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- defined function
- spectral theory
- problem can
- symmetrization problem
- operator defined
- tunnel effect
- transport operator
- klein-gordon equations
- branche
- given spectral

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STAR-SHAPED NETWORK: AN EXPLICIT FORMULA FOR THE

SPECTRAL REPRESENTATION

´F. ALI MEHMETI, R. HALLER-DINTELMANN, AND V. REGNIER

Abstract. We consider the Klein-Gordon equation on a star-shaped network composed of n

half-axes connected at their origins. We add a potential which is constant but diﬀerent on each

branch. The corresponding spatial operator is self-adjoint and we state explicit expressions

for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This

leads to a generalized Fourier type inversion formula in terms of an expansion in generalized

eigenfunctions. Furtherweprovethesurjectivityoftheassociatedtransformation, thusshowing

that it is in fact a spectral representation.

The characteristics of the problem are marked by the non-manifold character of the star-

shaped domain. Therefore the approach via the Sturm-Liouville theory for systems is not

well-suited. The considerable eﬀort to construct explicit formulas involving the tunnel eﬀect

generalized eigenfunctions is justiﬁed for example by the perspective to study the inﬂuence of

∞tunnel eﬀect on the L -time decay.

1. Introduction

This paper is motivated by the attempt to study the local behavior of waves near a node in

a network of one-dimensional media having diﬀerent dispersion properties. This leads to the

study of a star-shaped network with semi-inﬁnite branches. Results in experimental physics

[18, 20], theoretical physics [15] and functional analysis [8, 14] describe new phenomena created

in this situation by the dynamics of the tunnel eﬀect: the delayed reﬂection and advanced

transmission near nodes issuing two branches. It is of major importance for the comprehension

of the vibrations of networks to understand these phenomena near ramiﬁcation nodes i.e. nodes

with at least 3 branches. The associated spectral theory induces a considerable complexity (as

compared with the case of two branches) which is unraveled in the present paper.

Perturbation arguments do not seem to be suited for the above mentioned applications, as

they lead to Neumann series in the solution formulae that hide the interesting eﬀects.

The dynamical problem can be described as follows:

Let N ,...,N be n disjoint copies of (0,+∞) (n ≥ 2). Consider numbers a ,c satisfying1 n k k

0 < c , for k = 1,...,n and 0 ≤ a ≤ a ≤ ... ≤ a < +∞. Find a vector (u ,...,u ) ofk 1 2 n 1 n

functions u : [0,+∞)×N →C satisfying the Klein-Gordon equationsk k

2 2[∂ −c ∂ +a ]u (t,x) = 0 ,k = 1,...,n,k k kt x

on N ,...,N coupled at zero by usual Kirchhoﬀ conditions and complemented with initial1 n

conditions for the functions u and their derivatives.k

2000 Mathematics Subject Classiﬁcation. Primary 34B45; Secondary 42A38, 47A10, 47A60, 47A70.

Key words and phrases. networks, spectral theory, resolvent, generalized eigenfunctions, functional calculus,

evolution equations, dynamics of the tunnel eﬀect, Klein-Gordon equation.

Parts of this work were done, while the second author visited the University of Valenciennes. He wishes to

express his gratitude to F. Ali Mehmeti and the LAMAV for their hospitality.

1´2 F. ALI MEHMETI, R. HALLER-DINTELMANN, AND V. REGNIER

Reformulating this as an abstract Cauchy problem, one is confronted with the self-adjoint

2 2operator A = (−c ·∂ +a ) in L (N), with a domain that incorporates the Kirchhoﬀk k k=1,...,nx

transmission conditions at zero. For an exact deﬁnition of A, we refer to Section 2.

Invoking functional calculus for this operator, the solution can be given in terms of

√ √

±i At ±i Ate u and e v .0 0

The reﬁned study of transient phenomena thus requires concrete formulae for the spectral rep-

resentation of A. The seemingly straightforward idea to view this task as a Sturm-Liouville

problem for a system (following [27]) is not well-suited, because the resulting expansion for-

mulae do not take into account the non-manifold character of the star-shaped domain. The

ansatz used in [27] inhibits the exclusive use of generalized eigenfunctions satisfying the Kirch-

hoﬀ conditions. This is proved in Theorem 8.1 in the appendix of this paper, which furnishes

the comparison of the two approaches.

