Università degli Studi di Padova

DIPARTIMENTO DI MATEMATICA

Corso di Laurea Magistrale in Matematica

Tesi di laurea magistrale

Quantum graphs: spectrum and magnetic fields

Candidato:

Andrea Serio 1058580

Relatori:

Paolo Ciatti, Università degli Studi di Padova

Pavel Kurasov, Stockholm University

5 Dicembre 2014

Contents

1 Metric graphs and quantum graphs 31.1 Metric Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Notation and Vertex basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Differential operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Symmetry and Self-adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Matching conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Matching conditions via the vertex scattering matrix . . . . . . . . . . . . . . . . . . . 81.3.2 Properly Connecting Matching Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 S Matrix, special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Spectra of quantum graphs 122.1 Toolbox for spectrum investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Vertex Scattering Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.3 Edge Scattering Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 Characteristic Equation via Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Characteristic Equation via Scattering Matrices . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Removing the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Topology and Trace Formula 203.1 Some topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 Multiplicity of eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 The Trace formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Topological damping of Aharonov-Bohm effect 254.1 Graph setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 Spectrum computed via the Scattering matrices approach . . . . . . . . . . . . . . . . 264.1.2 Test via Trace formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1

Preface

I started to work on this thesis almost one year ago when I first met professor Pavel Kurasov in Stockholmduring an exchange study program. There, I attended his Ph.D. course on quantum graphs. At that time Istarted under his direction to work on the project that led to the result here presented in the last chapterof the thesis. The whole work is inspired by the lectures of the course, and most of the content of the firstchapters comes from the lecture notes that will be part of a book written by Kurasov.

Chapter 1 This chapter contains an overview of the main definitions and proofs about quantum graphs,matching conditions and differential operators, some examples are provided to explain all these ele-ments.

Chapter 2 We take a look at two of the three main methods to compute the characteristic equations ofa quantum graph, hence, to compute its spectrum. One of these methods is then used in the resultsshown in the following chapters, and for both examples are provided.

Chapter 3 Here we present two important results about quantum graphs: first, the relation between al-gebraic multiplicity of the eigenvalue zero and the Euler characteristic of the graph in the case of thestandard Laplace operator. Second, the trace formula that shows the connection of the spectrum of aquantum graph with the set of its closed paths.

Chapter 4 This chapter contains the result of our research concerning properly connecting matching condi-tions for the eight-shape graph such that the spectrum depends exclusively on the sum (or equivalentlyon the difference) of the magnetic fluxes around the two loops. As happens frequently, what we foundis slightly different from the original expectation: for particular matching conditions, it happens thatif one of the two fluxes is zero, then the spectrum does not depend on the other one. We give a doublecheck of this property, through the characteristic equation and the trace formula. Furthermore, thereis an interpretation of this property in terms of the set of closed paths along the graph. This effect hasbeen recognized as a topological damping of the Aharonov-Bohm effect [1].

Applications of quantum graphs There are applications of quantum graphs in a great variety of con-texts, the first one dates back to 1936 and was in chemistry [14]. There are also applications to electricalnetworks ([2] in [4, pp. 15]), physiology ([6] in [4, pp. 65]), quantum chaos [9], quantum wires, photoniccrystals and thin wave guidelines (see [4] and [3] for an extensive list of applications).

2

Chapter 1

Metric graphs and quantum graphs

A quantum graph is a triple (M, L, (A,B)), where

• M is a metric graph,

• L is a differential expression

• (A,B) is a couple of matrices determining the matching conditions.

Alternatively, we can say that a quantum graph is a metric graph equipped with a self-adjoint operator L.We think that our definition distinguishes better the elements that play a role in the thesis. We start

defining precisely these three objects.

1.1 Metric Graph

A metric graph is a graph with a metric defined on it. Formally it can be constructed as the pair (V,E) ofedges and vertices, but instead of starting from the set of vertices and connecting them through the edges, wefirst take the edges and then identify their boundaries as vertices. Consider Nc compact and Ni semi-infiniteintervals En. These form the set E = EiNi=1 of the edges,

En =

[x2n−1, x2n] 1 ≤ n ≤ Nc

[xn+2Nc ,∞ [ 1 ≤ n ≤ Ni,(1.1)

where N = Nc + Ni is the total number of the edges. Furthermore, let N = 2Nc + Ni denote the totalnumber of the endpoints. The set of vertices is realized by making a partition of the set of the endpointsE = xi|i = 1, . . . ,N, then xi and xj belong to the same vertex if xi ∼ xj . We call V = E/ ∼ the set ofvertices of the graph. We denote the graph with Γ = ∪iEi / ∼. Notice that since the edges are made bydistinct copies of R, metric graph could be defined in equivalent way by means of a weighted graph, wherethe weight correspond to the length of the edge ln = x2n − x2n−1 or ln =∞.

Definition 1.1 (Metric Graph). Consider N finite or semi-infinite closed intervals En, called edges, and apartition of the set E of their endpoints into equivalence classes V = V1, V2, . . . , VM called set of vertices.A metric graph is the pair M = E, V , where the union of the edges with the endpoints belonging to thesame vertex identified.

In this definition we are considering only finite graphs, but the infinite case is also important, see forexamples the alternative definitions of metric graph given in [15] and [3].

Example 1.2 (The lasso graph). We introduce now the simplest graph that will help the reader to under-stand some basic properties of quantum graphs. Let

E1 = [x1, x2] = [0, 1]

E2 = [x3,∞[= [0,∞[

with the following identifications of endpoints x1 ∼ x2 ∼ x3.

3

Figure 1.1: A visual representation of The Lasso graph

Compact finite Graphs These are graphs with only finite closed intervals as edges, i.e. Ni = 0. Noticethat in this case N = 2Nc = 2N . After this chapter we restrict the studied case to compact graphs, so sincechapter two we use only the symbol N for simplicity: N for number of edges and 2N for the number ofendpoints.

Remark 1.3 (Compact infinite graph). With a more general definition of metric graph it is possible to findcases of compact infinite graphs. An immediate example comes from considering a fractal structure: withsufficiently fast decreasing length of edges, hence the graph can be compact. See for examples [4].

1.1.1 Notation and Vertex basis

In this work according to the notation of [11] we will denote with the symbol u both the function defined onthe whole graph Γ and with ~u its restriction to the endpoints of the edges, that is a vector ~ui := u(xi) i =1, . . . ,N . This should make it clear which one we are referring to. When we are dealing with the functioncase we will write u(x) with x ∈ Γ. It is possible to see

~·|xi :W2,2 (Γ)→ CN (1.2)

u 7→ ~ui (1.3)

as an operator between Hilbert spaces, where W2,2 (Γ) =⊕N

n=1W2,2 (En), equipped with the sesquilinearform 〈·, ·〉L2 , is direct sum of the spaces of the W2,2-functions on the edges of the graph equipped with thesame product.

Here we prefer to follow the notation of Kurasov, but there are notations quite more formal, for detailssee [15].

We will use the vector space of the functions and their derivatives restricted to the enpoints, that inboth cases is CN endowed with three different bases: edge, vertex and odd-even bases. The edge basisfollows the numbering of the edges E1, . . . , EN, so that the endpoints, as given in the above definition,are ordered as x1, x2, . . . , xN . This is the basis we refer to unless otherwise specified. The vertex basisfollows the numbering of vertices V1, . . . , VM and inside the vertex the numbering of the endpoints (e.g.Vj = x1, x4, x5). The odd-even basis is used in compact graphs, it sorts before all the odd endpointsand then the even. It is x1, x3, . . . , x2, x4, . . . , x2N. The basis is usually specified with a superscript, e.g.Se,o, Se, Sv for the matrix S in even-odd, edge, vertex basis respectively.

1.2 Differential operator

On the metric graph we are going to define the Schrodinger operator which has the following differentialexpression:

τa,q :=

(id

dx+ a(x)

)2

+ q(x) (1.4)

Where the two functions a, q are called magnetic and electric potential. These functions must satisfycertain natural assumptions

1. q(x), a(x) ∈ R

2. q ∈ L2(Γ) and∫

Γ(1 + |x|)|q(x)| dx <∞ [Faddeev condition]

3. a ∈ C(Γ) ∩ C1(Γ \ V )

Remark 1.4. The first condition is required since a, q represent the electric and magnetic potentials, as itconcerns the second (or Faddeev) condition notice that in the case of a compact graph from q ∈ L2(Γ) itfollows ∫

Γ

(1 + |x|)|q(x)| dx ≤∫

1+|x|≤|q(x)||q(x)|2 dx+

(maxx∈Γ1 + |x|

)2 ∫1+|x|>|q(x)|

dx <∞

4

We give the definitions of normal derivative and its extension that is related to the differential expressionand that will be used widely.

Definition 1.5 (Normal Derivative).

∂N~u(xj) :=

limx→xj

ddxu(x), xj is the left endpoint

− limx→xjddxu(x), xj is the right endpoint

(1.5)

Definition 1.6 (Extended Normal Derivative).

∂~u(xj) :=

limx→xj

(ddxu(x)− ia(x)u(x)

), xj is the left endpoint

limx→xj −(ddxu(x)− ia(x)u(x)

), xj is the right endpoint

(1.6)

These definitions serve when we consider the derivatives at the endpoints of edges that are part of thesame vertex.

