The Klein-Gordon equation with multiple tunnel effect on a star-shaped network: Expansions in generalized eigenfunctions
Felix Ali Mehmeti, Robert Haller-Dintelmann, Virginie Régnier
aa r X i v : . [ m a t h . SP ] J un THE KLEIN-GORDON EQUATION WITH MULTIPLE TUNNEL EFFECTON A STAR-SHAPED NETWORK: EXPANSIONS IN GENERALIZEDEIGENFUNCTIONS
F. ALI MEHMETI, R. HALLER-DINTELMANN, AND V. R´EGNIER
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 eachbranch. The corresponding spatial operator is self-adjoint and we state explicit expressionsfor its resolvent and its resolution of the identity in terms of generalized eigenfunctions. Thisleads to a generalized Fourier type inversion formula in terms of an expansion in generalizedeigenfunctions.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 notwell-suited. Introduction
This paper is motivated by the attempt to study the local behavior of waves near a node ina network of one-dimensional media having different dispersion properties. This leads to thestudy of a star-shaped network with semi-infinite branches. Recent results in experimentalphysics [17, 19], theoretical physics [14] and functional analysis [8, 13] describe new phenomenacreated in this situation by the dynamics of the tunnel effect: the delayed reflection and advancedtransmission near nodes issuing two branches. It is of major importance for the comprehensionof the vibrations of networks to understand these phenomena near ramification nodes i.e. nodeswith at least 3 branches. The associated spectral theory induces a considerable complexity (ascompared with the case of two branches) which is unraveled in the present paper.The dynamical problem can be described as follows:Let N , . . . , N n be n disjoint copies of (0 , + ∞ ) ( n ≥ a k , c k satisfying0 < c k , for k = 1 , . . . , n and 0 ≤ a ≤ a ≤ . . . ≤ a n < + ∞ . Find a vector ( u , . . . , u n ) offunctions u k : [0 , + ∞ ) × N k → C satisfying the Klein-Gordon equations[ ∂ t − c k ∂ x + a k ] u k ( t, x ) = 0 , k = 1 , . . . , n, on N , . . . , N n coupled at zero by usual Kirchhoff conditions and complemented with initialconditions for the functions u k and their derivatives.Reformulating this as an abstract Cauchy problem, one is confronted with the self-adjointoperator A = ( − c k · ∂ x + a k ) k =1 ,...,n in L ( N ), with a domain that incorporates the Kirchhofftransmission 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 e ± i √ At u and e ± i √ At v . Mathematics Subject Classification.
Primary 34B45; Secondary 42A38, 47A10, 47A60, 47A70.
Key words and phrases. networks, spectral theory, resolvent, generalized eigenfunctions, functional calculus,evolution equations.Parts of this work were done, while the second author visited the University of Valenciennes. He wishes toexpress his gratitude to F. Ali Mehmeti and the LAMAV for their hospitality.
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-Liouvilleproblem 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. Theansatz 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 furnishesthe comparison of the two approaches.A first attempt to use well-suited generalized eigenfunctions in the ramified case but withouttunnel 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]. For n branches but with the same c k and a k on all branches a variant of the above problem has beentreated 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 familiesof generalized eigenfunctions of A , i.e. formal solutions F kλ for λ ∈ [ a , + ∞ ) of the equation AF kλ = λF kλ satisfying the Kirchhoff conditions in zero, such that e ± i √ λt F kλ ( x ) represent incoming or outgoingplane waves on all branches except N k for λ ∈ [ a n , + ∞ ). For λ ∈ [ a p , a p +1 ), 1 ≤ p < n we haveno propagation but exponential decay in n − p branches: this expresses what we call the multipletunnel effect, which is new with respect to [5]. Using variation of constants, we derive a formulafor the kernel of the resolvent of A in terms of the F kλ .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 theidentity 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( V f )( λ ) := (cid:0) ( V f ) k ( λ ) (cid:1) k =1 ,...,n := (cid:16)Z N f ( x )( F kλ )( x ) dx (cid:17) k =1 ,...,n in view of constructing its inverse. We show that it diagonalizes A and determine a metricsetting in which it is an isometry. This permits to express regularity and compatibility of f interms of decay of V f .Following [5] up to the end would induce a major defect in the last step of this program: thePlancherel type formula would read k f k H = Re (cid:26) n X j =1 Z σ ( A ) κ j ( λ ) (cid:0) V ( N j f ) (cid:1) j +1 ( λ ) ( V f ) j ( λ ) dλ (cid:27) , where the indices j, j + 1 are considered modulo n , and κ j is a suitable weight. This cyclicstructure stems from the underlying formula for the resolvent derived in Section 3, which reflectsthe invariance of a star-shaped network with respect to cyclic permutation of the indices of thebranches and thus the non-manifold character of the domain. This feature inhibits the analysisof the decay properties of the ( V f ) k : the finiteness of k f k H does not automatically imply thedecay of the terms on the right-hand side. In fact, the cutoff by the characteristic function N j causes a poor decay in λ .Consequently, the main objective of the present paper is the elimination of the cyclic structure,which is inevitable in the kernel of the resolvent, from the Plancherel type formula. To this end,we use a symmetrization procedure leading to a true formula of Plancherel type k f k H = n X j =1 Z σ ( A ) σ j ( λ ) | ( V f ) j ( λ ) | dλ. HE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 3
