Schrödinger Operators on Graphs and Geometry II. Integrable Potentials and an Ambartsumian Theorem

ISSN: 1401-5617

Schr¨ odinger Operators on Graphs and Geometry II. Integrable Potentials and an Ambartsumian

Theorem.

Jan Boman, Pavel Kurasov, Rune Suhr

Research Reports in Mathematics

Number 1, 2016

Electronic version of this document is available at http://www.math.su.se/reports/2016/1

Date of publication: Oktober 06, 2016.

2010 Mathematics Subject Classification:

Primary 34L25, 81U40; Secondary 35P25, 81V99.

Keywords: Ambartsumian Theorem, Euler characteristic, inverse problem, metric graph, quantum graph, Schr¨ odinger operator.

Postal address:

Department of Mathematics Stockholm University S-106 91 Stockholm Sweden

Electronic addresses:

http://www.math.su.se/

info@math.su.se

Schr¨odinger Operators on Graphs and Geometry II: Integrable Potentials and an Ambartsumian

Theorem.

Jan Boman, Pavel Kurasov, Rune Suhr

Abstract

In this paper we study Schr¨ odinger operators with absolutely inte- grable potentials on metric graphs. Uniform bounds – i.e. depending only on the graph and the potential – on the difference between the n

^th

eigenvalues of the Laplace and Schr¨ odinger operators are obtained. This in turn allows us to prove an extension of the classical Ambartsumian Theorem which was originally proven for Schr¨ odinger operators with Neu- mann conditions on an interval. We also extend a previous result relating the spectrum of a Schr¨ odinger operator to the Euler characteristic of the underlying metric graph.

1 Introduction

Quantum graphs — i.e. Schr¨ odinger operators acting on metric graphs — have become an increasingly important branch of mathematical physics in the last 20 years or so. They serve as models of branched thin networks, e.g. nanotubes, and complex molecules. Apart from applications they also serve as a rich source of objects suitable for mathematical inquiry. In particular Schr¨ odinger operators on graphs exhibit spectral properties both of partial differential and ordinary differential operators.

The aim of this paper is twofold. First we work towards proving a spectral es- timate, i.e. a comparison between the spectra of the Laplacian and Schr¨ odinger operators acting on the same metric graph. The main motivation for obtaining such estimates is that the spectrum of the Laplacian is much easier to calculate.

We will prove that just as in the case of a single interval the difference between

the Laplace and Schr¨ odinger eigenvalues is uniformly bounded, provided the

potential is just absolutely integrable. The classical proof of this relied heavily

on the explicit formula for the resolvent kernel of the Laplacian [7]. The corre-

sponding kernel for metric graphs is not in general given explicitly and we will

instead work with general perturbation theory. The second goal is to extend

results relating the Laplace spectrum to geometric properties of the graph to

Schr¨ odinger operators with L -potentials. In particular we obtain an inverse

spectral theorem that generalizes the celebrated theorem of Ambartsumian (see section 7), and show that the Euler characteristic of the underlying graph is reflected in the spectrum of the Schr¨ odinger operator in the case of standard vertex conditions (see (1) below).

1.1 Spectral estimates and inverse spectral theory.

Inverse spectral theory for the Schr¨odinger equation in R

ⁿ

has classically had as a goal to determine the potential given a spectrum. To solve the inverse problem for a Schr¨odinger operator on a metric graph completely one has in general to determine not only the potential but the underlying metric graph and vertex conditions. It appears that this complete inverse problem is rather difficult, especially since the set of spectral data is not obvious. Therefore it appeared attractive to start investigations assuming that the metric graph is known and that the vertex conditions are fixed. The simplest graph to be considered is the star graph formed by a finite number of compact edges and the corresponding inverse problem resembles very much the inverse problem on a single interval, where the potential is determined by two spectra [13, 30, 33, 34, 35] The case of trees has also been studied and we have rather good understanding of the problem [8, 37, 2, 31]. The case of graphs with cycles is much more involved - major difficulties are related to reconstruction of the potential on the cycles. To assure uniqueness one may either use the dependence of the spectral data on the magnetic fluxes [20, 22], or add extra spectral data like the Dirichlet spectrum, which cannot be measured without destructing the graph [38, 39, 40, 41].

The problem to reconstruct the metric graph has not been addressed in full generality. Topological characteristics of the graph may be reconstructed [18, 19]. Assuming that the edge lengths are rationally independent one may even reconstruct the graph using trace formula [14, 17], but assuimg that the potential is zero. Explicit examples of isospectral graphs have been constructed [14, 4]. Employing boundary control method one may reconstruct metric trees [2] without assuming zero potential. The problem of reconstructing vertex con- ditions, as well the influence of vertex conditions on the solvability of the inverse problem, is even less understood [3]. For example a metric tree is not always reconstructable for other than standard vertex conditions [21]. On the other hand more general vertex conditions may help to solve the inverse problem like it is done in [22, 23].

One result in the inverse spectral theory – the celebrated Ambartsumian

theorem from 1929 stating that the spectrum of a Schr¨ odinger operator on an

interval coincides with the spectrum of the (Neumann) Laplacian if and only if

the potential is zero – is of great importance. This theorem is rather special,

since in order to reconstruct a non-zero potential knowledge of two spectra

is required. Different authors generalized this theorem for the case of metric

graphs. The results can be divided into two categories. Several authors proved

that if the spectrum of the standard Schr¨ odinger operator on a fixed metric

graph Γ coincides with the spectrum of the standard Laplacian on Γ, then

potential is zero. This result was first proved for trees [29], [36], [27], [10] and

[30], and then for arbitrary graphs by B. Davies [11]. It was also noted that the Laplacian on a metric graph Γ is isospectral to the Laplacian on interval if and only if Γ is formed by a single interval. This can be seen as a geometric version of Ambartsumian’s theorem and is based on the fact that single interval maximizes the spectral gap for the Laplacian among graphs of fixed total length [28, 12, 25].

The inverse spectral result that we obtain – Theorem 7.5 – is that the spec- trum of a finite interval is unique among connected finite compact quantum graphs with standard conditions. More precisely, the spectrum of a Schr¨ odinger operator on a connected metric graph coincides with the spectrum of the Lapla- cian with Neumann conditions on an interval, then the graph is the interval, and the potential of the Schr¨odinger operator is zero. Here we assume that standard conditions are imposed on the graph - see section 2.

In the case of the graph being a compact interval, standard conditions coin- cide with Neumann conditions at the end-points. The result can therefore be seen as an extension of Ambartsumian’s (1929) celebrated theorem ([1]), which asserts the uniqueness of the spectrum of the free Neumann-Neumann problem for a compact interval (see Theorem 7.1).

1.2 Outline of the paper

Section 2 contains some fundamentals on quantum graphs. In Section 3 we give some elementary spectral estimates and prove that normalized eigenfunctions of the Laplacian L

^st₀

are uniformly bounded in the L

_∞

norm. Section 4 contains the definition of the operator L

^st_q

that we associate with the formal expression L

^st₀

+ q, for q ∈ L

¹

(Γ). The operator is defined via quadratic form methods.