A ﬁrst attempt to use well-suited generalized eigenfunctions in the ramiﬁed case but without

tunnel eﬀect [5] leads to a transformation whose inverse formula is diﬀerent on each branch.

The desired results for two branches but with tunnel eﬀect are implicitly included in [27]. Forn

branches but with the same c and a on all branches a variant of the above problem has beenk k

treated in [7] using Laplace transform in t.

In the present paper we start by following the lines of [5]. In Section 3, we deﬁne n families

kof generalized eigenfunctions of A, i.e. formal solutions F for λ∈ [a ,+∞) of the equation1λ

k kAF =λFλ λ

√

±i λt ksatisfyingtheKirchhoﬀconditionsinzero,suchthate F (x)representincomingoroutgoing

λ

plane waves on all branches except N for λ∈ [a ,+∞). For λ∈ [a ,a ), 1≤p<n we havek n p p+1

nopropagationbutexponential decayinn−p branches: this expresseswhatwe callthe multiple

tunnel eﬀect, which is new with respect to [5]. Using variation of constants, we derive a formula

kfor the kernel of the resolvent of A in terms of the F .

λ

Following the classical procedure, in Section 4 we derive a limiting absorption principle for

A, and then we insert A in Stone’s formula to obtain a representation of the resolution of the

identity of A in terms of the generalized eigenfunctions.

The aim of the paper, attained in Section 7, is the analysis of the Fourier type transformation

Z

k(Vf)(λ) := (Vf) (λ) := f(x)(F )(x)dxk k=1,...,n λ

k=1,...,nN

in view of constructing its inverse. We show that it diagonalizes A and determine a metric

setting in which it is an isometry. This permits to express regularity and compatibility of f in

terms of decay of Vf.

A major task in this context is to overcome the cyclic structure that the cyclic nature of

the n-star induces in the underlying resolvent formula derived in Section 3. To this end, we

use a symmetrization procedure that is carried out in Section 5. It combines the expression

for the resolution of the identity E(a,b) found in Section 4 with an ansatz for an expansion in

generalized eigenfunctions:

Z nb X

lf(x) = q (λ)F (x)(Vf) (λ)dλ.lm kλ

a

l,m=1

2 2This creates a (3n + 1)×n linear system for the q , whose solution leads to the result inlm

Theorem 5.3 and to the Plancherel type formula.THE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 3

A direct approach to the same symmetrization problem, carried out in Section 6, yields a

closed formula for the matrix q based on n×n matrices. This approach is to a great extent

independent of the special setting and is thus supposed to be generalizable.

In Section 7 the desired inversion formula as well as the Plancherel type theorem are stated.

Finally the domains of the powers ofA are characterized using the decay properties ofVf. We

show thatV is an ordered spectral representation (see Deﬁnition XII.3.15, p. 1216 of [17]). The

spectrum has n layers and it is p-fold on the frequency band [a ,a ). This reﬂects a kindp p+1

of continuous Zeemann eﬀect caused by the constant, semi inﬁnite potentials on the branches

N given by the terms a u . On this frequency band the generalized eigenfunctions have anj j j

exponential decay on n−p branches, expressing the multiple tunnel eﬀect.

Ourresultsaredesignedtoserveastoolsinapplicationsconcerningthedynamicsofthetunnel

eﬀect at ramiﬁcation nodes. In particular, we think of retarded reﬂection (following [8, 20]),

∞advanced transmission at barriers (following [18, 15, 14]), L -time decay (following [2, 3]), the

study of more general networks of wave guides (for example microwave networks [25]), causality

and global existence for nonlinear hyperbolic equations (following [9]) and the generalization to

coupled transmission conditions (following [12]).

Finally, let us comment on some related results. The existing general literature on expansions

in generalized eigenfunctions ([11, 24, 27] for example) does not seem to be helpful for our kind

of problem: their constructions start from an abstractly given spectral representation. But in

concrete cases you do not have an explicit formula for it at the beginning.

∞In[10]therelationoftheeigenvaluesoftheLaplacianinanL -settingoninﬁnite,locallyﬁnite

networks to the adjacency operator of the network is studied. The question of the completeness

2ofthecorrespondingeigenfunctions, viewedasgeneralizedeigenfunctionsinanL -setting, seems

to be open.