This is carried out in Section 5 combining the expression for the resolution of the identity E ( a, b )found in Section 4 with an ansatz for an expansion in generalized eigenfunctions: f ( x ) = Z ba n X l,m =1 q lm ( λ ) F lλ ( x )( V f ) k ( λ ) dλ. This creates an (3 n + 1) × n linear system for the q lm , whose solution leads to the result inTheorem 5.3 and to the Plancherel type formula.A direct approach to the same symmetrization problem, carried out in Section 6, yields aclosed formula for the matrix q based on n × n matrices. This approach is to a great extentindependent 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 of A are characterized using the decay properties of V f . Wesee that V exhibits all features of an ordered spectral representation (see Definition XII.3.15, p.1216 of [16]) except for the surjectivity, which is not essential for applications. The spectrumhas n layers and it is p -fold on the frequency band [ a p , a p +1 ). This reflects a kind of continuousZeemann effect due to the perturbation caused by the constant, semi infinite potentials on the N j given by the terms a j u j . On this frequency band the generalized eigenfunctions have anexponential decay on n − p branches, expressing the multiple tunnel effect.Our results are designed to serve as tools in some pertinent applications concerning thedynamics of the tunnel effect at ramification nodes. In particular, we think of retarded reflection(following [8, 19]), advanced transmission qt barriers (following [17, 14, 13]), L ∞ -time decay(following [2, 3]), the study of more general networks of wave guides (for example microwavenetworks [25]) and causality and global existence for nonlinear hyperbolic equations (following[9]).Finally, let us comment on some related results. The existing general literature on expansionsin generalized eigenfunctions ([11, 24, 27] for example) does not seem to be helpful for our kindof problem: their constructions start from an abstractly given spectral representation. But inconcrete cases you do not have an explicit formula for it at the beginning.In [10] the relation of the eigenvalues of the Laplacian in an L ∞ -setting on infinite, locally finitenetworks to the adjacency operator of the network is studied. The question of the completenessof the corresponding eigenfunctions, viewed as generalized eigenfunctions in an L -setting, couldbe asked.In [21], the authors consider general networks with semi-infinite ends. They give a constructionto compute some generalized eigenfunctions from the coefficients of the transmission conditions(scattering matrix). The eigenvalues of the associated Laplacian are the poles of the scatteringmatrix and their asymptotic behaviour is studied. But no attempt is made to construct explicitinversion formulas for a given family of generalized eigenfunctions.Spectral theory for the Laplacian on finite networks has been studied since the 1980ies forexample by J.P. Roth, J.v. Below, S. Nicaise, F. Ali Mehmeti. A list of references can be foundin [1].In [22] the transport operator is considered on finite networks. The connection between thespectrum of the adjacency matrix of the network and the (discrete) spectrum of the transportoperator is established. A generalization to infinite networks is contained in [15].Gaussian estimates for heat equations on networks have been proved in [23].For surveys on results on networks and multistructures, cf. [4, 18].Many results have been obtained in spectral theory for elliptic operators on various types ofunbounded domains for example [20, 12, 6, 3], cf. especially the references mentioned in [3]. F. ALI MEHMETI, R. HALLER-DINTELMANN, AND V. R´EGNIER 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 n be n disjoint sets identified with (0 , + ∞ ) ( n ∈ N , n ≥
2) and put N := S nk =1 N k , identifying the endpoints 0, see [1] for a detailed definition.Furthermore, we write [ a, b ] N k for the interval [ a, b ] in the branch N k . For the notation offunctions two viewpoints are used: • functions f on the object N and f k is the restriction of f to N k . • n -tuples of functions on the branches N k ; then sometimes we write f = ( f , . . . , f n ). Transmission conditions. ( T ): ( u k ) k =1 ,...,n ∈ n Y k =1 C ( N k ) satisfies u i (0) = u k (0) ∀ i, k ∈ { , . . . , n } . This condition in particular implies that ( u k ) k =1 ,...,n may be viewed as a well-defined functionon N . ( T ): ( u k ) k =1 ,...,n ∈ n Y k =1 C ( N k ) satisfies n X k =1 c k · ∂ x u k (0 + ) = 0 . Definition of the operator.
Define the real Hilbert space H = Q nk =1 L ( N k ) with scalarproduct ( u, v ) H = n X k =1 ( u k , v k ) L ( N k ) and the operator A : D ( A ) −→ H by D ( A ) = n ( u k ) k =1 ,...,n ∈ n Y k =1 H ( N k ) : ( u k ) k =1 ,...,n satisfies ( T ) and ( T ) o ,A (( u k ) k =1 ,...,n ) = ( A k u k ) k =1 ,...,n = ( − c k · ∂ x u k + a k u k ) k =1 ,...,n . Note that, if c k = 1 and a k = 0 for every k ∈ { , . . . , n } , A is the Laplacian in the sense of theexisting literature, cf. [10, 21, 23]. Notation for the resolvent.
The resolvent of an operator T is denoted by R , i.e. R ( z, T ) =( zI − T ) − for z ∈ ρ ( T ). Proposition 2.1.