Section 5 deals with spectral estimates, and we prove that there exists a uniform bound on the difference of eigenvalues of Laplace and Schr¨ odinger operators on metric graphs. This is done by using the Max-min and Min-max principles, along with a Sobolev estimate for functions ψ ∈ W

2¹

(Γ). Section 6 gives a result on the zeros of trigonometric polynomials, namely that if the zeros asymptoti- cally tend to the integers, then all the zeros are in fact exactly the integers. By combining this fact with the spectral estimate of Section 5 we are able to prove the inverse spectral theorem given in Section 7. Section 8 extends a previous result that the Euler characteristic of a graph is reflected in the spectrum of a Schr¨ odinger operator with L

_∞

potential to the case of L

1

potentials.

2 Preliminaries - basics on quantum graphs

A quantum graph is a metric graph equipped with a Schr¨ odinger operator, or

more formally, a triple (Γ, L, vc) with Γ a metric graph, L a differential operator,

and vc a set of vertex conditions imposed to connect the edges and ensure the

self-adjointness of L. In this paper we limit ourselves to compact finite graphs

(see below). We give a brief overview of these - for a thorough treatment of the theory of quantum graphs, see for example [6] and [26].

Metric graphs: A compact finite metric graph is a finite collection of compact intervals of R glued together at the endpoints. More precisely, let, {E

ⁿ

}

^Nn=1

be a finite set of compact intervals, each E

n

considered as a subset of a separate copy of R,

E

n

= [x

2n−1

, x

2n

], 1 ≤ n ≤ N.

Let V = {x

j

} = {x

2n−1

, x

2n

}

^Nn=1

denote the set of endpoints of the intervals.

Fix a partition of V into equivalence classes V = V

₁

∪ · · · ∪ V

M

.

Identifying the endpoints yields a graph with vertices given by the equivalence classes V

_i

.

We let `

n

= x

2n

− x

²ⁿ−1

denote the length of the edge E

n

and define the total length L of a graph as the sum of its edge lengths:

L = X

N n=1

`

n

.

Differential operators on graphs: L = L

q

acts as a differential operator on each edge separately, in our case −d

²

/dx

²

+q for some real potential function q - subscripts will denote the potential of the operator, and L

0

means that the potential is identically 0, so that L

₀

is the Laplacian. The action is in the Hilbert spaces L

₂

(E

_i

) of square integrable functions on the edges E

_i

. The measure on each edge is given naturally by the identification of the edge with an interval.

L then acts on

L

2

(Γ) = M

N i=1

L

2

(E

i

).

Note that if q ∈ L

1

(Γ) then the formal expression −d

²

/dx

²

+ q should be un- derstood as a sum of quadratic forms (see Section 4).

Vertex Conditions: We shall solely study quantum graphs with stan- dard conditions - also known as Kirchoff, Neumann, natural or free conditions.

Operators with standard conditions imposed will be written as L

^st_q

. Let us first discuss the case of a bounded potential q ∈ L

∞

(Γ), where the domain of the Schr¨ odinger operator can be given explicitly. For any ψ ∈ L

2

(Γ), also qψ ∈ L

2

(Γ), so L

_q

ψ = −(d/dx)

²

ψ + qψ ∈ L

2

(Γ) if an only if ψ ∈ W

2²

(Γ \ V ).

Let V

_j

= {x

j1

, . . . , x

_jk

} and let ψ(x

ji

) denote the limit ψ(x

_ji

) = lim

_x→xji

ψ(x), where the limit is taken over x inside the interval with x

ji

as an end-point.

Standard conditions are then given by imposing the following relations at each

vertex V

j

(

ψ(x

j1

) = ψ(x

j2

) = · · · = ψ(x

^jk

) P

xj∈Vm

∂

n

ψ

En(j)

(x

j

) = 0, (1) where ∂

n

denotes the normal derivative, i.e.

∂

_n

ψ(x

_j

) =

 ψ

⁰

(x

j

) x

j

left end-point

−ψ

⁰

(x

j

) x

j

right end-point.

In other words, ψ is required to be continuous in each vertex V

j

and the sum of normal derivatives should vanish there. This yields a self-adjoint operator L

^st_q

on Γ.

We will study L

^st_q

for q ∈ L

1

(Γ). This means that in general qψ 6∈ L

2

(Γ), so the formal expression L

_q

= −d

²

/dx

²

+ q lacks an immediate meaning as a sum of operators from L

2

(Γ) to L

2

(Γ). We will however establish - via a Sobolev estimate in Section 3 - that the perturbation by q is infinitesimally form-bounded (see Proposition 4.3) with respect to the quadratic form of L

0

. More precisely, the quadratic form Q

0

(ψ, ψ) = (L

^st₀

ψ, ψ) = (ψ

⁰

, ψ

⁰

) is defined on the domain of functions from W

₂¹

(Γ \ V ), which are in addition continuous at all vertices.

Then the expression L

^st₀

+ q = L

^st_q

can be assigned a meaning via the KLMN theorem (see e.g. [32]), as a self-adjoint operator with the same form domain as L

^st₀

. Note that the domain of the quadratic form only includes the continuity condition from (1). In the case of an L

_∞

potential, the second condition in (1) may be recovered from the quadratic form via partial integration, but for L

1

potentials in general only the continuity part of the vertex conditions can be given explicitly.

The spectrum: We denote the spectrum of L by σ(L(Γ)). For finite com- pact graphs the spectrum of L

₀

is discrete, for any self-adjoint boundary con- ditions, and furthermore satisfy Weyl asymptotics (see Corollary 3.3). This also holds for L

q

, for q ∈ L

¹

(Γ) (see Section 5). The n’th eigenvalue, counting multiplicities, of L

^st_q

(Γ) is denoted by λ

n

(L

^st_q

) = λ

n

(L

^st_q

(Γ)). When we write σ(L

^st_q1

(Γ

1

)) = σ(L

^st_q2

(Γ

2

)) we mean not only equality as sets, but also that the multiplicity of all eigenvalues are equal.

3 Elementary spectral properties.

In this section we start by proving upper and lower bounds on the Lapla-

cian eigenvalues λ

n

(L

^st₀

(Γ)), and in particular show that λ

n

(L

^st₀

) satisfies Weyl

asymptotics. This is of course well known, but for later purposes we need an

explicit estimate on the highest possible multiplicity of an eigenvalue. We show

in addition that normalized eigenfunctions of L

^st₀

are uniformly bounded in the

L

_∞

-norm.

Definition 3.1. For a bounded from below self-adjoint operator A with discrete spectrum, define the eigenvalue counting function E

_A

: R → N, by

E

A

(λ) = # {λ

j

∈ σ(A) | λ

j

≤ λ}.

Theorem 3.2. Let Γ be a finite compact graph of total length L with N edges and M vertices, then the eigenvalues of the standard Laplacian satisfy the fol- lowing estimates

 π L



2

(n − M)

²

≤ λ

ⁿ

(L

^st₀

) ≤  π L



2

(n + N − 1)

²

. (2) Proof. The standrad Laplacian is positive (hence the lower estimate in (2) is interesting only if n > M ) therefore when calculating the eigenvalue counting function we assume that λ ≥ 0.