In the simpler cases when one or both potential functions are identically zero we get the following:(id

dx+ a(x)

)2

q ≡ 0 (1.7)

−∆ + q(x) a ≡ 0 Schrodinger operator (1.8)

−∆ a, q ≡ 0 Laplace operator (1.9)

We will see in section (2.3) that via a unitary transformation it is always possible to get to the case a ≡ 0,but the presence of the magnetic field will remain in terms of fluxes, i.e. integrals of a(x) along loops in thegraph.

With the described differential expression it is natural to associate two domains and then the two oper-ators:

dom(Lminq,a

)= C∞0 (Γ \ V ) =

N⊕n=1

C∞0 (En)

dom(Lmaxq,a

)=W2,2 (Γ \ V ) =

N⊕n=1

W2,2 (En)

Where W2,2 (Ei) is the Sobolev space l = p = 2.

1.2.1 Symmetry and Self-adjointness

Here we have two operators defined on the metric graph and both do not take into account the actualstructure of the graph. Anyway they are different: the minimal operator is symmetric, but the maximaloperator is not. We find that1 〈Lmaxq,a u, v〉L2(Γ)−〈u, Lmaxq,a v〉L2(Γ) = 〈∂~u,~v〉CN −〈~u, ∂~v〉CN . It follows that theminimal operator is symmetric as defined on functions with value 0 at all endpoints, while for the maximaldomain functions can have different and independent boundary values.

Here we compute the above formula for compact graphs, but the result does not change in general: wewould just need to distinguish two sums, one for the compact edges and the other for infinite edges. Theresult does not change,

〈Lmaxq,a u, v〉 − 〈u, Lmaxq,a v〉 = (1.10)

=

N∑n=1

−∫En

(d

dx− ia(x)

)2

u(x) · v(x) dx+

∫En

u(x) ·(d

dx− ia(x)

)2

v(x) dx

The term including q is canceled

1Note that in the RHS term u, ∂~u,~v, ∂~v are vectors of CN , i.e. ~ui := u(xi), ∂~ui := ∂u(xi) according to (1.1.1)

5

=

N∑n=1

∫En

(d

dx− ia(x)

)u(x) ·

(d

dx− ia(x)

)v(x) dx+

−∫En

(d

dx− ia(x)

)u(x) ·

(d

dx− ia(x)

)v(x) dx

+

+

−(d

dx− ia(x)

)u(x)v(x) + u(x)

(d

dx− ia(x)

)v(x)

|x2nx2n−1

=

=

N∑n=1

−∂u(x2n)v(x2n) + u(x2n)∂v(x2n)−

(∂u(x2n−1)v(x2n−1)− u(x2n−1)∂v(x2n−1)

)=

= 〈∂~u,~v〉C2N − 〈~u, ∂~v〉C2N

1.3 Matching conditions

In order to define a self-adjoint extension of the minimal operator, we look for a subset of the maximaldomainW2,2 (Γ) so that for all possible couples of functions the expression (1.10) vanishes. This means thatthe operator is symmetric on such subset and we shall prove later in Proposition 1.12 that this is sufficientto deduce that the operator is even self-adjoint.

The solution comes by considering linear constraints that involve both the function and its derivative.We start from

〈Lmaxq,a u, v〉L2 − 〈u, Lmaxq,a v〉L2 = 〈∂~u,~v〉CN − 〈~u, ∂~v〉CN = 0 (1.11)

If we denote by E :=( O I−I O

)the 2N × 2N symplectic matrix and by ~U :=

(~u∂~u

), then the formula (1.11)

can be written as= 〈E~U, ~V 〉C2N , (1.12)

which is the symplectic form on the space C2N . So the maximal space where the operator is symmetric cor-responds to a linear Lagrangian submanifold M, dim(M) = N of C2N with d(·, ·) := 〈E·, ·〉C2N as symplecticform. From this perspective, the search for matching conditions is equivalent to look for a parametrizationof M . This can be done by means of a bijective linear map Ψ : CN →M , equivalent to a 2N ×N complexmatrix with maximal rank.

Let(B∗

A∗)

be a complex 2N ×N matrix, where A,B are square N ×N complex matrices. If u, v ∈ CN

are such that ~U =(B∗

A∗)u, ~V =

(B∗

A∗)v, then from (1.12) it follows that

〈E(B∗

A∗)u,(B∗

A∗)v〉C2N = 0 (1.13)

v∗(B A

)( O I−I O

)(B∗

A∗)u = 0

v∗(BA∗ −AB∗

)u = 0

Since this must hold for all u, v, we haveBA∗ = AB∗ (1.14)

This means that the subspace of W2,2(Γ) that makes the operator self-adjoint is identified by a coupleof complex square matrices A,B as above, i.e. such that

1. Rank(A,B) is maximum (that is required to get the self-adjointness).

2. AB∗ is Hermitian.

By the substitution(~u∂~u

)= U =

(B∗

A∗)u,

~u = B∗u, A∗u = ∂~u

A~u = AB∗u = BA∗u = B∂~u

A~u = B∂~u

So the (1.11) holds if we restrict the operator domain tof ∈ W2,2(Γ) |A~f = B∂ ~f

in place of W2,2(Γ)

6

Standard matching conditions The first and most important matching conditions we encounter isthe Standard Matching condition, hereafter sometimes called SMC. This condition comes from naturalassumptions, for each vertex V of Γ,

f is continuous∑xj∈V

∂f(xj) = 0 (1.15)

δ-type interactions A more general set of matching conditions is the so called δ-type conditions that aredefined as follows

f(x) is continuous at v∑xi∈v ∂f(xi) = αvf(v),

(1.16)

where αv is a fixed number for each vertex.

Example 1.7 (Lasso graph). We consider again the graph from the example (1.2). In order to respectcontinuity and zero sum of derivatives, we must have

A =

1 −1 00 1 −10 0 0

B =

0 0 00 0 01 1 1

Notice that AB∗ = BA∗ = O3. Then1 −1 0

0 1 −10 0 0

( f(x1)f(x2)f(x3)

)=

0 0 00 0 01 1 1

( ∂f(x1)∂f(x2)∂f(x3)

)

That is exactly f(x1)− f(x2) = 0

f(x2)− f(x3) = 0

∂f(x1) + ∂f(x2) + ∂f(x3) = 0

Example 1.8. In the case of a generic metric graphM with N edges and M vertices, for the vertex basis2,the SMC is given by N ×N complex matrices A,B with M diagonal blocks,

Av =

A1 0 0

0. . . 0

0 0 AM

Bv =

B1 0 0

0. . . 0

0 0 BM

,

where for each i = 1, . . . ,M ,

Ai =

1 −1 0 0 00 1 −1 0 0

0 0. . .

. . . 00 0 0 1 −10 0 0 0 0

Bi =

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 01 1 . . . . . . 1

The Standard Matching conditions is known in literature also with the name of Free, Kirchoff or Neumann

conditions [4]. Free, because a vertex connecting two edges can be removed, since in the vertex the function

is continuous and ∂ ~f(xi) + ∂ ~f(xj) = 0, which means that at this point the right and the left derivative existand they are the same. Neumann, because if we consider a vertex connected with just one edge they areequivalent to Neumann conditions i.e. zero derivative.

2See vertices basis in subsection (1.1.1)

7

1.3.1 Matching conditions via the vertex scattering matrix

At this point we can notice that, despite a couple of matrices (A,B) gives a unique matching condition, theopposite is not true. In fact if we take any C ∈ GlN (C), the couple (CA,CB) still gives the same conditionson the function u.

Then it is natural to be interested in looking for a subset of matrices that identify in a unique wayeach matching condition. Notice that property (1) of (A,B) implies that (A− iB) and (A+ iB) havemaximal rank and then are invertible, see the next lemma and Proposition (1.9) for the proof of this. Let

C := −2i (A− iB)−1

and substitute A = 12 ((A− iB) + (A+ iB)), B = − i

2 ((A− iB)− (A+ iB)), then

CA = −i(A+ iB)−1 ((A− iB) + (A+ iB)) = −i((A+ iB)−1(A− iB) + I

),

and similarlyCB =

(I− (A+ iB)−1(A− iB)

)If we denote by

S := −(A+ iB)−1(A− iB) (1.17)

we can rewrite the matching conditions in the form

i(S − I)~u = (S + I)∂~u. (1.18)

Proposition 1.9. The S matrix in equation (1.17) is well defined and unitary.

Proof. 1. S is well defined. In order to prove that (1.17) is a good definition we just need to show that(A+ iB) is invertible.

We prove a slightly more general lemma that will be useful later.

Lemma 1.10. Let A,B ∈ Mn(C) be such that

(a) rank (A,B) = n is maximal

(b) AB∗ = BA∗

Then the matrix (A+ ikB) is invertible for all k ∈ R \ 0.

Proof. Let ~v ∈ Cn \ 0

‖(AkB

)~v‖2 = ~v∗(A∗, kB∗)

(AkB

)~v =

= ~v∗(A∗A+ k2B∗B)~v =~v∗(A∗ − ikB∗)(A+ ikB)~v =

=~v∗(A+ ikB)∗(A+ ikB)~v = ‖(A+ ikB)~v‖2

By the first condition (1) on (A,B) ⇒ ∀~v ∈ Cn \ 0(AB

)~v 6= 0, without any substancial difference we

can multiply B by k(AkB

)~v 6= 0 i.e. ‖

(AkB

)~v‖ = ‖(A+ ikB)~v‖ 6= 0, that is (A+ ikB) is invertible.

In our case we take k = ±1 to get (A− iB) and (A+ iB) invertible.

