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MULTIPLE TUNNEL EFFECT FOR DISPERSIVE WAVES ON A STAR SHAPED NETWORK: AN EXPLICIT FORMULA FOR THE

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MULTIPLE TUNNEL EFFECT FOR DISPERSIVE WAVES ON A 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. Further we prove the surjectivity of the associated transformation, thus showing 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.

  • defined function

  • spectral theory

  • problem can

  • symmetrization problem

  • operator defined

  • tunnel effect

  • transport operator

  • klein-gordon equations

  • branche

  • given spectral


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MULTIPLE TUNNEL EFFECT FOR DISPERSIVE WAVES ON A
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