The operator A : D ( A ) → H defined above is self-adjoint and satisfies σ ( A ) ⊂ [ a , + ∞ ) .Proof. Consider the Hilbert space V = (cid:26) ( u k ) k =1 ,...,n ∈ n Y k =1 H ( N k ) : ( u k ) k =1 ,...,n satisfies ( T ) (cid:27) with the canonical scalar product ( · , · ) V . Then the bilinear form associated with A + ( ε − a ) I is a ε : V × V → C with a ε ( u, v ) = n X k =1 (cid:2) c k ( ∂ x u k , ∂ x v k ) L ( N k ) + ( a k + ε − a )( u k , v k ) L ( N k ) (cid:3) . Then clearly there is a
C > a ε ( u, u ) ≥ C ( u, u ) V for all u ∈ V and all ε >
0. By partialintegration one shows that the Friedrichs extension of ( a ε , V, H ) is ( A + ( ε − a ) I, D ( A )). Thus HE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 5 the operator A + ( ε − a ) I is self-adjoint and positive. Hence σ ( A + ( ε − a ) I ) ⊂ [0 , + ∞ ) for all ε >
0, what implies the assertion on the spectrum. (cid:3) Expansion in generalized eigenfunctions
The aim of this section is to find an explicit expression for the kernel of the resolvent of theoperator A on the star-shaped network defined in the previous section. Definition 3.1.
An element f ∈ Q nk =1 C ∞ ( N k ) is called generalized eigenfunction of A , if itsatisfies ( T ), ( T ) and formally the differential equation Af = λf for some λ ∈ C . Lemma 3.2 (Green’s formula on the star-shaped network) . Denote by V l ,...,l n the subset of thenetwork N defined by V l ,...,l n = { } ∪ n [ k =1 (0 , l k ) N k . Then u, v ∈ D ( A ) implies Z V l ,...,ln u ′′ ( x ) v ( x ) dx = Z V l ,...,ln u ( x ) v ′′ ( x ) dx − n X k =1 u ( l k ) v ′ ( l k ) + n X k =1 u ′ ( l k ) v ( l k ) . 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 ). So n X k =1 u k (0) v ′ k (0) = u (0) n X k =1 v ′ k (0) = 0 . Idem for P nk =1 u ′ k (0) v k (0). (cid:3) This Green formula yields now as usual an expression for the resolvent of A in terms of thegeneralized eigenfunctions. Proposition 3.3.
Let λ ∈ C with Im( λ ) = 0 be fixed and let e λ , e λ be generalized eigenfunctionsof A , such that the Wronskian w λ , ( x ) satisfies for every x in Nw λ , ( x ) = det W ( e λ ( x ) , e λ ( x )) = e λ ( x ) · ( e λ ) ′ ( x ) − ( e λ ) ′ ( x ) · e λ ( x ) = 0 . If for some k ∈ { , . . . , n } we have e λ | N k ∈ H ( N k ) and e λ | N m ∈ H ( N m ) for all m = k , thenfor every f ∈ H and x ∈ N k [ R ( λ, A ) f ]( x ) = 1 c k w λ , ( x ) · "Z ( x, + ∞ ) Nk e λ ( x ) e λ ( x ′ ) f ( x ′ ) dx ′ + Z N \ ( x, + ∞ ) Nk e λ ( x ) e λ ( x ′ ) f ( x ′ ) dx ′ . (1)Note that by the integral over N , we mean the sum of the integrals over N k , k = 1 , . . . , n . Proof.
Let λ ∈ 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 k . Then I λ := Z ( x, + ∞ ) Nk e λ ( x ) e λ ( x ′ )( λI − A ) u ( x ′ ) dx ′ + Z N \ ( x, + ∞ ) Nk e λ ( x ) e λ ( x ′ )( λI − A ) u ( x ′ ) dx ′ = e λ ( x ) lim l k →∞ Z l k x e λ ( x ′ )( λI − A ) u ( x ′ ) dx ′ + e λ ( x ) lim l m →∞ ,m = k Z V l ,...,lk − ,x,lk +1 ,...,ln e λ ( x ′ )( λI − A ) u ( x ′ ) dx ′ , F. ALI MEHMETI, R. HALLER-DINTELMANN, AND V. R´EGNIER due to the dominated convergence Theorem, the integrands being in L ( R ) by the hypotheses.We have u ∈ D ( A ) ⊂ Q nj =1 H ( N j ) and e λ | N k ∈ H ( N k ) , e λ | N m ∈ H ( N m ) , m = k by hypothesis and thus ∂ x u | N k ( x ) · e λ | N k ( x ) −→ x → + ∞ , u | N k ( x ) · ∂ x e λ | N k ( x ) −→ x → + ∞ ,∂ x u | N m ( x ) · e λ | N m ( x ) −→ x → + ∞ , u | N m ( x ) · ∂ x e λ | N m ( x ) −→ x → + ∞ , m = k, all