2. S is unitary.

SS∗ =(A+ iB)−1(A− iB)((A+ iB)−1(A− iB)

)∗=(A+ iB)−1(A− iB)(A∗ + iB∗)(A∗ − iB∗)−1

=(A+ iB)−1(AA∗ +BB∗)(A∗ − iB∗)−1

=(A+ iB)−1(A+ iB)(A∗ − iB∗)(A∗ − iB∗)−1 = I

Definition 1.11 (MSO - Magnetic Schrodinger Operator). We denote by MSO and by LSa,q the differentialexpression τa,q equipped with the domain

dom(LSa,q

)=f ∈ W2,2 (Γ) | i (S − I) ~f = (S + I) ∂ ~f

8

Proposition 1.12 (MSO is self-adjoint). The differential operator LSa,q is self-adjoint.

Proof. We have already chosen the set of the matching conditions so that LSa,q is a symmetric operator.

Now it remains to show that this implies that LSa,q is also self-adjoint. Let g ∈ W2,2 (Γ) be fixed. Using the

symmetry we prove that g must satisfy the matching condition given by S and then to be in dom(LSa,q

).

∀f ∈ dom(LSa,q

)〈∂ ~f,~g〉CN − 〈~f, ∂~g〉CN = 0

⇒ i (S − I)~g = (S + I) ∂~g (1.19)

Firstly use the change of variables ~F = i ~f − ∂ ~f, ∂ ~F = i ~f + ∂ ~f , and do the same with ~G and ∂ ~G. Thetwo matrices that give this transformation are invertible, this follows from the fact that it is the inversetransformation with matrix A+ iB that we showed in Proposition 1.9 to be invertible.

Notice that the (1.19) is equivalent to

∀~F ∈ CN s.t. S ~F = ∂ ~F

〈~F , ~G〉CN − 〈∂ ~F , ∂ ~G〉CN = 0

⇒ S ~G = ∂ ~G (1.20)

This is easy to prove. By substituting 〈S ~F , ∂ ~G〉 = 〈~F , S∗∂ ~G〉,

〈~F , S∗∂ ~G− ~G〉 = 0

and since ~F can be chosen arbitrarily∂ ~G− S ~G = 0

Then ∂ ~G = S ~G, that isi (S − I)~g = (S + I) ∂~g (1.21)

In order to finish the proof we explicitly compute the change of variables from (1.19) to (1.20).

〈∂ ~f,~g〉CN − 〈~f, ∂~g〉CN = 0⇒

〈2∂ ~f + i ~f − i ~f, 2i~g + ∂~g − ∂~g〉+ 〈2i ~f + ∂ ~f − ∂ ~f, 2∂~g + i~g − i~g〉 =

= 〈~F − ∂ ~F , ~G+ ∂ ~G〉+ 〈~F + ∂ ~F , ~G− ∂ ~G〉 =

= 〈~F , ~G〉+ 〈~F , ∂ ~G〉 − 〈∂ ~F , ~G〉 − 〈∂ ~F , ∂ ~G〉+ 〈~F , ~G〉 − 〈~F , ∂ ~G〉+ 〈∂ ~F , ~G〉 − 〈∂ ~F , ∂ ~G〉 =

= 2〈~F , ~G〉 − 2〈∂ ~F , ∂ ~G〉 = 0

〈~F , ~G〉 = 〈∂ ~F , ∂ ~G〉,

while the matching conditions become

i (S − I)~g = (S + I) ∂~gS i~g − i~g = S∂~g + ∂~g

S (i~g − ∂~g) = i~g + ∂~g

1.3.2 Properly Connecting Matching Conditions

From what we have seen above it is clear which are formally the matching conditions that make the operatorsymmetric, i.e. all the possible S ∈ UN , but in this way we are not taking into account the structure of theassociated metric graph M. Consider the following example

9

Example 1.13 (3-star graph). Consider a metric graph made by 3 segments with one endpoint in common(E1 = x1, x2, E2 = x3, x4, E3 = x5, x6, V1 = x1, x3, x5 , V2 = x2 , V3 = x4 , V4 = x6),and two different matching conditions defined by the scattering matrix S1 and S2, in the vertex basis3

Bv = (1, 3, 5, 2, 4, 6).

Sv1 :=

− 1

323

23 0 0 0

23 − 1

323 0 0 0

23

23 − 1

3 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

Sv2 :=

1 0 0 0 0 00 0 1 0 0 00 1 0 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

(1.22)

While S1 is a unitary matrix that gives a properly connecting matching condition, this is not true for S2,since it does not take into account that in the metric graph x1 ∈ V1.

Figure 1.2: On the left: 3-stars graphs with vertices labeled. For the Γ graph, S1 is a properly connectingmatching condition, but not S2. S2 is a properly connecting matching conditions for the Γ metric graphinstead.

Then we have to consider two kinds of restrictions on the set of all possible matrices S. The value of thefunction of each vertex must depend on all and only all the other values of the function and its derivativesat each endpoint that belong to the vertex.

Formally, each diagonal block of S corresponding to any vertex of the metric graph must be an irreduciblematrix.

If we take back (1.10) then we split the sum, separating the vertices of the graph (V1, . . . , VM ) that is

=

M∑m=1

∑xj∈Vm

−∂u(xj)v(xj) + u(xj)∂v(xj)

(1.23)

So that each block of S involves one term of the left sum, ∀Vm SVm = S|Vm . A properly connecting matchingconditions requires each SVm to be an irreducible matrix.

Definition 1.14 (Properly connecting matching condition). Given a metric graph Γ with vertices V1, . . . , Vmand matching conditions given by a scattering matrix S, we say that S gives a properly connecting matchingconditions if S|Vi is irreducible for all i = 1, . . . ,m.

1.3.3 S Matrix, special cases

Here we show how the S matrix looks like for the SMC and the class of non-resonant matching conditions.

SMC Consider the block of an S matrix relative to a single vertex made of n endpoints connected into it.Remember condition (1.15) (continuity of the function and zero sum of the derivatives). Then, it appearsthat all endpoints are equivalent and so the matrix is also invariant for all possible permutations of the basis:∀P permutation PSP−1 = S. It follows that we need just two parameters to define the matrix: the valueof diagonal d and the value outside of the diagonal h, and these are fixed by the continuity of f , that is

if(x) · (d− 1 + (n− 1)h) = 0

∀i (d+ 1)∂f(xi) +∑j 6=ixj∈v h ∂f(xj) = 0

(1.24)

Since the norm of each column is one and the scalar product of two different column is zero,d2 + (n− 1)h2 = 1

2hd+ (n− 2)h2 = 0,(1.25)

3so that the top-left 3× 3 block refers to V1

10

d = h+ 1 and so d = 2

n − 1

h = 2n ,

(1.26)

like in each block of the S matrix of example (1.13).The simplest cases are

S1 = (1), S2 = ( 0 11 0 ) , S3 =

1

3

(−1 2 22 −1 22 2 −1

), Sn =

1

n

2− n 2

. . .

2. . . 2

. . . 2 2− n

S Hermitian, matching conditions It is important to consider the case when S is also Hermitian. Bydefinition we know that the eigenvalues of S satisfy |λj | = 1 since the matrix is unitary. Moreover, we knowby the spectral theorem that these eigenvalues must be real because S is Hermitian. Hence, λj = ±1 andit follows that S can be decomposed into the sum of two projector S = P+1 − P−1. The first implication isthe separation of the matching conditions, to see it we substitute in the matching conditions I = P+1 +P−1

i(P+1 − P−1 − I)~u = (P+1 − P−1 + I)∂~u (1.27)

−2iP−1~u = 2P+1∂~u (1.28)

and, since eigenspaces corresponding to different eigenvalues are complementary, we obtainP−1~u = 0

P+1∂~u = 0(1.29)

The second implication involves the vertex scattering matrix Sv(k) that becomes energy independent, thiswill be shown later in Proposition (2.3).

11

Chapter 2

Spectra of quantum graphs

Hereafter we consider only compact graphs, Ni = 0 and N = 2N . In this chapter we present the instrumentsand the tecniques used to solve the eigenvalue problem on quantum graphs, i.e. how to solve the Schrodingerequation in presence of a magnetic field(

−(d

dx− ia(x)

)2

+ q(x)

)g(λ, x) = λg(λ, x). (2.1)

What we look for in this section is a characteristic equation, that is a simpler way to compute the spectrumof the operator without explicitly solving the eigenvalue-equation.

Theorem 2.1 (discreteness of the spectrum). The spectrum of a compact quantum graph equipped with aSchrodinger operator consists exclusively of isolated eigenvalues of finite multiplicity, with λj →∞.

Proof. See [Kuchment] Theorem 3.1.1

Proposition 2.2 (Positiveness of the spectrum for the standard Laplacian). The spectrum of the standardLaplace operator Lst = −∆ on a quantum graph is positive.

Proof. Notice that

〈u, Lstu〉 =

N∑n=1

∫En

|u′(x)|2 dx ≥ 0. (2.2)

Thus, in the case in which u is an eigenfunction of Lst

〈u, Lstu〉 = λ〈u, u〉, (2.3)

then λ is necessarily positive because 〈u, u〉 ≥ 0, with equality if and only if u = 0 a.e. .