Consider first the Laplace operator L

^D₀

on a single interval I of length ` with Dirichlet conditions at the end-points. The eigenvalues are λ

n

=

^πn_`

2

, n = 1, 2, . . . So the eigenvalue counting function for λ ≥ 0 is in fact given by

E

_L^D

0(I)

(λ) =

" √ λ π `

# ,

where square-brackets denote the integer-part of the argument. Returning to Γ we note that if we impose Dirichlet conditions on the vertices of Γ - denote the operator by L

^D₀

(Γ) - then we really just have a decoupled set of intervals and therefore the set of eigenvalues is just the union of the eigenvalues for each interval (counting multiplicities). Therefore the corresponding counting function E

_LD

0(Γ)

is given by E

_L^D

0(Γ)

(λ) = X

N n=1

E

_L^D

0(En)

(λ) =

" √ λ π `

1

# +

" √ λ π `

2

# + · · ·+

" √ λ π `

N

#

≤

" √ λ π L

# , (3) since taking integer parts may only decrease the value. First adding and then taking integer parts may - compared to adding the integer parts - at most raise the value by the number of terms −1, i.e. the number of edges −1, so conversely we also have " √

λ π L

#

− N + 1 ≤ E

L^D₀(Γ)

(λ). (4) Formulas (3) and (4) give effective two-sided bounds for the eigenvalues of the Dirichlet Laplacian on Γ.

We now show that ((L

^D₀

(Γ) − λ)

⁻¹

− (L

^st0

(Γ) − λ)

⁻¹

) is of finite rank, for λ ∈ ρ(L

^D0

(Γ)) ∩ ρ(L

^st0

(Γ)). Take =λ 6= 0, and suppose that

(L

^D₀

− λ)u

^D

= f, (L

^st₀

− λ)u

^st

= f.

Then for the differential operator −d

²

/dx

²

we have that



− d

²

dx

²

− λ



u

^D

− u

^st

= 0.

Since functions in the domain of L

^D

and L

^st

are continuous, to determine the rank of the resolvent difference, we need to determine dim ker 

−

dx^d22

− λ  on continuous functions. Prescribing values u

j

at V

j

for each vertex V

j

in Γ, we see that a unique continous solution to this boundary value problem is given as follows: on an edge E

n

= [x

2n−1

, x

2n

] between V

i

and V

j

, set

u(x) = u

i

sin k(x − x

²ⁿ

) sin k(x

2n−1

− x

²ⁿ

) + u

j

sin k(x − x

²ⁿ−1

) sin k(x

2n

− x

²ⁿ−1

)

for k

²

= λ. Then u is continous on Γ and solves ( −d

²

/dx

²

− λ)u = 0, so dim ker( −d

²

/dx

²

− λ) = M. Therefore we have

E

_L^st_0(Γ)

(λ) ≤ E

L^D₀(Γ)

(λ) + M ≤

" √ λ π L

#

+ M. (5)

The lower estimate estimate (4) can be modified in a similar way, but instead we shall take into account that L

^st₀

(Γ) ≤ L

^D0

(Γ). Really Dirichlet conditions in particular imply the continuity of functions in the domain of the quadratic form.

Passing to standard conditions means weakening the conditions on functions in the domain of the quadratic form, since now only continuity is required at the vertices. Therefore the domain of the quadratic form Q

_L^st₀

is larger than that of Q

_L^D₀

, so by the Min-Max principle (see Proposition 5.1) eigenvalues can only go up when imposing Dirichlet conditions. In particular, the lower bound (4) on the eigenvalue counting function E

_L^D

0(Γ)

is also valid for E

_L^st_0(Γ)

. Putting lower and upper estimates together we have

" √ λ π L

#

− N + 1 ≤ E

L^st₀(Γ)

(λ) ≤

" √ λ π L

#

+ M (6)

Setting λ =

^π_L²2

n

²

we obtain

n − N + 1 ≤ E

L^st₀(Γ)

 π

²

L

²

n

²



≤ n + M, so

λ

n−N+1

≤ π

²

L

²

n

²

≤ λ

n+M

.

Setting n

⁰

= n + M we get λ

_n0

≥

^π_L²²

(n

⁰

− M)

²

and similarly we find λ

_n0

≤

π²

L²

(n

⁰

+ N − 1)

²

, which proves the theorem.

Observe that in general no asymptotics of the form λ

_n

=

^π_L

2

n

²

+an+o(n) is valid for some a. Such an asymptotics would allow us to simplify several tedious calculations below.

The Weyl asymptotics follows as a consequence of the above theorem:

Corollary 3.3. The eigenvalues λ

n

(L

^st₀

) satisfy Weyl asymptotics, i.e.

n→∞

lim λ

_n

(L

^st₀

)

πn L

2

= 1. (7)

The multiplicities of the eigenvalues of L

^st₀

(Γ) are less than M + N , where M and N are the number of vertices and edges respectively.

Proof. The Weyl asymptotics follow from the relation

 π L



2

(n − M)

²

≤ λ

n

(L

^st₀

) ≤  π L



2

(n + N − 1)

²

The bound on the multiplicity follows from the relation

λ

n−N+1

≤ π

²

L

²

n

²

≤ λ

^n+M

.

The above corollary is widely used in quantum graph community, but we added it here to sake of completeness. The estimate on the eigenvalues multi- plicity can even be improved if one takes bridges into count [16, 24].

We shall also need the following uniform bound on the amplitude of eigen- functions of the Laplacian on finite metric graphs:

Theorem 3.4. Let Γ be a compact, finite metric graph. Then there exists a constant c = c(Γ) such that for any eigenfunction ψ of L

^st₀

(Γ)

kψk

L∞

≤ ckψk

²L2

Proof. For ψ corresponding to λ = k

²

we have kψk

²L2(Γ)

≥

Z

En

|ψ(x)|

²

dx ≥



x∈E

max

n

|ψ(x)|



2

1 2

 `

n

k 2π

 2π k ,

where [ ·] denotes the integer part of the argument. [`

n

k/2π] may be equal to zero for only finitely many k since the eigenvalues satisfy Weyl asymptotics. Since [`

n

k/2π]/k is bounded for k ∈ R this implies the existence of a k-independent bound c(Γ).

4 Definition of the operator.

The following Sobolev estimate will be used repeatedly. It will be extended in an obvious way to graphs in Corollary 4.2.

Theorem 4.1. Let ψ ∈ W

2¹

(0, `), with ` < ∞, then ψ(x) is uniformly bounded and in particular satisfies

k|ψk

²L∞

≤ kψ

⁰

k

²L2

+ 2

 kψk

²L2

(8)

for sufficiently small  > 0.

The constants, though sufficient for our needs, may be improved if one is willing to sacrifice some elegance in the proof.

Proof. Let x

min

denote a global minimum for ψ. Then in particular |ψ(x

min

) |

²

≤ kψk

²L2

/`. We then have

|ψ(x)|

²

= |ψ(x

^min

) |

²

+ 2 Z

x

xmin

ψ(y)ψ

⁰

(y)dy

≤ |ψ(x

min

) |

²

+ 2 Z

`

0

|ψ(y)ψ

⁰

(y) |dy

≤ |ψ(x

min

) |

²

+  Z

`

0

|ψ

⁰

(y) |

²

dy + 1

 Z

`

0

|ψ(y)|

²

dy

≤ kψk

²2

/` +  kψ

⁰

k

²2

+ 1

 kψk

²2

=  kψ

⁰

k

²2

+

 1

 + 1

`

 kψk

²2

. For  < ` we have 1/ > 1/` and the claim follows.