In[22],theauthorsconsidergeneralnetworkswithsemi-inﬁniteends. Theygiveaconstruction

to compute generalized eigenfunctions from the coeﬃcients of the transmission conditions and

the asymptotic behaviour of eigenvalues is studied. Thegenerality ofthe aproach, however, does

not allow for explicit inversion formulas for a given family of generalized eigenfunctions.

Spectral theory for the Laplacian on ﬁnite networks has been studied since the 1980ies for

example by J.P. Roth, J.v. Below, S. Nicaise, F. Ali Mehmeti. A list of references can be found

in [1].

In [23] the transport operator is considered on ﬁnite networks. The connection between the

spectrum of the adjacency matrix of the network and the (discrete) spectrum of the transport

operator is established. A generalization to inﬁnite networks is contained in [16].

For surveys on results on networks and multistructures, cf. [4, 19].

Many results have been obtained in spectral theory for elliptic operators on various types of

unbounded structures for example [21, 13, 6, 3], cf. especially the references mentioned in [3].

2. Data and functional analytic framework

Let us introduce some notation which will be used throughout the rest of the paper.

Domain and functions. Let N ,...,N be n disjoint sets identiﬁed with (0,+∞) (n ∈ N,1 nSnn ≥ 2) and put N := N , identifying the endpoints 0, see [1] for a detailed deﬁnition.kk=1

Furthermore, we write [a,b] for the interval [a,b] in the branch N . For the notation ofN kk

functions two viewpoints are used:

• functions f on the object N and f is the restriction of f to N .k k

• n-tuples of functions on the branches N ; then sometimes we write f = (f ,...,f ).k 1 n´4 F. ALI MEHMETI, R. HALLER-DINTELMANN, AND V. REGNIER

Transmission conditions.

nY

0(T ): (u ) ∈ C (N ) satisﬁes u (0) =u (0) ∀ i,k∈{1,...,n}.0 k k=1,...,n k i k

k=1

This condition in particular implies that (u ) may be viewed as a well-deﬁned functionk k=1,...,n

on N.

n nY X

1 +(T ): (u ) ∈ C (N ) satisﬁes c ·∂ u (0 ) = 0.1 k k=1,...,n k k x k

k=1 k=1

Q

n 2Deﬁnition of the operator. Deﬁne the real Hilbert space H = L (N ) with scalarkk=1

product

nX

(u,v) = (u ,v ) 2H k k L (N )k

k=1

and the operator A :D(A)−→H by

nn oY

2D(A) = (u ) ∈ H (N ) : (u ) satisﬁes (T ) and (T ) ,k k=1,...,n k k k=1,...,n 0 1

k=1

2A((u ) ) = (A u ) = (−c ·∂ u +a u ) .k k=1,...,n k k k=1,...,n k k k k k=1,...,nx

Note that, if c = 1 and a = 0 for every k ∈{1,...,n}, A is the Laplacian in the sense of thek k

existing literature, cf. [10, 22].

Proposition2.1. The operatorA :D(A)→H deﬁned above is self-adjoint and satisﬁesσ(A)⊂

[a ,+∞).1

Proof. Consider the Hilbert space

nY

1V = (u ) ∈ H (N ) : (u ) satisﬁes (T )k k=1,...,n k k k=1,...,n 0

k=1

with the canonical scalar product (·,·) . Then the bilinear form associated with A+(ε−a )IV 1

is a :V ×V →C withε

nX

a (u,v) = c (∂ u ,∂ v ) 2 +(a +ε−a )(u ,v ) 2 .ε k x k x k k 1 k kL (N ) L (N )k k

k=1

Then clearly there is a C > 0 with a (u,u)≥C(u,u) for all u∈V and all ε> 0. By partialε V

integration one shows that the Friedrichs extension of (a ,V,H) is (A+(ε−a )I,D(A)). Thusε 1

the operatorA+(ε−a )I is self-adjoint and positive. Henceσ(A+(ε−a )I)⊂ [0,+∞) for all1 1

ε> 0, what implies the assertion on the spectrum.

3. Expansion in generalized eigenfunctions

The aim of this section is to ﬁnd an explicit expression for the kernel of the resolvent of the

operator A on the star-shaped network deﬁned in the previous section.