2.1 Toolbox for spectrum investigation

We start this part by defining three different matrices that form the basis for all the instruments for solvingthe eigenvalue problem and further investigations, these are

Sv(k): The Vertex Scattering Matrix

Tq(λ): The Transfer Matrix

Se(k): The Edge Scattering Matrix

12

2.1.1 Vertex Scattering Matrix

The vertex scattering matrix is an instrument based on a physical interpretation of how do the vertexworks in presence of incoming and outgoing waves along its edges. Consider a n-star graph, E := Ej =[xj ,∞) s.t. xj = 0, j = 1 . . . N provided with the Dirichlet operator −∆. On each half line, the solutionof the eigenvalue equation (2.4)

−∆ψ = λψ λ = k2 (2.4)

(let Ej := [xj ,∞) and k ≥ 0) is ψ(x)|Ej = ajeikx + bje

−ikx with derivative ∂ψ(x)|Ej = ik(ajeikx − bje−ikx),

and it can be written in this vertex form

~ψ = ~aeikx +~be−ikx|x=0 = ~a+~b (2.5)

∂ ~ψ = ik(~aeikx −~be−ikx)|x=0 = ik(~a−~b), (2.6)

where each component represents the function on the corresponding edge. Substituting this into the matchingconditions, we obtain

i(S − I)~ψ = ki(S + I)∂ ~ψ

i(S − I)(~a+~b) = ki(S + I)(~a−~b)

[S(k + 1) + I(k − 1)]~b = [S(k − 1) + I(k + 1)]~a

This is a good definition for all reals k 6= 0; since Lemma (1.10) shows that with A = i(S − I), B = (S + I),for k ∈ R \ 0 the matrix (i(S − I) + ik(S + I)) is invertible.

This permits to write ~a, the vector of the outgoing wave amplitudes, in terms of ~b, the vector of theincoming wave. Furthermore, in this expression these two matrices commute and this allow us to write with(abusing notation a bit).

~a =S(k + 1) + I(k − 1)

S(k − 1) + I(k + 1)~b (2.7)

The matrix that gives this correspondence between ~a and ~b is called Vertex Scattering Matrix

Sv(k) :=S(k + 1) + I(k − 1)

S(k − 1) + I(k + 1)(2.8)

There are a couple of properties of Sv(k) we will need in the sequel.

Proposition 2.3 (Vertex Scattering Matrix properties).

1. Sv(1) = S.

2. If S is Hermitian, then Sv(k) ≡ S.

Proof.

1.

Sv(1) =2S + I · 0S · 0 + 2I

= S, (2.9)

2. If S is also Hermitian, then S = S∗ ⇒ SS = I and

Sv(k) =S(k + 1) + I(k − 1)

S(k − 1) + I(k + 1)=S · (I(k + 1) + S(k − 1))

S(k − 1) + I(k + 1)= S (2.10)

When the vertex scattering matrix has no singularities we say that the matching conditions are non-resonant. This happens when Sv does not depend on the energy: Sv(k) ≡ S ∀k ∈ R. Under this hypothesis,

S = Sv(k) =S(k + 1) + I(k − 1)

S(k − 1) + I(k + 1)(2.11)

13

SS(k − 1) + S(k + 1) = S(k + 1) + I(k − 1)

SS(k − 1) = I(k − 1)

It is sufficient the existence of just one value of k 6= 1 that makes the matrix S(k− 1) + I(k+ 1) invertible toconclude that S2 = I. Moreover the S matrix is unitary by definition and this implies S2 = I = SS∗, thenS is Hermitian

S = S∗ (2.12)

2.1.2 Transfer Matrix

Like in dynamical systems theory it is useful to define a transfer matrix. This matrix put in relation thevalues of the function and its derivative at the endpoints of a compact edge.

Consider the one-dimensional Schrodinger equation (2.1) defined on a single compact interval E = [x1, x2]and suppose that g is a solution,

Tq(λ)x2x1

:(gλ(x1)

g′λ(x1)

)7→(gλ(x2)

g′λ(x2)

)(2.13)

It is well known that in the case q ≡ 0 when (2.1) is simply the Laplacian, the transfer matrix becomes

T0(λ)x2x1

=

(cos k(x2 − x1) sin k(x2−x1)

k−k sin k(x2 − x1) cos k(x2 − x1)

)(2.14)

Proposition 2.4 (Transfer Matrix properties). detTq(λ)x2x1≡ 1

Sketch. For details see [11]. Consider two solutions g and f to the eigenvalue equation Lqf = λf for real λ.Start with the following identity and use integration by parts

0 = 〈(Lq − λ)f, g〉L2(E1) − 〈f, (Lq − λ)g〉L2(E1)

= f ′(x1)g(x1)− f(x1)g′(x1)− f ′(x2)g(x2) + f(x2)g′(x2)

This leads to

T tq (λ)

(0 −11 0

)Tq(λ) =

(0 −11 0

)(2.15)

then the conclusion.

2.1.3 Edge Scattering Matrix

We make now a construction to establish a relation between the incoming and the outgoing waves along theedge.

We deal with a compact edge E = [x1, x2], x1 ≤ x2 with the Schrodinger equation defined on it and letψ be the solution to the problem (2.1).

Extend the edge E, the equation and the solution on all R, provided a = q ≡ 0 outside E, let ψ be thesolution, then on the rest of the real line

ψ(x) =

a1e−ik|x−x1| + b1e

ik|x−x1| If x ≤ x1

a2e−ik|x−x2| + b2e

ik|x−x2| If x ≥ x2,(2.16)

here the a-term is the incoming wave and the b-term is the outgoing one.

From the previous subsection we know that(ψ(x2)∂ψ(x2)

)= Ta,q(λ)x2

x1

(ψ(x1)∂ψ(x1)

), so we can substitute by

continuity of construction the extended values of ψ. Let tij :=(Ta,q(λ)x2

x1

)ij

In general ψ(x) = aeikx+be−ikx

with derivative ∂ψ(x) = ik(aeikx − be−ikx), so that the solution becomes(a2+b2

ik(b2−a2)

)=(t11 t12t21 t22

) ( a1+b1ik(b1−a1)

)(2.17)

We define Se as the matrix such that (b1b2

)= Se(k) ( a1a2 ) (2.18)

14

From the above definitions, as long as k 6= 0, we obtain

Se(k) =

(k2t12−ik(t11−t22)+t21k2t12+ik(t11+t22)−t21

2ikk2t12+ik(t11+t22)−t21

2ikk2t12+ik(t11+t22)−t21

k2t12+ik(t11−t22)+t21k2t12+ik(t11+t22)−t21

)(2.19)

This matrix in the particular case a = q ≡ 0 is

Se(k) =

(0 eikl

eikl 0

)(2.20)

2.2 Characteristic Equation

There are three principal ways to compute the spectrum of a quantum graph. Here we present two of these,even if we will just use the Scattering matrix approach. yet another method is the so called Titchmarsh-Weylm-function, that is the analogue of the Dirichlet-to-Neumann map (see [11] and [3]).

2.2.1 Characteristic Equation via Transfer Matrix

If we consider the transfer matrix on a finite interval E = [x2n−1, x2n] it happens

Tnq,a(λ) :(

~ψ2n−1

∂ ~ψ2n−1

)7→(

~ψ2n

−∂ ~ψ2n

)(2.21)

So, it is possible to give a relation to the vector of the function and its derivative at the endpoints of oneedge. Let tij := Tnq,a(λ) when it is clear which edge and eigenvalue we are referring to.

(t11 1 t12 0t22 0 t21 1

)~ψ2i−1

~ψ2i

∂ ~ψ2i−1

∂ ~ψ2i

=

(00

)(2.22)

and in the same fashion for the whole graph we obtainC1 D1

. . .. . .

Ci Di

. . .. . .

(~ψ

∂ ~ψ

)=

0...0

(2.23)

If this system has a solution, then the eigenvalue problem has a solution for λ. But from this solutionswe have to select the functions which are eligible for matching conditions. That is i(S− I)~ψ = (S+ I)∂ ~ψ, soif we merge together these conditions

. . .. . .

Ci Di

. . .. . .

i(I− S) (I + S)

(~ψ

∂ ~ψ

)=

0......0

(2.24)

and this is a square linear system with non trivial solution if and only if detM = 0, with M the wholematrix of equation (2.24).