products being in some H ( N j ). Recall that Z ba f ′′ g = Z ba f g ′′ − f ( b ) g ′ ( b ) + f ′ ( b ) g ( b ) + f ( a ) g ′ ( a ) − f ′ ( a ) g ( a )for f, g ∈ H (( a, b )). Now Lemma 3.2 and ( λI − A ) e λr = 0 for r = 1 , I λ = e λ ( x ) lim l k →∞ hZ l k x ( λI − A ) e λ ( x ′ ) u ( x ′ ) dx ′ + c k (cid:16) − u ( l k ) ∂ x e λ ( l k ) + ∂ x u ( l k ) e λ ( l k ) + u ( x ) ∂ x e λ ( x ) − ∂ x u ( x ) e λ ( x ) (cid:17)i + e λ ( x ) h lim l m →∞ ,m = k Z V l ,...,lk − ,x,lk +1 ,...,ln ( λI − A ) e λ ( x ′ ) u ( x ′ ) dx ′ + X j = k c j lim l j →∞ (cid:16) − u ( l j ) ∂ x e λ ( l j ) + ∂ x u ( l j ) e λ ( l j ) (cid:17) + c k (cid:16) − u ( x ) ∂ x e λ ( x ) + ∂ x u ( x ) e λ ( x ) (cid:17)i = c k (cid:16) e λ ( x ) ∂ x e λ ( x ) − ∂ x e λ ( x ) e λ ( x ) (cid:17) u ( x )= c k w λ , ( x ) u ( x ) . Now the invertibility of λI − A implies the result. (cid:3) Definition 3.4 (Generalized eigenfunctions of A ) . For k ∈ { , . . . , n } and λ ∈ C let ξ k ( λ ) := s λ − a k c k and s k := − P l = k c l ξ l ( λ ) c k ξ k ( λ ) . Here, and in all what follows, the complex square root is chosen in such a way that √ r · e iφ = √ re iφ/ with r > φ ∈ [ − π, π ).For λ ∈ C and j, k ∈ { , . . . , n } , F ± ,jλ : N → C is defined for x ∈ N k by F ± ,jλ ( x ) := F ± ,jλ,k ( x )with ( F ± ,jλ,j ( x ) = cos( ξ j ( λ ) x ) ± is j ( λ ) sin( ξ j ( λ ) x ) ,F ± ,jλ,k ( x ) = exp( ± iξ k ( λ ) x ) , for k = j. Remark . • F ± ,jλ satisfies the transmission conditions ( T ) and ( T ). • Formally it holds AF ± ,jλ = λF ± ,jλ . • Clearly F ± ,jλ does not belong to H , thus it is not a classical eigenfunction. • For Im( λ ) = 0, the function F ± ,jλ,k , where the +-sign (respectively − -sign) is chosen ifIm( λ ) > λ ) < H ( N k ) for k = j . This feature is usedin the formula for the resolvent of A . HE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 7 ☛ ❏❏❫✻ ❏❏❏❏❏❏❏❏❏❏ ✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁ ✲❩❩❩❩❩❩⑥ ❩❩❩❩❩❩❩❩❩❩❩ ✑✑✑✑✑✑✑✑✑✑✑✑✑✑✰ ✑✑✑✑✑✑✑✑✑✑✑✑ ✑✑✑✑✑✑✑✑✑✑✑✑✑✑✑ ✄✄✄✄✄✄✄✄ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) x ∈ N x ′ ∈ N x ∈ N x ∈ N x ′ ∈ N x ′ ∈ N diag N × N diag N × N diag N × N Figure 1. N × N in the case n = 3 Definition 3.6 (Kernel of the resolvent) . Let w : C → C be defined by w ( λ ) = ± i · P nj =1 c j ξ j ( λ ).For any λ ∈ C such that w ( λ ) = 0, j ∈ { , . . . , n } and x ∈ N j , we define K ( x, x ′ , λ ) = w ( λ ) F ± ,jλ,j ( x ) F ± ,j +1 λ,j ( x ′ ) , for x ′ ∈ N j , x ′ > x, w ( λ ) F ± ,j +1 λ,j ( x ) F ± ,jλ ( x ′ ) , for x ′ ∈ N k , k = j or x ′ ∈ N j , x ′ < x. In the whole formula + (respectively − ) is chosen, if Im( λ ) > λ ) ≤ 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 maindiagonals, where the kernel is not smooth . Note that in particular, if c j = c and a j = 0 for all j ∈ { , . . . , n } , then w ( λ ) = ± inc √ λ for all j ∈ { , . . . , n } , which only vanishes for λ = 0. Onthe other hand, if there exist i and j in { , . . . , n } , such that a i = a j , then it is clear that w ( λ )never vanishes on R , but we need to know, if it vanishes on C .We will show in Theorem 3.8 that K is indeed the kernel of the resolvent of A . In order todo so, we collect some useful observations in the following lemma. Lemma 3.7. i) For a ≤ λ and ε ≥ holds | w ( λ − iε ) | ≥ P nj =1 c j | λ − a j | . ii) For λ ∈ ρ ( A ) such that Re( λ ) ≥ a , the Wronskian w only vanishes at λ = α , if a k = α for all k ∈ { , . . . , n } .Proof. We first prove i).