In particular any function ψ ∈ W

2¹

(Γ \ V ) will satisfy the estimate (8) on each edge E

_n

of Γ, with ` = `

_n

. As the set of edges is finite there exists an edge that has minimal length so we obtain a global estimate of |ψ(x)|

²

on Γ:

Corollary 4.2. Let Γ be a finite compact graph and ψ ∈ W

2¹

(Γ). Then kψk

²L∞(Γ)

≤ kψ

⁰

k

²L2(Γ)

+ 2

 kψk

²L2(Γ)

(9)

for sufficiently small  > 0 and x ∈ Γ.

Proposition 4.3. Let q ∈ L

¹

(Γ), and let Q

q

be the quadratic form given by Q

q

(ψ, ψ) =

Z

Γ

q(x) |ψ(x)|

²

dx,

and Q

_L^st₀

the quadratic form associated with L

^st₀

. Then for sufficiently small

 > 0 there exists b() such that

|Q

q

(ψ, ψ) | ≤ Q

L0

(ψ, ψ) + b()(ψ, ψ).

In other words Q

q

is infinitesimally bounded by Q

_L^st₀

. Proof. Theorem 4.1 shows that for sufficiently small  > 0

|ψ(x)|

²

≤ kψ

⁰

k

²L2(Γ)

+ 2

 kψk

²L2(Γ)

.

Multiplying by |q| and integrating we obtain  

  Z

Γ

q(x) |ψ(x)|

²

dx     ≤

Z

Γ

|q(x)||ψ(x)|

²

dx

≤ kψ

⁰

k

²L2(Γ)

kqk

L1(Γ)

+ 2

 kψk

²L2(Γ)

kqk

L1(Γ)

=  kqk

L1(Γ)

Q

_L^st

0

(ψ, ψ) + 2

 kqk

L1(Γ)

kψk

²L2(Γ)

. Substituting  with / kqk

L1(Γ)

, we may chose b() =

²_

kqk

²L1(Γ)

.

The KLMN theorem [32] now let us conclude that there is a unique bounded from below self-adjoint operator associated with the form Q

_L^st

0

+ Q

_q

:

Definition 4.4. For q ∈ L

¹

(Γ) we denote by L

^st_q

(Γ) the operator associated with the form Q

_L^st₀

+ Q

q

.

We note that the form domains of L

^st_q

and L

^st₀

coincide, and (L

^st_q

ψ, φ) = (L

^st₀

ψ, φ) +

Z

Γ

q(x) |ψ(x)|

²

dx, for all ψ, φ ∈ Dom(Q

Lq

) = Dom(Q

_L^st₀

).

5 Spectral estimates.

We recall the following standard variational theorems, see e.g. [32] for proofs.

Proposition 5.1 (Min-Max). Let A be a self-adjoint, bounded from below, op- erator with discrete spectrum, then the n

^th

eigenvalue of A is given by

λ

n

(A) = min

Vⁿ

max

u∈Vⁿ kukL2=1

Q

A

(u, u),

where V

ⁿ

ranges over all n-dimensional subspaces of Dom(Q

A

), the domain of the quadratic form Q

A

associated with A.

Proposition 5.2 (Max-Min). Let A be a self-adjoint bounded from below, op- erator with discrete spectrum, then the n

^th

eigenvalue of A is given by

λ

n

(A) = max

Vn−1 u⊥V

min

n−1 kukL2=1

Q

A

(u, u),

where V

n−1

ranges over all (n − 1)-dimensional subspaces of Dom(Q

A

), the

domain of the quadratic form Q

A

associated with A.

In order to apply Propositions 5.1 and 5.2 as we do in the following it is of course required that the spectrum of L

^st_q

(Γ) be discrete. For Schr¨ odinger operators with L

₁

potentials on finite intervals this is well known, and so it is true for finite metric graphs as well since these are just finite-rank perturbations – in the resolvent sense – of the Dirichlet Schr¨ odinger operators (defined by Dirichlet conditions at all vertices similar to L

^D₀

), which is nothing else than an orthogonal sum of Dirichlet Schr¨odinger operators on a collection of finite intervals.

We now proceed to prove the spectral estimate for finite compact graphs, i.e.

we show that the difference between the Laplace and the Schr¨ odinger eigenvalues is uniformly bounded:

Theorem 5.3. Let Γ be a finite compact metric graph, and let q ∈ L

1

(Γ).

Then the difference between the eigenvalues λ

_n

(L

^st₀

) and λ

_n

(L

^st_q

) is bounded by a constant, i.e.

|λ

ⁿ

(L

^st₀

) − λ

ⁿ

(L

^st_q

) | ≤ C, (10) where C = C(Γ, kqk

L1(Γ)

) is independent of n.

We are going to prove the Theorem using Propositions 5.1 and 5.2. To illustrate the strategy let us first try to derive an upper estimate for λ

_n

(L

^st_q

) using a more naive approach. In the case of standard conditions the quadratic form is

Q

L^st_q

(u, u) = Z

Γ

|u

⁰

(x) |

²

dx + Z

Γ

q(x) |u(x)|

²

dx.

It can be estimated from above by Q

_L^st_q

(u, u) ≤ Q

L^st_q+

(u, u) =

Z

Γ

|u

⁰

(x) |

²

dx + Z

Γ

q

₊

(x) |u(x)|

²

dx, (11) where q

₊

is the positive part of the potential q:

q(x) = q

₊

(x) − q

−

(x), q

_±

(x) ≥ 0. (12) This step cannot be improved much, since the new estimate coincides with the original one in the case where q is nonnegative.

The idea how to proceed is to choose a concrete n-dimensional subspace V

n⁰

, then the Rayleigh quotient will give not an exact value for λ

n

(L

^st_q

), but an upper estimate when Proposition 5.1 is used

λ

_n

(L

^st_q

) = min

Vn u∈V

max

n

Q

_L^st_q

(u, u) kuk

²L2

≤ max

u∈Vn⁰

Q

_L^st_q

(u, u) kuk

²L2

.

The only reasonable candidate for V

n⁰

we have at hand is the linear span of the Laplacian eigenfunctions corresponding to the n lowest eigenvalues

V

n⁰

= L n

ψ

₁^Lst0

, ψ

₂^Lst0

, . . . , ψ

^Ln^st0

o

. (13)

If q ≡ 0 then this estimate gives the exact value for λ

n

. Therefore it is natural to split the quadratic form as follows:

λ

n

(L

^st_q

) ≤ max

u∈Vn⁰

Q

_L^st_q

(u, u) kuk

²L2

≤ max

u∈Vn⁰

R

Γ

|u

⁰

(x) |

²

dx kuk

²L2

+ max

u∈Vn⁰

R

Γ

q

+

(x) |u(x)|

²

dx kuk

²L2

.

Then the first quotient is equal to λ

n

(L

^st₀

) and the maximum is attained on u = ψ

^Ln^st0

.