Theorem 2.5 (Characteristic equation). Let Γ be a finite compact graph, then the spectrum of the corre-sponding magnetic Schrodinger operator LSa,q is pure discrete with a unique accumulation point +∞. Theeigenvalues are solutions to the characteristic equation detM = 0

Proof. See [11] Theorem 4.1

15

Now we can notice that the above expression is quite large even for the small graph, because it involvesa matrix 4N × 4N = 16N2 terms with N the number of edges, but there is redundant information. In factthis value can be reduced by a simple consideration, if we separate the even by the odd terms in the functionvector and its derivative, then there is a relation between these elements. Let

~ψodd = ψ(x2n−1)Nn=1, ∂ ~ψodd = ψ(x2n−1)Nn=1

~ψeven = ψ(x2n)Nn=1, ∂ ~ψeven = ψ(x2n)Nn=1

we denote1 and Tij = diagtnijNn=1. Then equation (2.21) can be re-written as(T11 T12

T21 T22

)(~ψodd

∂ ~ψodd

)=

(~ψeven

−∂ ~ψeven

)(2.25)

Then we are allowed to write everything in term of only half variables, we denote ~ψo,e = (~ψodd, ~ψeven) and

similarly for ∂ ~ψo,e, that is just a permutation2 of the terms ~ψ and ∂ ~ψ

~ψo,e =

(IN ONT11 T12

)(~ψodd

∂ ~ψodd

),

∂ ~ψo,e =

(ON IN−T21 −T22

)(~ψodd

∂ ~ψodd

)

By substitution into the matching conditions we obtain a new equation equivalent to the (2.24)

i(So,e − I2N )

(IN ONT11 T12

)(~ψodd

∂ ~ψodd

)= (So,e + I2N )

(ON IN−T21 −T22

)(~ψodd

∂ ~ψodd

)(2.26)

In this way, in place of detM we use

det

i(So,e − I2N )

(IN ONT11 T12

)+ (So,e + I2N )

(ON −INT21 T22

)= 0 (2.27)

Example 2.6 (Characteristic equation - Single loop (part 1)). Consider a graph made of a single edge[x0, x1] of length l with its endpoints identified (i.e. the circumference S1), let it be equipped with the Laplaceoperator and with SMC. Then So,e = S = ( 0 1

1 0 ) and the transfer matrix is the (2.21) with |x1 − x0| = l.Then expression (2.27) becomes

detM(k) = −4i(cos kl1 − 1) (2.28)

Which has solution for k = 2nπl1∀n ∈ Z. In fact, let x0 = 0, x1 = l, the corresponding eigenfunctions are

ψ(x) = sin kx for k =2nπ

l1n ∈ Z (2.29)

Example 2.7 (Characteristic equation - Compact Lasso Graph (Part 1)). For this example we considera variant of the Lasso Graph with a finite length interval in place of the infinite one. E1 := [x1, x2],E2 := [x3, x4], V1 = x1, x2, x3, V2 = x4. Furthermore let us take l1 = (x2 − x1) and l2 = (x4 − x3). Wetake it provided by standard matching conditions

So,e = S =

− 1

323

23 0

23 − 1

323 0

23

23 − 1

3 00 0 0 1

1that is just a permutation of the original ~ψ vector.2P o,e

ij = δj2i−1 + δj2i−N ∈ C2N×2N

16

7

Figure 2.1: The Lasso graph

and equipped with a Laplace operator L = −∆. Then the transfer matrices

T 1(λ) =

(cos kl1

sin kl1k

−k sin kl1 cos kl1

)T 2(λ) =

(cos kl2

sin kl2k

−k sin kl2 cos kl2

)(2.30)

T11 = T22 =

(cos kl1 0

0 cos kl2

)T12 =

(sin kl1k 0

0 sin kl2k

)T21 =

(−k sin kl1 0

0 −k sin kl2

)and the expression (2.27) becomes

detM(k) =8

3(sin (l1 + l2)k − 4(cos l1k − 1) cos l2k) (2.31)

Notice that in the case l2 = 0 we get the same solutions of the previous example (2.6).

2.2.2 Characteristic Equation via Scattering Matrices

An alternative way to find the characteristic equation exploits the scattering matrices Se and Sv. Thismethod is easier, but a little less general. We will see that it works for strictly positive eigenvalues only.

Let Γ be a compact finite graph with the Laplace operator defined on it −∆. Then take ψ a solution tothe corresponding eigenvalue problem −∆ψ = λψ. The solution restricted to the vertices can be written asa combination of plain waves as in the formula (2.16), so that

~ψ = ~a+~b (2.32)

∂ ~ψ = ik(~a−~b) (2.33)

and it must be~b = Se(k)~a (see (2.18)). Recall that from Section (2.1.1) we know that ~a = Sv(k)~b. Combiningthese two formulas we get the identity ~a = Sv(k)Se(k)~a. Then, a solution to the eigenvalue(positive) problemexists if and only if

det (S(k)− I) = 0, (2.34)

where S(k) = Sv(k)Se(k).

Observation 2.8. Notice that in (2.34) Sv and Se can be exchanged: ~b = Se(k)Sv(k)~b.

Observation 2.9. Whenever we deal with an hermitian scattering matrix S, the vertex scattering matrix isenergy independent and the equation (2.34) depends on k only through Se(k) so there are just the eikl terms.

Example 2.10 (Characteristic equation - Single loop (part 2)). Recall the example (2.6), Sv(k) = S = ( 0 11 0 ).

We are in the case of the Laplace operator, so the edge scattering matrix is

Se(k) =

(0 eikl

eikl 0

)(2.35)

Then expression (2.34) becomesdet (Se(k)Sv(k)− I) = (eikl − 1)2, (2.36)

which has solution for k = 2nπl ∀n ∈ Z, the same solution as in the Part 1.

17

Example 2.11 (Characteristic equation - Compact Lasso Graph (Part 2)). Recall Sv(k) = S from Part 1and again notice that, since we deal with the Laplace operator, then

Se(k) =

0 eikl1 0 0

eikl1 0 0 00 0 0 eikl2

0 0 eikl2 0

. (2.37)

The characteristic equation (2.34) is

det (Se(k)Sv(k)− I) =1

3

(eikl1 − 1

) (eikl1 + 3ei(a+2l2)k − e2ikl2 − 3

)(2.38)

Furthermore, notice again that in the case l2 = 0 we get the same solutions of the previous example (2.6).

2.3 Removing the Magnetic Field

Consider the case of a quantum graph Γ with the magnetic Schrodinger operator La,q, a(x) 6≡ 0 and matchingconditions given by the couple (A,B) or the relative unitary matrix S. It is always possible in such case tocancel the term a(x) thanks to a unitary tranformation. Let Ua be an operator on Γ such that on each edgeEi = [0, li] it acts in the following way Ua : u(·) 7→ u(·)e−i

∫ ·0a(y) dy, where u is a function defined on Ei ⊆ Γ.

In general Ua acts on each edge Ei = [x2i−1, x2i] of Γ with a parametrization dependent by the edge: let

u : Ei → R, then Uau(x) := u(x)e−i

∫ xx2i−1

a(y) dyfor x ∈ [x2i−1, x2i].

We are going to prove thatU−1a (L0,q)Ua = La,q (2.39)

where L0,q = −∆ + q(x), a ≡ 0.Since it is obvious that U−1

a (q)Ua(x) = q(x), we can consider the case q ≡ 0, then we need to prove that

U−1a (−∆)Ua = −

(d

dx− ia(x)

)2

(2.40)

Again, take a generic function u(x) ∈ W2,2(Γ) and look at each single edge Ei parametrized with zero inthe left endpoint and li in the right one.(

La,0U−1a

)u(x) =

−(d

dx− ia(x)

)2

u(x)ei∫ x0a(y) dy = −

(d

dx− ia(x)

)(iu′(x)ei

∫ x0a(y) dy

)= −u′′(x)ei

∫ x0a(y) dy

= −(U−1a ∆

)u(x)

from this it follows that if u(x) is a λ−eigenfunction for the operator −∆, then u := Uau is a λ−eigenfunctionfor the operator La,

Lau = λu (2.41)(UaLaU

−1a

)Uau = λUau

−∆(Uau) = λ(Uau)

−∆u = λu. (2.42)

Then we have to pay attention to the matching conditions. The transformation Ua has consequences also onthe matrices A and B (or similarly on S) because we need to move the condition from u on u. This requiresto define (A, B) as the matrices such that

A~u = B∂~u

A~u = B∂N ~u.

18

If we look at the definition of vector ~u and its extended derivative we can see that these last vectors canbe written as ~u = D(Ua)−1~u and ∂N ~u = D(Ua)−1∂~u where D(Ua), in the edge basis, is a N ×N diagonalmatrix with ei

∫ x0a(y) dy evaluated at the endpoints of the vertices on the diagonal

D(Ua)e =

1 0 0 0 0 . . .

0 ei∫ x2x1

a(y) dy0 0 0 . . .

0 0 1 0 0 . . .

0 0 0 ei∫ x4x3

a(y) dy0 . . .

0 0 0 0. . . . . .

(2.43)

so thatAD(Ua)−1~u = BD(Ua)−1∂~u.

Eventually we see that the changes that occur on the matching conditions matrices are A := D(Ua)−1AD(Ua)and B := D(Ua)−1BD(Ua) and that a similar change occurs for S = D(Ua)−1SD(Ua). As last observationnotice that

Ua(x) : dom(LSa,q

)→ dom

(LS0,q

)(2.44)

u(·) 7→ u(·)e−i∫ ·0a(y) dy

Change on Sv In the same way we see how the Vertex Scattering Matrix changes

Sv =S(k + 1) + I(k − 1)

S(k − 1) + I(k + 1)=

=D(La)−1 (S(k + 1) + I(k − 1))D(La)

D(La)−1 (S(k − 1) + I(k + 1))D(La)= D(La)−1Sv(k)D(La)

(2.45)

Example 2.12 (Lasso Graph with magnetic field). We consider again the graph from the example (1.2)

and we consider on it the operator La = −(ddx − ia(x)

)2,

A =

1 −1 00 1 −10 0 0

B =

0 0 00 0 01 1 1

In this particular case we have

D(La) =

1 0 00 eiΦ 00 0 1

, (2.46)

where Φ =∫ l

0a(y) dy and l is the length of the loop.

So the new matching conditions (A, B) will be

A =

1 −eiΦ 00 eiΦ −10 0 0

B =

0 0 00 0 01 eiΦ 1

If we were working with the scattering matrix S, we easily see that the correct transformation is S :=D(La)−1SD(La)

S =

1 1 11 1 11 1 1

S =

1 eiΦ 1e−iΦ 1 e−iΦ

1 eiΦ 1

19

Chapter 3

Topology and Trace Formula

Henceforth we deal with the standard Laplacian and so also the case of the Schrodinger operator with zeroelectric potential and with non-zero magnetic field. It is of great interest to study how topological andmetric properties are connected with the spectrum of a quantum graph. In this chapter we are going tosee in particular the trace formula that connects all the possible closed paths on a graph with the spectrumvalues. In general we refer to [10], see the other citations for details.