F. ALI MEHMETI, R. HALLER-DINTELMANN, AND V. R´EGNIER
Note that for z , z , . . . , z n ∈ C holds (cid:12)(cid:12)(cid:12) n X j =1 z j (cid:12)(cid:12)(cid:12) = n X j =1 | z j | + 2 n X k,l =1 k = l Re( z k z l ) . With z k := c k ξ k ( λ − iε ) and the abbreviation η k := √ λ − iε − a k it follows | w ( λ − iε ) | = n X j =1 c j | η j | + 2 n X k,l =1 k = l c k c l Re( η k η l ) . Thus it suffices to show Re( η k η l ) ≥ k, l = 1 , . . . , n with k = l . With our convention √ z = p | z | e i arg( z )2 , arg( z ) ∈ [ − π, π ), this meansarg( η k η l ) ∈ h − π , π i . Without loss of generality, let k < l . Then we have a k ≤ a l and there are three possible positionsof λ : • a l ≤ λ :Then for r = k and for r = l we have arg( λ − iε − a r ) ∈ [ − π ,
0] and thereforearg( η r ) = 12 arg( λ − iε − a r ) ∈ h − π , i . Using λ − a k > λ − a l , this impliesarg( η k η l ) = arg( η k ) − arg( η l ) ∈ h , π i . • a k ≤ λ < a l :Then arg( λ − iε − a k ) ∈ [ − π ,
0] and thereforearg( η k ) = 12 arg( λ − iε − a k ) ∈ h − π , i . Furthermore, arg( λ − iε − a l ) ∈ [ − π, − π ], and thereforearg( η l ) = 12 arg( λ − iε − a l ) ∈ h − π , − π i . Putting everything together we getarg( η k η l ) = arg( η k ) − arg( η l ) ∈ h , π i . • λ < a k :In this case we get again for r = k and r = l arg( λ − iε − a r ) ∈ [ − π, − π ] and thereforearg( η r ) = 12 arg( λ − iε − a r ) ∈ h − π , − π i . This yields arg( η k η l ) = arg( η k ) − arg( η l ) ∈ h − π , π i . Thus, in all three cases we have arg( η k η l ) ∈ [ − π , π ] and, hence, Re( η k η l ) ≥ k, l =1 , . . . , n with k = l . HE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 9
In order to prove ii), we note that the choice of the branch cut of the complex square roothas been made in such a way that √ µ = √ µ for all µ ∈ C . Thus w ( µ ) = w ( µ ) for all µ ∈ C .This implies that the first part of the lemma can be generalized to: | w ( µ ) | ≥ n X j =1 c j | Re( µ ) − a j | for every µ such that Re( µ ) ≥ a . Then w ( µ ) = 0 ⇔ | Re( µ ) − a j | = 0 for all j ∈ { , . . . , n } .This implies that w never vanishes, if there exist k, l in { , . . . , n } , with k = l such that a k = a l .Finally, if a k = α for all k ∈ { , . . . , n } , then | w ( µ ) | = (cid:0)P nk =1 √ c k (cid:1) | µ − α | , which onlyvanishes for µ = α . (cid:3) Theorem 3.8.
Let f ∈ H . Then, for x ∈ N and λ ∈ ρ ( A ) such that Re( λ ) ≥ a [ R ( λ, A ) f ]( x ) = Z N K ( x, x ′ , λ ) f ( x ′ ) dx ′ . Proof.
In (1), the generalized eigenfunction e λ can be chosen to be F ± ,jλ . Then e λ can be F ± ,lλ with any l = j so we have chosen j + 1 to fix the formula. The choice has been done so thatthe integrands lie in L (0 , + ∞ ) (cf. the last item in Remark 3.5). Simple calculations yield theexpression for the Wronskian w ( λ ).For λ ∈ ρ ( A ) such that Re( λ ) ≥ a , the Wronskian only vanishes at λ = α if a k = α for all k ∈ { , . . . , n } due to Lemma 3.7. But in this case, the Wronskian is w ( λ ) = (cid:0)P nk =1 √ c k (cid:1) √ λ − α and w ( λ ) − has an L -singularity at λ = α . (cid:3) Application of Stone’s formula and limiting absorption principle
Let us first recall Stone’s formula (see Theorem XII.2.11 in [16]).
Theorem 4.1.
Let E be the resolution of the identity of a linear unbounded self-adjoint operator T : D ( T ) → H in a Hilbert space H (i.e. E ( a, b ) = ( a,b ) ( T ) for ( a, b ) ∈ R , a < b ). Then, inthe strong operator topology h ( T ) E ( a, b ) = lim δ → + lim ε → + πi Z b − δa + δ h ( λ )[ R ( λ − εi, T ) − R ( λ + εi, T )] dλ for all ( a, b ) ∈ R with a < b and for any continuous scalar function h defined on the real line. To apply this formula we need to study the behaviour of the resolvent R ( λ, A ) for λ approach-ing the spectrum of A . Lemma 3.7 will be useful as well as the following results. Lemma 4.2.
Let δ > be fixed. For all a ≤ λ , < ε < δ and j = 1 , . . . , n holds | s j ( λ − iε ) | ≤ M ( λ, δ ) := max j =1 ,...,n n p | λ − a j | o n X k =1 (cid:0) ( λ − a k ) + δ (cid:1) / . Proof.