Q

n ∞Deﬁnition 3.1. An element f ∈ C (N ) is called generalized eigenfunction of A, if itkk=1

satisﬁes (T ), (T ) and formally the diﬀerential equation Af =λf for some λ∈C.0 1THE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 5

Lemma 3.2 (Green’s formula on the star-shaped network). Denote by V the subset of thel ,...,ln1

network N deﬁned by

n[

V ={0}∪ (0,l ) .l ,...,l k N1 n k

k=1

Then u,v∈D(A) implies

Z Z n nX X

00 00 0 0u (x)v(x)dx = u(x)v (x)dx− u(l )v (l )+ u(l )v(l ).k k k k

V Vl ,...,l l ,...,ln n1 1 k=1 k=1

Proof. Two successive integrations by parts are used and since both u and v belong to D(A),

they both satisfy the transmission conditions (T ) and (T ). So0 1

n nX X

0 0u (0)v (0) =u (0) v (0) = 0.k 1k k

k=1 k=1

Pn 0Idem for u (0)v (0). kk=1 k

This Green formula yields now as usual an expression for the resolvent of A in terms of the

generalized eigenfunctions.

λ λProposition 3.3. Let λ∈ρ(A) be ﬁxed and let e , e be generalized eigenfunctions of A, such1 2

λthat the Wronskian w (x) satisﬁes for every x in N1,2

λ λ λ λ λ 0 λ 0 λw (x) = detW(e (x),e (x)) =e (x)·(e )(x)−(e )(x)·e (x) = 0.1,2 1 2 1 2 1 2

λ 2 λ 2If for some k ∈{1,...,n} we have e | ∈ H (N ) and e | ∈ H (N ) for all m = k, thenN k N mm2 k 1

for every f ∈H and x∈Nk

" #Z Z

1 λ λ 0 0 0 λ λ 0 0 0[R(λ,A)f](x) = · e (x)e (x)f(x)dx + e (x)e (x)f(x)dx . (1)1 2 2 1λc w (x)k (x,+∞) N\(x,+∞)1,2 N Nk k

Note that by the integral over N, we mean the sum of the integrals over N , k = 1,...,n.k

Proof. Letλ∈ρ(A). We shall show that the integral operator deﬁned by the right-hand side of

(1) is a left inverse of λI−A. Let u∈D(A) and x∈N . Thenk

Z Z

λ λ 0 0 0 λ λ 0 0 0I := e (x)e (x)(λI−A)u(x)dx + e (x)e (x)(λI−A)u(x)dxλ 1 2 2 1

(x,+∞) N\(x,+∞)N Nk k

Z Zlk

λ λ 0 0 0 λ λ 0 0 0=e (x) lim e (x)(λI−A)u(x)dx +e (x) lim e (x)(λI−A)u(x)dx,1 2 2 1

l →∞ l →∞,m=kk mx Vl ,...,l ,x,l ,...,ln1 k−1 k+1

1due to the dominated convergence Theorem, the integrands being in L (R) by the hypotheses.Qn 2We have u∈D(A)⊂ H (N ) andjj=1

λ 2 λ 2e | ∈H (N ), e | ∈H (N ), m =kN k N m2 k 1 m

by hypothesis and thus

λ λ∂ u| (x)·e | (x) −→ 0, u| (x)·∂ e | (x) −→ 0,x N N N x N2 2k k k kx→+∞ x→+∞

λ λ∂ u| (x)·e | (x) −→ 0, u| (x)·∂ e | (x) −→ 0, m =k,x N N N x Nm 1 m m 1 m

x→+∞ x→+∞

2all products being in some H (N ). Recall thatj

Z Zb b

00 00 0 0 0 0f g = fg −f(b)g (b)+f (b)g(b)+f(a)g (a)−f (a)g(a)

a a

66666´6 F. ALI MEHMETI, R. HALLER-DINTELMANN, AND V. REGNIER

2 λfor f,g∈H ((a,b)). Now Lemma 3.2 and (λI−A)e = 0 for r = 1,2 implyr

Zh lk

λ λ 0 0 0I =e (x) lim (λI−A)e (x)u(x)dxλ 1 2

l →∞k x i

λ λ λ λ+c −u(l )∂ e (l )+∂ u(l )e (l )+u(x)∂ e (x)−∂ u(x)e (x)k k x k x k k x x2 2 2 2

Zh

λ λ 0 0 0+e (x) lim (λI−A)e (x)u(x)dx2 1

l →∞,m=km Vl ,...,l ,x,l ,...,ln1 k−1 k+1 iX

λ λ λ λ+ c lim −u(l )∂ e (l )+∂ u(l )e (l ) +c −u(x)∂ e (x)+∂ u(x)e (x)j j x j x j j k x x1 1 1 1

l →∞j

j=k

λ λ λ λ=c e (x)∂ e (x)−∂ e (x)e (x) u(x)k x x1 2 1 2

λ=c w (x)u(x).k 1,2

Now the invertibility of λI−A implies the result.