3.1 Some topological properties

3.1.1 Multiplicity of eigenvalues

Hereafter we distinguish two multiplicities for eigenvalues, the algebraic multiplicity and the spectral multi-plicity.

ms(λ) The Spectral multiplicity is the dimension of the eigensubspace of Lst(Γ) corresponding to the eigen-value λ

ma(λ) The Algebraic multiplicity is the dimension of ker(SvSe(k)− I), k2 = λ.

What we are going to prove is that these two numbers coincide for all eigenvalues, except for zero, becauseof the missed correspondence of the solution of the eigenvalue-equation with the wave function along theedges when k = 0. With zero energy (k) the edge scattering matrix Se(0) does not hold, but we can use itanyway to get a different value that involves the Euler characteristic of the graph.

Theorem 3.1 (Spectral and Algebraic multiplicity of zero ( [11] [13])). Let Γ be a finite compact metricgraph with d connected components and Euler characteristic χ, and let Lst(Γ) be the corresponding standardLaplace operator. Then λ = 0 is an eigenvalue with spectral multiplicity ms(0) = d and algebraic multiplicityma(0) = 2d− χ.

Proof.

Spectral multiplicity. Consider the equation ∆f = 0 on a single edge E = [x0, x1]. All the possible solutionsof this equation can be written in the form f(x) = ax+b. For a graph, we assign to each edge Ei the relativesolution f |Ei(x) = aix + bi. We prove by contradiction that there is only the solution ai = 0; bi = b∀i.Consider a connected component of the graph, since it is compact and the solution is continuous on it, theremust be a maximum. This maximum value is located at the endpoint of an edge, because f(x) is linear,hence it is on a vertex Vj . Then by linearity of f(x) it follows that the sum of normal derivatives on thevertex Vj is strictly less than zero ∑

xi∈Ek∩Vj

∂Nf(xi) =∑xi∈Vj

σkak ≤ 0, (3.1)

where the factor σk = ±1 compensates for the orientation of the edge Ek, ∀i σiai < 0 because xi is themaximum on each edge, contradicting the standard matching conditions

∑xi∈Vj ∂Nf(xi) = 0. So the only

solutions to the eigenvalue equation for λ = 0 are the constant functions, with independent value on differentconnected components of the graph Γ. We have ms(0) = d = ](connected components).

20

Figure 3.1: A visual example of the contradiction of a non-trivial solution of the equation −∆f = 0, thegraph is planar and the height is proportioned to the value of the function, from Vj the solution decreaseand the normal derivatives are negative.

Algebraic multiplicity. The equation (2.34) is written using the wave representation of the eigenfunctionsthat is not uniquely determined when k = 0 because

ψ(x, λ) = a2n−1e−ik|x−x2n−1| + b2n−1e

ik|x−x2n−1| ≡ a2ne−ik|x−x2n| + b2ne

ik|x−x2n| (3.2)

becomes ψ(x, 0) = a2n−1 + b2n−1 ≡ a2n + b2n and ψ′(x, 0) = 0, that is a system of two equations for fourvariables. Now assume first that Γ is connected and consider the system

SvSe(0)~a = ~a. (3.3)

We write it in a more extended way in order to better understand the link with the topology of Γ,~b = Se(0)~a

~a = Sv~b(3.4)

From the first equation, since the restriction of Se(0) to all edges Ei is Se(0)|Ei = ( 0 11 0 ), we can easily replace

~b witha2n−1 = b2n, a2n = b2n−1

while, regarding to the second equation, we recall the standard matching conditions in the more intuitiveform

ai + bi = aj + bj , xi, xj ∈ Vm∑xi∈Vm(ai − bi) = 0 ∀Vm.

(3.5)

Then the system (3.4) becomes a2i−1 + a2i = a2j−1 + a2j xi, xj ∈ Vm∑xi∈Vm(ai − ai−(−1)i) = 0 ∀Vm

(3.6)

First we notice that the solution ai = c ∀i corresponds to the constant solution eigenfunction ψ(x, 0) = 2c(that contributes 1 to the multiplicity for each connected component), but it is not the only one. We makea change of variables defining two associate values for each edge called flux and stab in the following way

Ei 7→

fi = a2i−1 − a2i

si = a2i−1 + a2i.(3.7)

This change of variables has inverse a2i = si−fi2 , a2i−1 = si+fi

2 , so the above equation becomessi = sj xi, xj ∈ Vm∑xi∈Vm(−1)ifb i+1

2 c= 0 ∀Vm

(3.8)

(where bxc is the integer part of x). The first equation means that si = c∀i constant, since the graph issupposed to be connected. The second equation instead admits non-trivial solution if the graph containscycles. In fact if the graph is a tree, then on each leaf edge the flux must be zero; continuing in this fashionwe can see that the only solution is fi ≡ 0. If instead there exists a cycle (Ei1 , Ei2 , . . . , Eik), where all ijare distinct, then a solution is provided also by fi1 = fi2 = · · · = fik = c 6= 0 and fi = 0 for the remainingindeces. In order to show that there is exactly one solution for each independent cycle we start consideringthe spanning tree of a generic graph. This is obtained by removing 1 + ]E − ]V = 1 − χ edges, where χis the Euler characteristic of the graph. Adding one edge at a time the system of equations (3.8) increasesof two variables (fi, si) and of one equation 1, so that there is one more solution. We conclude that thereare 1 + 1− χ solutions on a connected graph. On a general graph (with d connected components) the totalnumber of solutions is 2d− χ = ma(0), since the Euler-characteristic is additive.

1since the edge that was removed has both endpoints part of vertices still present on the spanning tree

21

Figure 3.2: A graph Γ with one cycle, χ = 0; f1 = f2 = f3 = 0, f4 = f5 = f6 = f7 = 1 is a solution to therelative system (3.8). The cycle is highlighed where the flux can be different from zero

3.2 Trace Formula

The formula first appeared in a paper concerning the standard Laplacian by J.P.Roth [16]. We show here amore general version with detailed proof taken from [12] (see also [5]).

3.2.1 The Trace formula

Theorem 3.2 (Trace Formula for the Laplacian LS). Let Γ be a finite compact metric graph with Eulercharacteristic χ and total length L, and let LS(Γ) be the Laplace operator in L2(Γ) determined by properlyconnecting non-resonant matching conditions at the vertices described by the unitary Hermitian matrix S.Then the following trace formula establish the relation between the spectrum k2

n of L(Γ) and the set P of theclosed paths on the metric graph Γ

u(k) := 2ms(0)δ(k) +∑kn 6=0

(δ(k − kn) + δ(k + kn)) (3.9)

= (2ms(0)−ma(0))δ(k) +Lπ

+1

2π

∑p∈P

l(p)(

Ω(p)eikl(p) + Ω∗(p)e−ikl(p)),

where

• ms(0) is the spectral multiplicity of the eigenvalue zero2;• p is any closed path on Γ;• P is the set of closed path p;• l(p) is the length of the closed path p;• p is one of the primitive paths for p;• Ω(p) is the product of all vertex scattering coefficients along the path p3.

Proof.Let f(k) := det(S(k) − I) = 0 be the characteristic equation (2.24) (recall from subsection (2.2.2) that

S = SeSv(k)).Consider the distribution u(k) defined above and notice that it depends only on the spectrum of the

operator. All the non-zero eigenvalues can be otbained as zeros of the analytic4 and then meromorphicfunction f(k). So we can apply the Argument Principle and see that for any test function ϕ ∈ C∞c (R) theaction of the distribution u is given by

(u, ϕ) = limε→0

1

i2π

∫R

(f ′(k − iε)f(k − iε)

− f ′(k + iε)

f(k + iε)

)ϕ(k) dk, (3.10)

which is the integral of the logarithmic derivative of f(k) around the real axis. Next calculations must beunderstood in the distributional sense, so that we can write∑

k2n∈σL

(δ(k − kn) + δ(k + kn)) =1

i2π

(d

dkln(f(k + i0+))− d

dkln(f(k + i0−))

)(3.11)

1

i2π

(d

dkln det(S(k + i0+)− I)− d

dkln det(S(k + i0−)− I)

)=

1

i2π

(Tr

d

dkln(S(k + i0+)− I)− Tr

d

dkln(S(k + i0−)− I)

)2that is the number d of connected components3Sv(p) in the original notation, here we prefer to better distinguish it from the matrix Sv4Since it is a polynomial in eikli , f(k) = det (S(k)− I)

22

We justify this last passage looking at A : R → Mn(C), A ∈ C1 by linearity of derivation, ddx TrA(x) =

Tr ddxA(x) and furthermore because ln detA = Tr lnA; this because all these three functional (logarithm,

trace and determinant) are basis-independent, so we can verify the equivalence in the diagonal matrix casethat is immediate:

ln det diag(a1, a2, . . . , an) = ln((a1 · a2 . . . an) =∑i ln ai = Tr ln diag(a1, a2, . . . , an).

