u ∈ H1,0(Γ) with bounded support are dense in this
subspace.
Along with H1,0(Γ), we need also the homogeneous Sobolev space
H1,0(Γ) formed by all continuous functions u on Γ, such that u(o) = 0,
u↾e ∈ H1(e) on each edge e ⊂ Γ, and u′∈ L2(Γ). By definition,
?u?H1(Γ)= ?u′?.
In contrast with the space H1,0(Γ), the functions u ∈ H1,0(Γ) with
bounded support are not always dense in H1,0(Γ). As a consequence,
the set H1,0(Γ), which is evidently contained in H1,0(Γ), is not always
dense in the latter space. This property depends on geometry of the
tree. A tree such that H1,0(Γ) is dense in H1,0(Γ) is called recurrent,
otherwise it is called transient. We shall denote the closure of H1,0(Γ)
in H1,0(Γ) by H◦(Γ). So, H◦(Γ) = H1,0(Γ) if and only if the tree Γ is
recurrent.
A simple necessary and sufficient condition of transiency is known
for the regular trees, see, e.g., [9], Theorem 3.2. Namely, such tree is

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4MICHAEL SOLOMYAK
transient if and only if its reduced height
trans
(1.3)l(Γ) :=
?
Γ
dt
g0(t)
is finite.
2. Schr¨ odinger operator on Γ.
schr
Our main object in this paper is the Schr¨ odinger operator on Γ,
sch
(2.1)
HαV = −∆D− αV,
with the non-negative potential V (x) and a large parameter (the cou-
pling constant) α > 0. In (2.1) the symbol ∆Dstands for the Dirichlet
Laplacian on Γ. In Section 6 we discuss also an analogue of (2.1) for
the Neumann Laplacian ∆N.
The operator (2.1) is defined via its quadratic form which is
?
We are interested in estimating the number of the negative eigenval-
ues of the operator HαV in terms of the potential V and the coupling
constant α.
shrqf
(2.2)
hαV[u] =
Γ(|u′|2− αV |u|2)dx,u ∈ H1,0(Γ).
Assume that the potential V ≥ 0 is such that the inequality
?
is satisfied. By the continuity, it extends to all u ∈ H◦(Γ). We assume
that CV is the least possible constant in (2.3). We call any such po-
tential V a Hardy weight on H◦(Γ), and we say that the Hardy weight
is normalized, if CV = 1. If V is a Hardy weight, the operator (2.1)
is well-defined at least for small values of α. Some further assump-
tions about V , that we impose later, imply finiteness of the negative
spectrum of this operator.
In general, for a self-adjoint, bounded from below operator H in
a Hilbert space H, whose negative spectrum is finite, we denote by
N−(H) the number of its negative eigenvalues counted according to
their multiplicities.
hw
(2.3)
Γ
V |u|2dx ≤ CV
?
Γ
|u′|2dx, ∀u ∈ H1,0(Γ)
For estimating the quantity N−(HαV) it is important to have a de-
scription of the Hardy weights on H◦(Γ). Such description, for general
trees and the Hardy weights on a wider class H1,0(Γ), was obtained
in [6]. A similar result for H◦(Γ) seems to be unknown so far. The

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NEGATIVE EIGENVALUES OF SCHR¨ODINGER OPERATOR ON GRAPHS5
situation changes if we restrict ourselves to the regular trees and to the
symmetric weights
V (x) = v(|x|),∀x ∈ Γ
where v ≥ 0 is a measurable function on [0,∞). For the regular trees
an exhaustive description of all symmetric Hardy weights can be eas-
ily derived from the classical Hardy-type inequalities, see, e.g., [10], or
Section 1.3 in the book [7]. The result of such reduction was presented
in [9], Theorem 5.2, but only for the transient regular trees. We re-
produce it below, including also the recurrent case. The proof for this
case remains the same.
prhardy
Proposition 2.1. Let Γ be a regular tree, and let g0(t) be its branching
function (1.2). Suppose V (x) = v(|x|) ≥ 0 is a symmetric weight
function on Γ.
1◦Let Γ be recurrent. Then V (x) is a Hardy weight on H◦(Γ) if and
only if
??∞
Moreover, the least possible constant CV in (2.3) satisfies
B0(v) := sup
t>0
t
v(s)g0(s)ds ·
?t
0
ds
g0(s)
?
< ∞.
B0(v) ≤ CV ≤ 4B0(v).
2◦Let Γ be transient. Then V (x) is a Hardy weight on H◦(Γ) if and
only if the following two conditions are satisfied:
??t
?
Moreover,
hardy
(2.4)B1(v) := sup
t>t1
t1
v(s)g0(s)ds ·
?∞
t
ds
g0(s)
?
< ∞;
hardy1
(2.5)B2(v) := sup
t<t1
t ·
?t1
t
v(s)ds
?
< ∞.
CV ≤ 4(B1(v) + B2(v));B1(v) ≤ CV; B2(v) ≤
?
1 +
b1t1
t2− t1
?
CV.
3. Classes of functions, of number sequences, and of
compact operators
clas
As we shall see, the sharp estimates of the function N−(HαV) in-
volve the weak Lp-spaces on the tree, rather than the classical Lebesgue
spaces Lp. Here we recall their definitions. See, e.g., [1], Section 1.3,
for more detail.