This follows directly from the definition of s j . (cid:3) Note that M ( · , δ ) ∈ L ([ a , + ∞ )). Furthermore, if a = . . . = a n , then s j ( µ ) = 1 √ c j n X k =1 k = j √ c k for all µ ∈ C , which means that s j is constant and we may take M ( λ, δ ) := max j =1 ,...,n n √ c j n X k =1 k = j √ c k o . Theorem 4.3 (Limiting absorption principle for A ) . Let δ > be fixed and let M ( λ, δ ) bedefined as in Lemma 4.2. Then for all a ≤ λ , < ε < δ and ( x, x ′ ) ∈ N we have i) lim α → K ( x, x ′ , λ − iα ) = K ( x, x ′ , λ ) , ii) | K ( x, x ′ , λ − iε ) | ≤ N ( λ, δ ) e γ ( x + x ′ ) , where N ( λ, δ ) := M ( λ,δ )( P nj =1 c j | λ − a j | ) / and γ := max j =1 ,...,n { c − j } max { (( a n − a ) + δ ) , , δ } . Proof. i) The complex square root is, by definition, continuous on { z ∈ C : Im( z ) ≤ } (cf.Definition 3.4), hence the continuity of K ( x, x ′ , · ) from below on the real axis. Note that x and x ′ are fixed parameters in this context.ii) For Im( µ ) ≤ , µ = λ − iε and x ∈ N j we have in concrete terms K ( x, x ′ , µ ) = 1 w ( µ ) (cid:2) cos (cid:0) ξ j ( µ ) x (cid:1) − is j ( µ ) sin (cid:0) ξ j ( µ ) x (cid:1)(cid:3) exp( − iξ j ( µ ) x ′ ) , x ′ ∈ N j , x ′ > x, exp (cid:0) − iξ j ( µ ) x (cid:1)(cid:2) cos (cid:0) ξ j ( µ ) x ′ (cid:1) − is j ( µ ) sin (cid:0) ξ j ( µ ) x ′ (cid:1)(cid:3) , x ′ ∈ N j , x ′ < x, exp (cid:0) − iξ j ( µ ) x (cid:1) exp (cid:0) − iξ k ( µ ) x ′ (cid:1) , x ′ ∈ N k , k = j. Now, let us first look at the case λ > a n . Then | exp( − iξ j ( µ ) x ) | ≤ exp (cid:0) | Im( ξ j ( µ ) x ) | (cid:1) = exp (cid:0) c − / j | Im( p λ − iε − a j ) | (cid:1) . Using the fact, that for z ∈ C with | arg( z ) | ≤ π/ | Im( √ z ) | ≤ max { , | Im( z ) |} ,we obtain | exp( − iξ j ( µ ) x ) | ≤ exp (cid:0) c − / j max { , δ } x (cid:1) . In the case a ≤ λ ≤ a n we find | ξ j ( µ ) | = s | µ − a j | c j = c − / j (cid:0) ( λ − a j ) + ε (cid:1) / ≤ c − / j (cid:0) ( a n − a j ) + δ (cid:1) / . Using these estimates and | e z | = e Re( z ) ≤ e | z | , we find for all λ ≥ a | K ( x, x ′ , µ ) | ≤ | w ( µ ) | ( (cid:0) | s j ( µ ) | (cid:1) exp( | ξ j ( µ ) | x ) exp( | ξ j ( µ ) | x ′ ) , x ′ ∈ N j , exp( | ξ j ( µ ) | x ) exp( | ξ k ( µ ) | x ′ ) , x ′ ∈ N k , k = j ≤ | w ( µ ) | (cid:0) | s j ( µ ) | (cid:1) exp( γ ( x + x ′ )) . The conclusion now follows using Lemma 3.7 and Lemma 4.2. (cid:3)
Note that these estimates in particular imply that N ( · , δ ) ∈ L ([ a , + ∞ )). In fact, if a = . . . = a n , then M ( λ, δ ) can be chosen to be constant, see Lemma 4.2, and the denominator of N causes only an L -type singularity. On the other hand, if there are two different a j , thedenominator of N is never zero and M ( · , δ ) ∈ L ([ a , + ∞ )) again by Lemma 4.2. Lemma 4.4.
For ( x, x ′ ) ∈ N and λ ∈ C , it holds K ( x, x ′ , λ ) = K ( x, x ′ , λ ) .Proof. The choice of the branch cut of the complex square root has been made such that √ λ = √ λ for all λ ∈ C . This implies e i √ λx = e i √ λx = e − i √ λx for all λ ∈ C and x ∈ R . Thus it holds F + ,jλ ( x ) = F − ,jλ ( x ) and F − ,jλ ( x ) = F + ,jλ ( x ) HE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 11 for all λ ∈ C , x ∈ N and j ∈ { , . . . , n } . In the same way we have w ( λ ) = − w ( λ ). Observe,that switching from λ to λ the sign of the imaginary part is changing, so in the definition of K ( x, x ′ , λ ) we have to take the other sign, whenever there is a ± -sign in the formula. This givesthe assertion. (cid:3) Now, we can deduce a first formula for the resolution of the identity of A . Proposition 4.5.