If nothing about q is known, then to estimate the second quotient one may use Z

Γ

q

+

(x) |u(x)|

²

dx ≤ kq

+

k

L1(Γ)



max

x∈Γ

|u(x)|



2

. (14)

We need to estimate |u(x)|

²

, provided u = P

n

j=1

α

j

ψ

^L_j^st0

. Since u(x) = P

n 1

α

j

ψ

_j^Lst0

and max |ψ

^Lj^st0

(x) | ≤ c (Theorem 3.4) we obtain with the Schwarz inequality

max

x∈Γ

|u(x)| ≤ X

n

1

|α

^j

| max |ψ

j^Lst0

(x) | ≤ c X

n

1

|α

^j

| ≤ c √ n

X

n 1

|α

^j

|

²

!

1/2

= c √ n kuk

L2

.

Hence R

Γ

q

₊

(x) |u(x)|

²

dx kuk

²L2

≤ kq

+

k

L1

c

²

n, and

Q

_L^st_q

(u, u) ≤ λ

n

(L

^st₀

) + c

²

n kq

+

k

L1

kuk

²L2

(15)

⇒ λ

n

(L

^st_q

) − λ

n

(L

^st₀

) ≤ kq

+

k

L1

cn, (16) i.e. we do not get an estimate uniform in n – the estimate grows linearly with n. The reason is the splitting of the quadratic form of L

^st_q

into two parts. To obtain the upper bounds we used two intrinsically different vectors: the first term is maximized if u = ψ

^Ln^st0

, while to estimate the second term we used u = ψ

₁^Lst0

+ ψ

^L₂^st0

+ · · · + ψ

^nLst0

. This is the reason that the estimate (16) is not optimal.

Proof. (Theorem 5.3) We divide the proof into two parts proving upper and

lower estimates separately. We shall also assume that n is sufficiently large,

since the precise value of a finite number of eigenvalues may affect the value of

C, but not the existence of the estimate (5.3).

Upper estimate.

As before we use the estimate λ

n

(L

^st_q

) ≤ max

u∈Vn⁰

R

Γ

|u

⁰

(x) |

²

dx + R

Γ

q

₊

(x) |u(x)|

²

dx kuk

²L2

, (17)

where V

n⁰

is defined by (13). Every function u = P

n

j=1

α

j

ψ

_j^Lst0

from V

n⁰

can be written as a sum u = u

1

+ u

2

, where

u

1

:= α

1

ψ

^L₁^st0

+ α

2

ψ

^L₂^st0

+ · · · + α

ⁿ−p

ψ

_n−p^Lst0

,

u

2

:= α

n−p+1

ψ

^L_n−p+1^st0

+ α

n−p+2

ψ

_n^L_−p+2^st0

+ · · · + α

ⁿ

ψ

n^Lst0

.

Here p is a natural number to be fixed later (independent of n, but depending on Γ and q), therefore as n increases the first function u

1

will contain an increasing number of terms, while the second function will always be given by a sum of p terms.

From the inequality R

|u

1

+ u

2

|

²

dx ≤ 2 R

|u

1

|

²

dx + 2 R

|u

2

|

²

dx and the fact that q

₊

is nonnegative we have

Z

Γ

q

₊

(x) |u

1

(x)+u

₂

(x) |

²

dx ≤ 2 Z

Γ

q

₊

(x) |u

1

(x) |

²

dx+2 Z

Γ

q

₊

(x) |u

2

(x) |

²

dx. (18) That u

1

, u

2

are orthogonal is clear, and from this the orthogonality of u

⁰₁

and u

⁰₂

also follow:

(u

⁰₁

, u

⁰₂

) = −(u

⁰⁰1

, u

2

) = −

n−p

X

i=1

λ

i

(L

^st₀

)(α

i

ψ

_i^Lst0

, u

2

) = 0.

Taking this into account we arrive at Q

L^st_q+

(u, u) ≤

Z

Γ

|u

⁰1

(x) |

²

dx + 2 Z

Γ

q

+

(x) |u

¹

(x) |

²

dx

| {z }

=: Q

_L^st

2q+

(u

1

, u

1

)

(19)

+ Z

Γ

|u

⁰2

(x) |

²

dx + 2 Z

Γ

q

₊

(x) |u

2

(x) |

²

dx

| {z }

=: Q

_L^st

2q+

(u

2

, u

2

)

.

To estimate the first form we use (14) and the Sobolev estimate (9) to estimate max |u(x)|. We get

Q

_L^st

2q+

(u

1

, u

1

) = Z

Γ

|u

⁰1

(x) |

²

dx + 2 Z

Γ

q

+

(x) |u

¹

(x) |

²

dx

≤ ku

⁰1

k

²L2

+ 2 kq

+

k

L1

max

x∈Γ

|u

1

(x) |

²

≤ ku

⁰1

k

²L2

+ 2 kq

⁺

k

^L1

( ku

⁰1

k

²L2

+ 2

 ku

¹

k

²L2

)

= (1 + 2 kq k ) ku

⁰

k

²

+ 4 kq

+

k

L1

ku k

²

.

Using

ku

⁰1

k

²L2

= (u

⁰₁

, u

⁰₁

) = −(u

⁰⁰1

, u

1

) =

n−p

X

j=1

λ

j

(L

^st₀

)(α

j

ψ

j

, α

j

ψ

j

) =

n−p

X

j=1

λ

j

(L

^st₀

) |α

j

|

²

≤ λ

n−p

(L

^st₀

)

n−p

X

j=1

|α

j

|

²

= λ

n−p

(L

^st₀

) ku

1

k

²L2

,

(20) we get

Q

_L^st

2q+

(u

₁

, u

₁

) ≤ (1 + 2kq

+

k

L1

) ku

⁰1

k

²L2

+ 4 kq

+

k

L1

 ku

1

k

²L2

≤



(1 + 2 kq

⁺

k

^L1

)λ

n−p

(L

^st₀

) + 4 kq

+

k

L1



 ku

¹

k

²L2

. The key point is that  and p can be chosen in such a way that

(1 + 2 kq

+

k

L1

)λ

_n−p

(L

^st₀

) + 4

 kq

+

k

L1

< λ

_n

(L

^st₀

) holds (see (23) below).

On the other hand, our naive approach (15) can be applied to the second form with the only difference being that the number of eigenfunctions involved is p, not n

Q

_L^st

2q+

(u

2

, u

2

) ≤ λ

n

(L

^st₀

) + 2c

²

p kq

+

k

L1

ku

2

k

²L2

. (21) Putting together the obtained estimates in (19) and using that ku

²

k

²L2

= kuk

²L2

− ku

¹

k

²L2

we get

Q

_L^st_q

(u, u) ≤



(1 + 2 kq

⁺

k

^L1

)λ

n−p

(L

^st₀

) + 4

 kq

⁺

k

^L1

 ku

¹

k

²L2

+ λ

_n

(L

^st₀

) + 2c

²

p kq

+

k

L1

ku

2

k

²L2

≤ λ

ⁿ

(L

^st₀

) kuk

²L2

+ 2c

²

p kq

⁺

k

^L1

p kuk

²L2

−



λ

n

(L

^st₀

) − (1 + 2kq

⁺

k

^L1

)λ

n−p

(L

^st₀

) − 4

 kq

⁺

k

^L1

 ku

¹

k

²L2

. We would get the desired estimate

λ

n

(L

^st_q

) ≤ max

u∈Vn⁰

Q

_L^st_q

(u, u) kuk

²L2

≤ λ

ⁿ

(L

^st₀

) + C (22)

with C = 2c

²

kq

⁺

k

^L1

p if we manage to prove that λ

n

(L

^st₀

) − (1 + 2kq

+

k

L1

)λ

n−p

(L

^st₀

) − 4

 kq

+

k

L1

> 0 (23)

for a certain  that may depend on n and p. We use the estimate for Laplacian eigenvalues given in Theorem 3.2:

 π L



2

(n − M)

²

≤ λ

ⁿ

(L

^st₀

) ≤  π L



2

(n + N − 1)

²

. (24) Substituting λ

n

(L

^st₀

) with the lower bound and λ

n−p

(L

^st₀

) with the upper and setting  = 1/n, we get the following inequality for the left-hand side of (23)

λ

n

(L

^st₀

) − (1 + 2/nkq

+

k

L1

)λ

n−p

(L

^st₀

) − 4nkq

+

k

L1

≥  π L



2

(n − M)

²

− (1 + 2/nkq

+

k

L1

)  π L



2

(n − p + N − 1)

²

− 4nkq

+

k

L1

= 2n  π L



2

p − M − N + 1 − 1 + 2

 L π



2

! kq

+

k

L1

! + O(1).

We see that for any integer p > M + N −1+(1+2(L/π)

²

) kq

⁺

k

^L1

the expression is positive for sufficiently large n and the difference between the eigenvalues possesses the uniform upper estimate:

λ

n

(L

^st_q

) − λ

ⁿ

(L

^st₀

) ≤ C. (25) If one is interested in the difference between the eigenvalues for large n only, then the constant C can be taken equal to

C = 2c

²

kqk

L1

(M + N − 1 + (1 + 2(L/π)

²

) kqk

L1

),

but this value of C may be too small in order to ensure that (25) holds for all n, since proving (23) we assumed that n is sufficiently large. The latter assumption does not affect the final result, since for a finite number of eigenvalues (25) is always satisfied, but the value of the constant C may be affected.

Lower estimate.

To obtain a lower estimate we are going to use the Max-Min principle (5.2).

The first step is to notice that Q

L^st_q

(u, u) ≥

Z

Γ

|u

⁰

(x) |

²

dx − Z

Γ

q

₋

(x) |u(x)|

²

dx. (26) Using the subspace V

n⁰−1

defined in (13) we get

λ

n

(L

^st_q

) ≥ min

u⊥Vn−1

Q

L^st_q

(u, u) kuk

²L2

. Since u is orthogonal to V

n⁰−1

it possesses the representation

u = X

∞

α

j

ψ

^L_j^st0

.

As before let us split the function u = u

1

+ u

2

u

₁

:= α

_n

ψ

^Ln^st0

+ α

_n+1

ψ

_n+1^Lst0

+ · · · + α

n+p−1

ψ

_n+p^Lst0 ₋₁

,

u

2

:= α

n+p

ψ

_n+p^Lst0

+ α

n+p+1

ψ

^L_n+p+1^st0

+ . . . . (27) Note two important differences:

• the function u

1

is given by the sum of p terms, where the number p independent of n will be chosen later, so the functions u

1

and u

2

exchange roles compared with the proof of the upper estimate;

• the function u

²

is given by an infinite series, not by an increasing number of terms as the function u

1

in the proof of upper estimate.

Using the fact that q

₋

is nonnegative we may split the quadratic form (compare (19))

Q

_L^st

q−

(u, u) ≥ Z

Γ

|u

⁰1

(x) |

²

dx − 2 Z

Γ

q

₋

(x) |u

1

(x) |

²

dx

| {z }

=: Q

_L^st

−2q−

(u

1

, u

1

)

(28)

+ Z

Γ

|u

⁰2

(x) |

²

dx − 2 Z

Γ

q

₋

(x) |u

2

(x) |

²

dx

| {z }

=: Q

_L^st

−2q−

(u

2

, u

2

)

. (29)

Now the function u

1

is given by a finite number of terms and we may similarly to (21) estimate

Q

_L^st

−2q−

(u

₁

, u

₁

) ≥ λ

n

(L

^st₀

) − c

²

kq

−

k

L1(Γ)

p

ku

1

k

²L2

. (30) To estimate the second form we use (14) and the Sobolev estimate (9) for max |u(x)|

²

. We get

Q

_L^st

2q+

(u

₂

, u

₂

) ≥ ku

⁰2

k

²L2

− 2kq

−

k

L1

max |u

2

(x) |

²

≥ ku

⁰2

k

²L2

− 2kq

−

k

L1



 ku

⁰2

k

²L2

+ 2

 ku

2

k

²L2



= (1 − 2kq

−

k

L1

) ku

⁰2

k

²L2

− 4 kq

−

k

L1

 ku

2

k

²L2

. Taking into account

ku

⁰2

k

²L2

= (u

⁰₂

, u

⁰₂

) = −(u

⁰⁰2

, u

2

) = X

∞ j=n+p

λ

j

(L

^st₀

)(α

j

ψ

j

, α

j

ψ

j

) = X

∞ j=n+p

λ

j

(L

^st₀

) |α

^j

|

²

≥ λ

n+p

(L

^st₀

) X

∞ j=n+p

|α

j

|

²

= λ

n+p

(L

^st₀

) ku

2

k

²L2

,

(31)

we arrive at Q

_L^st

2q+

(u

2

, u

2

) ≥



(1 − 2kq

−

k

L1(Γ)

)λ

n+p

(L

^st₀

) − 4 kq

−

k

L1(Γ)





ku

²

k

²L2

. (32) Summing the estimates (30) and (32) and taking into account that ku

2

k

²L2

= kuk

²L2

− ku

1

k

²L2

we get

Q

_L^st_q

(u, u) ≥ λ

n

(L

^st₀

) − c

²

kq

−

k

^L1

p
ku

¹

k

²L2

+



(1 − 2kq

−

k

L1

)λ

n+p

(L

^st₀

) − 4 kq

−

k

^L1



 ku

2

k

²L2

≥ λ

n

(L

^st₀

) kuk

²L2

− c

²

kq

−

k

L1

p kuk

²L2

+



(1 − 2kq

−

k

L1

)λ

n+p

(L

^st₀

) − 4 kq

−

k

^L1

 − λ

n

(L

^st₀

)

 ku

2

k

²L2

. As before, to prove the desired uniform estimate it is sufficient to show that for large enough n the following expression can be made positive by choosing an appropriate :

(1 − 2kq

−

k

^L1

)λ

n+p

(L

^st₀

) − 4 kq

−

k

L1

 − λ

ⁿ

(L

^st₀

) > 0. (33) Again we use (24): we substitute λ

_n+p

(L

^st₀

) with the lower bound and λ

_n

(L

^st₀

) with the upper. As before we choose  = 1/n, so the left-hand side of (33) becomes

(1 − 2kq

−

k

L1

)λ

_n+p

(L

^st₀

) − 4

 kq

−

k

L1

− λ

n

(L

^st₀

)

≥ (1 − 2kq

−

k

L1

/n)  π L



2

(n + p − M)

²

− 4nkq

−

k

L1

−  π L



2

(n + N − 1)

²

= 2n  π L



2

p − M − N + 1 − 1 + 2

 L π



2

! kq

−

k

L1

! + O(1).