Deﬁnition 3.4 (Generalized eigenfunctions of A). For k∈{1,...,n} and λ∈C let

s P

cξ (λ)λ−a l lk l=kξ (λ) := and s :=− .k k

c c ξ (λ)k k k

√

iφHere, and in all what follows, the complex square root is chosen in such a way that r·e =√

iφ/2re with r> 0 and φ∈ [−π,π).

±,j ±,j ±,jFor λ∈C and j,k ∈{1,...,n}, F :N →C is deﬁned for x∈N by F (x) := F (x)kλ λ λ,k

with (

±,jF (x) = cos(ξ (λ)x)±is (λ)sin(ξ (λ)x),j j jλ,j

±,j

F (x) = exp(±iξ (λ)x), for k =j.kλ,k

±,j

Remark 3.5. • F satisﬁes the transmission conditions (T ) and (T ) and formally it0 1λ

±,j ±,j ±,j

holdsAF =λF . Thus it is a generalized eigenfunction ofA, but clearlyF doesλ λ λ

not belong to H, so it is not a classical eigenfunction.

±,j• For Im(λ) = 0, the function F , where the +-sign (respectively −-sign) is chosen if

λ,k

2Im(λ)> 0 (respectively Im(λ)< 0), belongs to H (N ) for k =j. This feature is usedk

in the formula for the resolvent of A.

PnDeﬁnition3.6(Kerneloftheresolvent). Letw :C→Cbedeﬁnedbyw(λ) :=±i· c ξ (λ).j jj=1

For any λ∈C such that w(λ) = 0, j ∈{1,...,n}, and for every x∈N , we deﬁnej

1 ±,j ±,j+1 0 0 0 F (x)F (x), for x ∈N , x >x, jλ,j λ,jw(λ)0K(x,x,λ) =

1 ±,j+1 ±,j 0 0 0 0 F (x)F (x), for x ∈N ,k =j or x ∈N , x <x.jkλ,j λw(λ)

Inthewholeformula+(respectively−)ischosen,ifIm(λ)> 0(respectivelyIm(λ)≤ 0). Finally,

the index j is to be understood modulo n, that is to say, if j =n, then j +1 = 1.

Figure 1 shows the domain of the kernel K(·,·,λ) in the case n = 3 with its three main

diagonals, where the kernel is not smooth. We will show in Theorem 3.8 that K is indeed

the kernel of the resolvent of A. In order to do so, we collect some useful observations in the

following lemma.

66666666THE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 7

ZZ

Z

Z

0Z x ∈N1x∈N2 Z

Z

Z}Z 6Z Z Z

Z

Z

Z diagN×N1 1

-Z x∈N1Z JdiagN×N2 2 J

J

JdiagN×N3 3

J J^ 0 x ∈N3+ 0J x ∈N 2

x∈N J3

J

J

Figure 1. N×N in the case n = 3

√

Note that in particular, if c =c and a = 0 for all j ∈{1,...,n}, then w(λ) =±inc λ forj j

all j ∈ {1,...,n}, which only vanishes for λ = 0. On the other hand, if there exist i and j in

{1,...,n}, such that a = a , then it is clear that w(λ) never vanishes on R, but we need toi j

know, if it vanishes onC.

Pn2Lemma 3.7. i) For a ≤λ and ε≥ 0 holds |w(λ−iε)| ≥ c |λ−a |.1 j jj=1

ii) For λ∈ρ(A) such that Re(λ)≥a , the Wronskian w only vanishes at λ =α, if a =α1 k

for all k∈{1,...,n}.

Proof. We ﬁrst prove i). Note that for z ,z ,...,z ∈C holds1 2 n

n n n X X X2