=1

i2πTr

d

dk

(ln(S(k + i0+)− I)− ln(S(k + i0−)− I)

)=

1

i2πTr

(S′(k + i0+)

S(k + i0+)− I− S′(k + i0−)

S(k + i0−)− I

)

Now we write 1S−1 = 1

S1

1−S−1 and then we use the series expansion of 11−X−1

(X − I)−1 =

∑∞n=0−Xn If ‖X‖ < 1∑∞n=0X

−n−1 If ‖X−1‖ < 1(3.12)

Indeed ‖S(k + iε)‖ < 1 if ε > 0 and vice versa for ε < 0. ‖S(k + iε)‖ = ‖SvSe(k + iε)‖ = ‖Se(k + iε)‖ since

Sv is unitary. Furthermore, recall that Se(k) is composed by σj =(

0 eiklj

eiklj 0

)blocks, so ‖Sv(k + iε)‖ =

‖DεSv(k)‖ = ‖Dε‖, because also Se(k) is unitary5 where Dε = Diag(e−εl1 , e−εl1 , e−εl1 , . . . ). ‖Dε‖ < 1 ifε > 0 and vice versa if ε < 0, and so we can exploit both the series. For simplicity we write k± in place ofk ± ε

= limε→0

1

i2πTr

(S−1(k+)

∞∑n=0

S−n(k+)S′(k+)−∞∑n=0

Sn(k−)S′(k−)

)=

=1

i2πTr

( ∞∑n=−∞

Sn(k)S′(k)

).

Furthermore, notice that S′(k) = SvSe(k)iD = iS(k)D with D = Diag (l1, l1, l2, l2, l3, . . . ), so that at last wehave

=1

2πTr

( ∞∑n=−∞

Sn(k)D

). (3.13)

Then it remains to understand the link of this formula with the set of the closed paths P on the graph. Firstwe separate k = 0 from all the other eigenvalues and the case n = 0 from the sum,

2ms(0)δ(k) +∑k2n∈σLkn 6=0

(δ(k − kn) + δ(k + kn)) =Tr(D)

2π+

1

2πTr

∞∑n=−∞n 6=0

Sn(k)D

. (3.14)

From the above definition of D, TrD = 2L it follows that

2ms(0)δ(k) +∑k2n∈σLkn 6=0

(δ(k − kn) + δ(k + kn)) =Lπ

+1

2π

∞∑n=−∞n 6=0

Tr (Sn(k)D) . (3.15)

Now consider one of the terms of the right sum for a fixed n, here ej2Nj=1 is the canonical basis,

Tr (Sn(k)D) =

2N∑j=1

〈ej , Sn(k)Dej〉 =

2N∑j=1

l[ j+12 ]S

nj,j(k) (3.16)

We denote by P the set of all periodic orbits on the graph, with Pn the periodic orbits with discrete length6

n and Pnm the subset of periodic orbits of discrete length n going trough the edge Em = [x2m−1, x2m]. When

5we are still considering k ∈ R6Number of edges crossed by the path

23

the orbit goes from one edge to another it passes through a vertex and we will need to take into account thecorresponding scattering coefficient, that is the contribute from Sv in Sn(k). Then let us denote by T (p) theset of all scattering coefficients (Sv)ij along the orbit p, we find

∞∑n=1

Tr (Sn(k)D) =

2N∑m=1

l[ j+12 ]

∑p∈Pmn

eikl(p)∏

Sv(k)ij∈T (p)

(Sv)ij =

=∑p∈Pn

l(p)eikl(p)∏

Sv(k)ij∈T (p)

(Sv)ij ,

observing that Ω(p) =∏Sv(k)ij∈T (p)(Sv)ij we obtain

∞∑n=1

Tr (Sn(k)D) =∑p∈Pn

l(p)eikl(p)Ω(p). (3.17)

To conclude the proof we need to do the same computation for the negative exponents in the (3.13), weobtain

−∞∑n=−1

Tr (Sn(k)D) =∑p∈Pn

l(p)e−ikl(p)Ω(p) (3.18)

Corollary 3.3 (Trace formula for the standard Laplacian). In the case of the standard Laplacian Lst thetrace formula becomes

u(k) = χδ(k) +Lπ

+1

π

∑p∈P

l(p)Ω(p) cos(kl(p)) (3.19)

Proof. To prove this result we collect the Ω(p) = Ω∗(p) terms

1

π

∑p∈P

l(p)Ω(p) cos(kl(p)) =1

π

∑p∈P

l(p)Ω(p)

(eikl(p) + e−ikl(p)

2

), (3.20)

and recall from Theorem (3.1) that ms(0) = d and ma(0) = 2d− χ.

24

Chapter 4

Topological damping ofAharonov-Bohm effect

The result presented here does not answer completely to the question that was raised at the beginning. Theoriginal task was to find suitable matching conditions on the 8-shape graph that guaranteed the spectrumof the magnetic Schrodinger operator to depend only on the sum 1 of the magnetic fluxes through the twoloops. This was suggested by the existence of matching conditions on the compact 4-star graph such thatthe waves along one side does not interact with the opposite one.

Apparently this is impossible for properly connecting matching conditions, but not completely excludedyet. Instead, it was found that for the same matching conditions considered originally on the compact 4-stargraph, the spectrum does not depend on the two magnetic fluxes provided one of them is zero.

4.1 Graph setting

Let us consider the quantum 8-shape graph with the Schrodinger operator in the presence of a magneticfield a(x). We assume that the electric potential q is zero. The vertices are numerated from 1 to 4 clockwisestarting from the bottom left in the picture.

Figure 4.1: The 8-shape graph

The lenghts of the two loops are l1 = x2 − x1 and l2 = x4 − x3. La,0 =(i ddx + a(x)

)2. Let a(x) be such

that on the two loops its integral is equal to φ1 =∫ x2

x1a(x) dx and φ2 =

∫ x4

x3a(x) dx. Then the left loop has

phase φ1 and the right one has phase φ2.

We already know from Section 2.3 that through the multiplication by U(x) = e−i

∫ xx1a(x)dx

we can takeout the magnetic field from the operator provided that we modify the matching conditions in the followingway,

Sv =

s11 s12 s13 s14

s12 s22 s23 s24

s13 s23 s33 s34

s14 s24 s32 s44

, D =

1 0 0 00 eiφ1 0 00 0 1 00 0 0 eiφ2

, (4.1)

Sv = DSvD−1, (4.2)

then the edge scattering matrix Se becomes

Se(k) =

0 eil1k 0 0

eil1k 0 0 00 0 0 eil2k

0 0 eil2k 0

(4.3)

1without loss of generality, otherwise by the difference

25

4.1.1 Spectrum computed via the Scattering matrices approach

We can effectively calculate the matrix whose determinant yields the characteristic equation,

Se(k)Sv(k)− I4 =

eil1k+iφ1s12 − 1 eil1ks11 eil2k+iφ2s14 eil2ks13

eil1ks22 eil1k−iφ1s12 − 1 eil2k−iφ1+iφ2s24 eil2k−iφ1s23

eil1k+iφ1s23 eil1ks13 eil2k+iφ2s34 − 1 eil2ks33

eil1k+iφ1−iφ2s24 eil1k−iφ2s14 eil2ks44 eil2k−iφ2s34 − 1

(4.4)

Let us consider now the next more specific family of matching conditions and evaluate its characteristicequation

Figure 4.2: A visual representation of the connections given by the matching conditions to the four endpoints:each number correspond to one of the endpoints in the graph, the bold type bonds are the possible passageof a scattering wave from one endpoint to an other one depending by the α and β coefficients

S = S(α, β) =

0 0 α β0 0 −β αα −β 0 0β α 0 0

α2 + β2 = 1 (4.5)

We have considered an Hermitian matrix S so the vertex scattering matrix is energy independent as wehave seen in Proposition 2.3 , Sv(k) = Sv(1) = S,

Se(k)Sv − I4 =

−1 0 βeikl2+iφ2 αeikl2

0 −1 αeikl2+i(φ2−φ1) −βeikl2−iφ1

−βeikl1+iφ1 αeikl1 −1 0αeikl1+i(φ1−φ2) βeikl1−iφ2 0 −1

(4.6)

det (SvSe(k)− I4) = (4.7)

= 1 + α4e2ik(l1+l2) + 2α2β2e2ik(l1+l2) − α2eik(l1+l2)+i(φ1−φ2) − α2eik(l1+l2)−i(φ1−φ2)++β4e2ik(l1+l2) + β2eik(l1+l2)−i(φ1+φ2) + β2eik(l1+l2)+i(φ1+φ2) (4.8)

= 1 + (α4 + β4 + 2α2β2)e2ik(l1+l2) − α2eik(l1+l2)(e+i(φ1−φ2) + e−i(φ1−φ2)

)+

+β2eik(l1+l2)(e−i(φ1+φ2) + e+i(φ1+φ2)

)(4.9)

= 1 + (α2 + β2)2e2ik(l1+l2) − α2eik(l1+l2) (2 cos (φ1 − φ2)) ++β2eik(l1+l2) (2 cos (φ1 + φ2)) (4.10)

= 1 + e2ik(l1+l2) − 2α2eik(l1+l2) cos (φ1 − φ2) + 2β2eik(l1+l2) cos (φ1 + φ2) (4.11)

= eik(l1+l2)(e−ik(l1+l2) + eik(l1+l2) − 2α2eik(l1+l2) cos (φ1 − φ2) + 2β2eik(l1+l2) cos (φ1 + φ2)

)(4.12)

= 2eik(l1+l2)(cos (k(l1 + l2))− α2 cos (φ1 − φ2) + β2 cos (φ1 + φ2)

). (4.13)

In the case α = 1√2

we have:

Se(k)Sv − I4 = eik(l1+l2) (2 cos (k(l1 + l2))− cos (φ1 − φ2) + cos (φ1 + φ2)) . (4.14)

From this equation we deduce that in the case cos(φ1−φ2) = cos(φ1 +φ2) the determinant does not dependon the magnetic field. This happens when φ1 − φ2 = ±(φ1 + φ2) + 2nπ. In other words

φi = nπ i = 1, 2. (4.15)

We also write the final equation in the form

det (SvSe(k)− I4) = 2eik(l1+l2) (cos (k(l1 + l2))− sin (φ1) sin (φ2)) (4.16)

26

k = ±arccos (sin (φ1) sin (φ2)) + 2nπ

l1 + l2n ∈ Z (4.17)

This means in particular that in the case φ1 = 0 the spectrum does not depend on the other flux φ2.This is observed just in the case α = β = 1√

2. Notice that the trivial cases α = 1 and β = 1 refers to the

single loop, that means the four vertices are not properly connected.