If p > M + N − 1 + (1 + 2(L/π)

²

) kq

−

k

^L1

, then for sufficiently large n the expression is positive, hence the following lower estimate holds

λ

n

(L

^st_q

) − λ

n

(L

^st₀

) ≥ C, (34) where the exact value of C is determined by the difference between the first few eigenvalues as described above.

Theorem 5.3 allows us to conclude that the spectra of Schr¨ odinger operators satisfy Weyl asymptotics as well:

Corollary 5.4. Let q ∈ L

1

(Γ), then λ

n

(L

^st_q

(Γ)) satisfies Weyl asymptotics.

Proof. This is an immediate consequence of Corollary 3.3 and Theorem 5.3.

More importantly we may now show that the effect of an L

1

-perturbation on the eigenvalues will tend to zero in n in the scale of square roots. This step is critical in the proof of Theorem 7.5.

Corollary 5.5. Let λ

_n

(L

^st_q

) = k

_n,q²

and λ

_n

(L

^st₀

) = k

_n,0²

. If |λ

n

(L

^st_q

) −λ

n

(L

^st₀

) | ≤ C ∈ R then |k

n,q

− k

n,0

| ≤

^C_n⁰

, for some constant C

0

∈ R. In particular,

|k

n,q

− k

n,0

| → 0, n → ∞. (35)

Proof. Since the eigenvalues satisfy Weyl asymptotics that depend only on the length of Γ we have that k

_n,q

, k

_n,0

≥ nD for some constant D and sufficiently large n. We have

|k

^n,q

− k

^n,0

| =  

  (k

n,q

+ k

n,0

)(k

n,q

− k

n,0

) k

n,q

+ k

n,0

    =

    

k

_n,q²

− k

n,0²

k

n,q

+ k

n,0

    

=    

λ

n

(L

^st_q

) − λ

n

(L

^st₀

) k

n,q

+ k

n,0

 

  ≤ C

|k

^n,q

+ k

n,0

| ≤ C

0

n , for some constant C

₀

∈ R.

6 On the zeros of trigonometric polynomials.

In this section we recall the secular equation for the spectrum σ(L

^st₀

(Γ)): it is given as the squares of the zeros of a trigonometric polynomial. We then prove that if the zeros k

_m

of such a finite trigonometric polynomial with constant coefficients are close to a certain equispaced sequence, i.e. satisfy |k

^m

−mπ/L| → 0 then in fact k

m

= mπ/ L for all m (Theorem 6.2). From this we then prove the Ambartsumian Theorem 7.5.

Theorem 6.1. The eigenvalues λ

n

(L

^st₀

) are given by the squares of the zeros of a certain trigonometric polynomial

p(k) = X

N n=1

a

n

e

^iωnk

with k-independent coefficients a

n

∈ C and ω

n

∈ R: λ

n

(L

^st₀

) = k

_n,0²

if and only if p(k

_n,0

) = 0.

Proof. For each non-zero eigenvalue the corresponding eigenfunction is edge- wise just a sum of sine and cosine functions, on the edge E

_i

: ψ(x) = a

_i

cos(kx)+

b

_i

sin(kx). The solutions have to satisfy the vertex conditions. Continuity can at each V be written as

a

i

cos kx + b

i

sin kx = a

j

cos ky + b

j

sin ky,

if x, y ∈ V . This yields 2N − M equations, where N is the number of edges and M is the number of vertices. The conditions on the normal derivatives can at each vertex V be written as

X

xj∈V

( −1)

^j

−a

[(j+1)/2]

sin kx

j

+ b

_[(j+1)/2]

cos kx

j

= 0,

where k has been factored out. This yields an additional M equations, so that we in total have 2N equations, which may be written in the form:

T (k)~c = 0,

with ~c a vector of the coefficients a

i

, b

i

and T (k) a matrix with trigonometric entries depending on k. A real number λ = k

²

is an eigenvalue of L

^st₀

if and only if k is a root of the trigonometric polynomial det T (k) = 0. We refer to [6, 17] for details. See also eg. [14] and [26].

Note that it is crucial that the vertex conditions were standard: in the case of more general vertex conditions the secular equation is given by a quasipoly- nomial instead of the trigonometric polynomial.

Theorem 6.2. Let f be the trigonometric polynomial p(k) =

X

J j=1

a

j

e

^iωjk

(36)

with all ω

j

∈ R, a

^j

∈ C. If the zeros k

^m

of f satisfy

m

lim

→∞

(k

_m

− m) = 0 then k

m

= m for all m.

First we need a Lemma:

Lemma 6.3. Given ω

1

, . . . , ω

J

∈ R there exists a subsequence {m

ⁿ

} of the natural numbers such that

e

^iωjmn

→ 1 for each ω

j

.

Proof. For ~ ω := (ω

1

, . . . , ω

J

) ∈ R

^J

let [~ ω] := ([ω

1

], . . . , [ω

J

]) denote the image

of ω under the standard projection to the J-torus: R

^J

→ (R/2πZ)

^J

. The

statement of the Lemma is equivalent to the existence of an increasing sequence

of integers m

n

such that [m

n

ω] → ~0 := (0, . . . , 0) ∈ (R/2πZ)

^J

. Consider the set

of points m~ ω, with m ∈ N and its projection [m~ω] = ([mω

1

], . . . , [mω

J

]). Since

the J-torus is compact this set has a limit point ~z and an increasing subsequence

(m

_i

) ⊂ N such that [m

i

~ ω] → ~z. This is a Cauchy sequence so for any  > 0

there exists I() such that for any i

₁

, i

₂

≥ I()

where d( ·, ·) denotes the metric on the J-torus. Taking a sequence 

i

→ 0 we may chose i

₁

(

_i

) = I(

_i

) and in each step i

₂

(

_i

) > I(

_i

) so large that the difference

m

i

:= n

_i2(i)

− n

i1(i)

, is an increasing sequence. It follows that

[m

_i

~ ω] = [(n

_i2(i)

− n

i1(i)

)~ ω, ] → ~0 which proves the claim.

Remark. If 2π, ω

1

, . . . , ω

J

had been rationally independent then Kronecker’s classical theorem (see e.g. [15]) asserts that the integer multiples [mω] are dense in ( R/2πZ)

^J

. But as in fact each ω

_j

arises as a sum of lengths of edges in the graph (this follows from the proof of Theorem 6.1) this would impose a severe, seemingly arbitrary, and in view of Lemma 6.3, unnecessary restriction on the class of graphs under consideration.

Proof, Theorem 6.2. Consider the trigonometric polynomial p(k) in (36). De- note k

m

− m =: γ

m

so that γ

m

tends to zero as m → ∞. We have k

m

= m + γ

m

so for each ω

j

:

e

^iωjkm

= e

^iωjm

e

^iωjγm

.

Choose a subsequence {m

ⁿ

} as described in Lemma 6.3 and pass, for an arbi- trary r ∈ N, to the (r + m

n

):th zero of p. We have

0 = p(k

_r+mn

) = X

J j=1

a

_j

e

^iωj(r+mn)

e

^iωjγ(r+mn)

= X

J j=1

a

j

e

^iωjr

e

^iωjmn

e

^iωjγ(r+mn)

→ X

J j=1

a

j

e

^iωjr

= p(r),

as n → ∞. The limit follows from the choice of m

n

and the fact that γ

_(r+mn)

tends to 0.