Take f ∈ H = Q nj =1 L ( N j ) , vanishing almost everywhere outside a compactset B ⊂ N and let −∞ < a < b < + ∞ . Then for any continuous scalar function h defined onthe real line and for all x ∈ N (cid:0) h ( A ) E ( a, b ) f (cid:1) ( x )= Z ( a,b ) ∩ [ a , + ∞ ) h ( λ ) n X j =1 N j ( x ) nZ N f ( x ′ ) h ( x, + ∞ ) Nj ( x ′ ) · Im (cid:16) w ( λ ) F − ,jλ ( x ) F − ,j +1 λ ( x ′ ) (cid:17) + N \ ( x, + ∞ ) Nj ( x ′ ) · Im (cid:16) w ( λ ) F − ,j +1 λ ( x ) F − ,jλ ( x ′ ) (cid:17)i dx ′ o dλ, where E is the resolution of the identity of A (cf. Theorem 4.1) and the index j is to beunderstood modulo n , that is to say, if j = n , then j + 1 = 1 .Proof. The proof is analogous to that of Lemma 3.13 in [2]. Let g ∈ H be vanishing outside B .Then( h ( A ) E ( a, b ) f, g ) H = (cid:18) lim δ → + lim ε → + πi Z b − δa + δ h ( λ ) (cid:2) R ( λ − εi, A ) − R ( λ + εi, A ) (cid:3) dλ f, g (cid:19) H (2)= lim δ → + lim ε → + πi (cid:18)Z b − δa + δ h ( λ ) (cid:2) R ( λ − εi, A ) − R ( λ + εi, A ) (cid:3) dλ f, g (cid:19) H (3)= lim δ → + lim ε → + πi Z b − δa + δ h ( λ ) (cid:0)(cid:2) R ( λ − εi, A ) − R ( λ + εi, A ) (cid:3) f, g (cid:1) H dλ (4)= lim δ,ε → + πi Z ( a + δ,b − δ ) ∩ [ a , + ∞ ) h ( λ ) (cid:18)Z N f ( x ′ ) (cid:2) K ( · , x ′ , λ − iε ) − K ( · , x ′ , λ + iε ) (cid:3) dx ′ , g ( · ) (cid:19) H dλ (5)= lim δ,ε → + πi Z ( a + δ,b − δ ) ∩ [ a , + ∞ ) h ( λ ) (cid:18)Z N f ( x ′ ) (cid:2) K ( · , x ′ , λ − iε ) − K ( · , x ′ , λ − iε ) (cid:3) dx ′ , g ( · ) (cid:19) H dλ (6)= lim δ,ε → + πi Z ( a + δ,b − δ ) ∩ [ a , + ∞ ) h ( λ ) (cid:18)Z N f ( x ′ ) 2 i Im( K ( · , x ′ , λ − iε )) dx ′ , g ( · ) (cid:19) H dλ (7)= lim δ → + π Z ( a + δ,b − δ ) ∩ [ a , + ∞ ) h ( λ ) (cid:18)Z N f ( x ′ )[ lim ε → + Im( K ( · , x ′ , λ − iε ))] dx ′ , g ( · ) (cid:19) H dλ (8)= π Z ( a,b ) ∩ [ a , + ∞ ) h ( λ ) Z N f ( x ′ )Im( K ( · , x ′ , λ − i dx ′ dλ, g ( · ) ! H (9)= Z N π Z ( a,b ) ∩ [ a , + ∞ ) h ( λ ) nZ N f ( x ′ )Im h w ( λ ) n X j =1 N j ( x ) (cid:16) ( x, + ∞ ) Nj ( x ′ ) F − ,jλ ( x ) F − ,j +1 λ ( x ′ ) (10)+ N \ ( x, + ∞ ) Nj ( x ′ ) F − ,j +1 λ ( x ) F − ,jλ ( x ′ ) (cid:17)i dx ′ o dλ g ( x ) dx. Here, the justifications for the equalities are the following: (2): Stone’s formula (Theorem 4.1).(3): After applying the operator valued integral to f , the two limits are in H . So theycommute with the scalar product in H .(4): ( · f, g ) H is a continuous linear form on L ( H ), and can therefore be commuted withthe vector-valued integration. Note that λ R ( λ, A ) is continuous on the half-plane { λ ∈ C : Re( λ ) < a } , since the resolvent is holomorphic outside the spectrum, cf.Proposition 2.1.(5): Theorem 3.8.(6): Lemma 4.4.(7): z − z = 2 i · Im z ∀ z ∈ C .(8): Dominated convergence. Since supp f , supp g and [ a, b ] are compact, we use the limitingabsorption principle (Theorem 4.3).(9): Fubini’s Theorem.(10): Definition 3.6.The assertion follows, since g was arbitrary with compact support. (cid:3) The unpleasant point about the formula in the above proposition is the apparent cut alongthe diagonal { x = x ′ } expressed by the characteristic functions in the variable x ′ . In fact, thereis no discontinuity and the two integrals recombine with respect to x ′ . This is a consequence ofthe next lemma that gives an explicit representation of the integrand above.In the following, we use the convention a n +1 := + ∞ , in order to unify notation and we set ξ ′ j := iξ j . Lemma 4.6.