4.1.2 Test via Trace formula

Now we use the trace formula applied to this case to explain why the spectrum does not depend on themagnetic field if one of the two fluxes is zero.

Recall the Trace formula from the previous chapter, Theorem 3.2

u(k) := 2ms(0) +∑kn 6=

(δ(k − kn) + δ(l + l(p)) =

= (ma(0)− 2ms(0))δ(k) +Lπ

+1

π

∑p∈P

l(p)Ω(p) cos(kl(p)) (4.18)

The hypotheses are verified since the scattering matrix S = Sv is Hermitian, hence gives non-resonantmatching conditions and the vertex scattering matrix is energy independent.

Multiplicities of the zero eigenvalue The algebraic and spectral multiplicities of zero, that in this casedepend on the particular given matching conditions, appear in the first part of the trace formula. From(4.16) we can see that det (S(0)− I4) = 2(1− sin(φ1) sin(φ2)), so

ma(0) =

2 if sin(φ1) sin(φ2) = 1

0 otherwise.(4.19)

In fact, it is easy to verify that rank (S(0)− I4) = 2.As it concerns the spectral multiplicity we follows the same idea given in the proof of the Theorem 3.1:

let

f(x) =

a1x+ b1 If x ∈ E1

a2x+ b2 If x ∈ E2

(4.20)

This is the only solution to the equation ∆f(x) = 0 on the single edges. Then at the endpoints we have

~f =

b1

a1l1 + b1b2

a2l2 + b2

∂ ~f =

a1

−a1

a2

−a2

(4.21)

The S matrix is Hermitian, then we are allowed to use the simplified matching conditions from (1.29), thenthe matching conditions equations become

− 12 0 1

2√

2eiφ2

2√

2

0 − 12 − e

−iφ1

2√

2e−iφ1+iφ2

2√

21

2√

2− e

iφ1

2√

2− 1

2 0e−iφ2

2√

2eiφ1−iφ2

2√

20 − 1

2

a1

−a1

a2

−a2

= 0 (4.22)

12 0 1

2√

2eiφ2

2√

2

0 12 − e

−iφ1

2√

2e−iφ1+iφ2

2√

21

2√

2− e

iφ1

2√

212 0

e−iφ2

2√

2eiφ1−iφ2

2√

20 1

2

b1a1l1 + b1

b2a2l2 + b2

= 0 (4.23)

We omit the computations and give a sketch of the proof. From the first equation we obtain a1 = a2 = 0,then we plugging this result in the second equation we obtain b1 = b2 = 0 for all possible choices of φ1 andφ2. This proves that ms(0) = 0.

27

Analysis of paths We need to check that the RHS does not depend on the fluxes. Observe that theunique term that involves the fluxes is Ω(p).

Fix one of the two loops, say φ1. We can subdivide the closed paths in two groups:

• The paths on which the wave get a zero magnetic flux from φ1, which are the paths passing an evennumber of times on the first loop, half times in both ways (see figure (4.1.2)).

• The paths passing a different number of times in each way on the loop φ1, so that the wave get anon-zero magnetic flux (see figure (4.1.2)).

From this partition it is clear that on the case in which the flux on the other loop (e.g. φ2) is zero, thenthe sum of the terms indexed by the paths gives a contribution independent of φ1. Instead, the sum of theterms of the second group is zero, since they cancel each others.

So, let p be a path, we necessary have l(p) = n(l1 + 12) with n ∈ N, since the scattering matrix allowsonly the waves to move from a loop to the other, without reflections (Svii = 0 ∀i) or repeated self-loops(Sv12 = Sv34 = 0).

Figure 4.3: Two different path of length l1 + l2

Figure 4.4: A path of length 2(l1 + l2) with zero magnetic field contribution from the right loop

Now we are going to consider the sum ∑p∈P l(p)=n(l1+l2)

SV (p). (4.24)

Without loss of generality we can place the start-end point of the close path on one of the two loops, say[x1, x2]. Then, we see that the sequence of vertex scattering coefficients, that must be multiplied in orderto get Ω(p), starts with one coefficient of the left-bottom block of the Sv matrix. Then it continues with acoefficient of the top-right block. That is the coefficient are provided first by blocks then by the other andso on. In such a way that, if we take a path starting from s13, the next one must be one among s41 or s42,(since from the 3-end interval the path can move only through [x3, x4] until the s41 or the s42 crossing).

Sv =1√2

0 0 1 e−iφ2

0 0 −eiφ1 ei(φ1−φ2)

1 −e−iφ1 0 0eiφ2 ei(φ2−φ1) 0 0

(4.25)

It is not difficult to guess that all the possible combination of Ω(p) are of the type (for simplicity we call sijthe coefficient of the matrix Sv):

Ω(p) =

n∏i=1

sj2i−1l2i−1sl2ij2i (4.26)

where ji ∈ 1, 2, li ∈ 3, 4 and l2i−1 6= l2i, j2i 6= j2i+1 and j0 6= j2n. It is possible to get all these productsfrom the trace of a matrix. Let us call

A =1√2

(1 −e−iφ1

eiφ2 ei(φ2−φ1)

)(4.27)

so that

Sv =

(O2 A∗

A O2

). (4.28)

Let P =

(0 11 0

). Then we have that

28

Proposition 4.1 (trace formula for 8-shape graph).∑p∈P l(p)=n(l1+l2)

Ω(p) = Tr ((PA∗PA)n) . (4.29)

Furthermore, the matrix PA∗PA has eigenvalue e1, e2 independent from the magnetic field if any of the twofluxes φ1 or φ1 valishes. The same holds in (4.29) for any n ∈ N.

Proof. We show that each closed path p of length n(l1 + l2) starting from z0 and passing through s13 is ofthe type: Ω(p) = s13s∗∗ . . . s∗∗s∗2. Then we show that these terms are summed up in (PA∗PA)n11. In thesame way one can see that the terms corresponding to the path starting with s24 or s23 are summed up in(PA∗PA)n22. To prove the result, we compute the eigenvalue of the matrix

PA∗PA =

((i cos(φ1) + sin(φ1)) sin(φ2) − cos(φ2)

cos(φ2) (−i cos(φ1) + sin(φ1)) sin(φ2)

), (4.30)

that are

e1, e2 =1

4

(cos(φ1 − φ2)− cos(φ1 + φ2)±

√sin2(φ1) sin2(φ2)− 1

). (4.31)

If φ1 or φ2 = 2nπ, n ∈ N, then

e1 =i

4, e2 = − i

4. (4.32)

Hence, the trace formula does not depend on the other magnetic flux.

4.2 Conclusions

In the above computations we found that the S matrix (4.5) for α = β = 1√2

gives a particular matching

condition to the 8-shape graph such that the spectrum of a magnetic Schrodinger operator with zero electricpotential becomes independent of the magnetic fluxes φ1, φ2 if conditions (4.15) are verified.

The Aharonov-Bohm effect, described in [1] tells that a charged particle can be affected by the magneticfield, even if the particle is confined into a region in which the magnetic field is zero. Here we observe thecontrary: the spectrum is not influenced by the magnetic field even though the field is non-zero.

We are planning to look for other matching conditions exhibiting similar behavior and look for moregeneral graphs, or class of graphs with similar properties. Now it seems that an analogue effect cannot beachieved on the 3-loops graphs.

29

Bibliography

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Symbol Table

Symbol Descriptionχ Euler characteristic of a graph (V,E), χ = ]V − ]EΓ Quantum graph∂ Extended normal derivative

∂N Normal derivativeE Set of all endpointsEj j-Edge of a graphEn Symplectic matrix of order nφ magnetic flux inside a loop

l(p) Length of a path pL Total length of a graph

Lq,a Magnetic Schrodinger operatorLS General Laplace operator with matching conditions given by SLSt Standard Laplace operator, i.e. a Laplacian with SMCM metric graphp Closed Path on a graphp Primitive Path of p, one of the shortest paths such that p is a multiple of itP Set of all closed path on a graphP Permutation matrixVj j-Vertex of a graphS Unitary matrix for Matching Condition

S(k) The product of Vertex and Edge scattering matricesSe(k) Edge Scattering MatrixSMC Standard Matching ConditionSv Vertex Scattering Matrix energy independent

Sv(k) Vertex Scattering MatrixSVj Matching conditions restricted to the single vertex Vj

Tq(λ)x2x1

Transfer Matrix from x1 to x2 for the λ-eigenvalue equationΩ(p) Product of scattering terms alogn a path p

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