The above calculation shows that the expressions for 0 = p(k

_r+αnν

) converges to the trigonometric polynomial P

M

m=1

a

m

e

^iωmr

as n → ∞ for any r ∈ N. But this is just p(r) which shows that p(r) = 0. But p(r) = 0 and k

r

− r → 0 together imply that k

r

= r, since even a single extra zero would make the asymptotic behavior impossible. This proves the theorem.

Remark. It is not important in the above theorem that k

n

tends to the integers.

A scaling argument allows one to extend it to the case where k

n

are close to

integer-multiples of an arbitrary real number.

7 An Ambartsumian Theorem

With the result of the previous two sections we are now in a position to prove an inverse spectral theorem that may be seen as a generalization of Ambartsumian’s classical theorem. For the proof we recall the classical result as well as its geometric version for Laplacians.

The following theorem has been a source of inspiration for researchers in inverse problems for almost a century. In the original article [1] it was assumed that the potential is continuous, but the result holds even if q ∈ L

1

. We adjusted the formulation to our notations.

Proposition 7.1. [Ambartsumian’s theorem [1]] Let q be a real-valued abso- lutely integrable function on an interval I. Then the spectrum of the standard Schr¨ odinger operator L

^st_q

(I) coincides with the spectrum of the standard Lapla- cian L

^st₀

(I) if and only if the potential q is identically equal to zero.

Standard conditions on a single compact interval is of course just the classical Neumann conditions at both end-points. It appears that the theorem is still valid if instead of the interval I we have arbitrary connected finite compact metric graph Γ. This result was proven step-by-step by several authors [29], [36], [27], [10] and [30], but the most general version was given by B. Davies [11]

(Davies proved it for q ∈ L

∞

(Γ) but noted that this condition surely can be weakened).

The second theorem is a geometric version of Ambartsumian theorem for standard Laplacians. We start by recalling the result that among graphs with fixed total length the spectral gap is minimized by the single interval [28], [12]

and [25]:

Proposition 7.2 (Theorem 3 from [25]). Let L

^st₀

(Γ) be the standard Laplace operator on a connected finite compact metric graph Γ of total length L(Γ).

Assume that the first (nonzero) eigenvalue of L

^st₀

(Γ) coincides with the first (nonzero) eigenvalue of the Laplacian on the interval I of length L(Γ)

λ

1

(L

^st₀

(Γ)) = λ

1

(L

^st₀

(I));

then the graph Γ coincides with the interval I.

The asymptotics of the spectrum determines the total length of the graph, hence the above proposition implies:

Theorem 7.3. Let Γ be a finite connected compact metric graph. The spectrum of the standard Laplacian on Γ coincides with the spectrum of the standard Laplacian on the interval I

λ

_j

(L

^st₀

(Γ)) = λ

_j

(L

^st₀

(I)), j = 0, 1, 2, . . . , (37)

if and only if the graph Γ coincides with the interval I.

The assumptions of the theorem can be weakened, since to ensure that Γ and I have the same total length it is enough to check the asymptiotics, we are going to use this observation.

Our goal is to prove that if the spectrum of a Schr¨ odinger operator on a metric graph coincides with the spectrum of the Laplacian on an interval then the graph coincides with the interval and the potential is zero. This statement cannot be proven as a simple combination of the above mentioned results. The main difficulty is to show that the graph coincides with the interval. Theorem 7.3 cannot be applied directly, since it requires q ≡ 0.

Theorem 7.4. Let L

^st_q

(Γ) be standard Schr¨ odinger operator with L

1

-potential on a finite compact connected metric graph Γ and suppose that

|λ

n

(L

^st_q

(Γ)) − λ

n

(L

^st₀

(I)) | < C, (38) for some C ∈ R, then Γ = I.

Proof. Assume that estimate (38) holds. The length of the interval as well as the total length L of Γ is reflected in the asymptotics of the spectrum L(Γ) =



lim

_n→∞^π_λ²ⁿ²

n



1/2

(Corollary 5.4), and therefore have to be equal. Then the spectrum of the Laplacian on I is

σ(L

^st₀

(I)) =  πn L



2



, n ∈ N.

Taking into account that the difference between the eigenvalues of the Schr¨ odinger and Laplace operators on the same metric graph is O(1) (Theorem 5.3), we con-

clude that  

 λ

ⁿ

(L

^st₀

) −  πn L



2

 

  = O(1), which on the level of square roots imply

 

k

n,0

− πn L

 

 → 0, as n → ∞. (39)

But k

_n,0

are given as the zeros of a trigonometric polynomial (Theorem 6.1).

Hence the spectrum of Laplacian on Γ given by zeroes of the trigonometric polynomial is asymptotically close to a set of equidistant points and Theorem 6.2 can be applied. We conclude that in fact k

n,0

= πn/ L, for all n. This means that the spectrum of the Lapalcian on Γ coincides with the spectrum of the Laplacian on an intreval. We may finally apply the geometric version of Ambartsumian theorem for Laplacians (Theorem 7.3) to conclude that the graph Γ coincides with the interval I.

An immediate consequence is then a generalisation of Ambartsumian’s the-

orem (Proposition 7.1):

Theorem 7.5. Let Γ be a finite compact metric graph and q ∈ L

1

(Γ). The spec- trum of the standard Schr¨ odinger operator L

^st_q

(Γ) coincides with the spectrum of the standard (i.e. Neumann) Laplacian on an interval

σ(L

^st_q

(Γ)) = σ(L

^st₀

(I)), (40) if and only if Γ = I and q ≡ 0.

Proof. Equation (40) implies that (38) is satisfied for any C > 0, so we conclude that Γ = I. But then Ambartsumian’s classical Theorem 7.1 implies that the potential is identically equal to zero q ≡ 0.

8 Euler Characteristic

The spectral estimate (10) allows us to extend a previous result of one of the authors regarding the Euler characteristic χ = M −N. In [18], [19] it was proven that for the standard Laplacian the Euler characteristic can be calculated by the following explicit formula:

χ(Γ) = 2 lim

t→∞

X

∞ n=0

cos q

λ

_n

(L

^st_q

)/t



 sin q

λ

n

(L

^st_q

)/2t q λ

_n

(L

^st_q

)/2t





2

, (41)

with the convention (45) below assumed.

Later it was shown that essentially the same formula holds for the Schr¨ odinger operator L

^st_q

(Γ), provided q ∈ L

∞

(Γ), see [19]. The proof is based on the observation that small perturbations of the eigenvalues do not change the limit (41), in particular:

Lemma 8.1 (Lemma 2 from [19]). Let k

n

and k

_n⁰

be two real sequences satisfying the following conditions

|k

ⁿ

− k

⁰n

| = O

 1 n



, k

_n⁰

= π

L n + O(1), (42)

and assume also the existence of the limit

t

lim

→∞

X

∞ n=0

cos k

⁰_n

/t

 sin k

⁰_n

/2t k

⁰_n

/2t



2

. (43)

Then the following limits coincide:

t

lim

→∞

X

∞ n=0

cos k

n

/t

 sin k

_n

/2t k

_n

/2t



2

= lim

t→∞

X

∞ n=0

cos k

⁰_n

/t

 sin k

⁰_n

/2t k

⁰_n

/2t



2

.