Let j, k ∈ { , . . . , n } and let λ be fixed in ( a p , a p +1 ) , with p ∈ { , . . . , n } . Then Im (cid:2) w (cid:0) F − ,j +1 λ (cid:1) j ( x ) (cid:0) F − ,jλ (cid:1) k ( x ′ ) (cid:3) is given by the following expressions, respectively: • If k ≥ j > p or j ≥ k > p (Case (a)) Im (cid:16) w (cid:17) e − ξ ′ j x − ξ ′ k x ′ , • If j < k ≤ p or k < j ≤ p (Case (b)) Im (cid:16) w (cid:17) cos( ξ j x ) cos( ξ k x ′ ) − Im (cid:0) w (cid:17) sin( ξ j x ) sin( ξ k x ′ ) − Re (cid:16) w (cid:17) cos( ξ j x ) sin( ξ k x ′ ) − Re (cid:16) w (cid:17) sin( ξ j x ) cos( ξ k x ′ ) , • If j = k ≤ p (Case (b), j = k ) Im (cid:16) w (cid:17) cos( ξ j x ) cos( ξ k x ′ ) − Im (cid:16) s k w (cid:17) sin( ξ j x ) sin( ξ k x ′ ) − Re (cid:16) w (cid:17) cos( ξ j x ) sin( ξ k x ′ ) − Re (cid:16) w (cid:17) sin( ξ j x ) cos( ξ k x ′ ) , • If j ≤ p < k (Case (c)) Im (cid:16) w (cid:17) e − ξ ′ k x ′ cos( ξ j x ) + Im (cid:16) iw (cid:17) e − ξ ′ k x ′ sin( ξ j x ) , • If k ≤ p < j (Case (d)) Im (cid:16) w (cid:17) e − ξ ′ j x cos( ξ k x ′ ) − Im (cid:16) iw (cid:17) e − ξ ′ j x sin( ξ k x ′ ) . HE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 13
Proof.
Since λ belongs to ( a p , a p +1 ), ξ j ( λ ) is a real number, if and only if j ≤ p . Otherwise, ξ j is a purely imaginary number and we have ξ j =: − iξ ′ j .Now, the proof is pure calculation, using the following expressions for the generalized eigen-functions in the case j = k Re (cid:0) F − ,jλ (cid:1) k ( x ) = ( cos( ξ k x ) , if ξ k ∈ R , i.e. k ≤ p,e − ξ ′ k x , if ξ k ∈ i R , i.e. k > p, Im (cid:0) F − ,jλ (cid:1) k ( x ) = ( − sin( ξ k x ) , if ξ k ∈ R , i.e. k ≤ p, , if ξ k ∈ i R , i.e. k > p, and for j = k Re (cid:0) F − ,kλ (cid:1) k ( x ) = (cid:26) cos( ξ k x ) + Im( s k ) sin( ξ k x ) , if ξ k ∈ R , i.e. k ≤ p, Re (cid:0) (1 + s k ) (cid:1) e − ξ ′ k x + Re (cid:0) (1 − s k ) (cid:1) e ξ ′ k x , if ξ k ∈ i R , i.e. k > p, Im (cid:0) F − ,kλ (cid:1) k ( x ) = (cid:26) − Re( s k ) sin( ξ k x ) , if ξ k ∈ R , i.e. k ≤ p, Im (cid:0) (1 + s k ) (cid:1) e − ξ ′ k x + Im (cid:0) (1 − s k ) (cid:1) e ξ ′ k x , if ξ k ∈ i R , i.e. k > p, respectively. (cid:3) Theorem 4.7.
Take f ∈ H = Q nj =1 L ( N j ) , vanishing almost everywhere outside a compact set B ⊂ N and let −∞ < a < b < + ∞ . Then for any continuous scalar function h defined on thereal line and for all x ∈ N (cid:0) h ( A ) E ( a, b ) f (cid:1) ( x ) = Z [ a,b ] ∩ [ a , + ∞ ) h ( λ ) n X j =1 N j ( x ) nZ N f ( x ′ )Im h w ( λ ) F − ,j +1 λ ( x ) F − ,jλ ( x ′ ) i dx ′ o dλ, (11) where again the index j is to be understood modulo n , i.e, if j = n , then j + 1 = 1 .Proof. All the work has already been done. It remains to inspect the formulae for j = k in thecases (a) and (b) of Lemma 4.6, to observe that the expressions are symmetric in x and x ′ . Sofor x, x ′ ∈ N j with x < x ′ we find F − ,jλ ( x ) F − ,j +1 λ ( x ′ ) = F − ,jλ ( x ′ ) F − ,j +1 λ ( x ), which implies theassertion. (cid:3) Symmetrization
As was already explained in the introduction, the aim of this section will be to find complexnumbers q l,m , l, m ∈ { , . . . , n } , such that the resolution of identity of A can be written as( E ( a, b ) f ) ( x ) = Z ba n X l,m =1 q lm ( λ ) F − ,lλ ( x ) Z N F − ,mλ ( x ′ ) f ( x ′ ) dx ′ dλ, (12)in order to eliminate the cyclic structure of the formula in Proposition 4.5.In this section we shall often suppress the dependence on λ of several quantities for the easeof notation, so s j = s j ( λ ), q l,m = q l,m ( λ ), ξ j = ξ j ( λ ), ξ ′ j = ξ ′ j ( λ ) and w = w ( λ ). Lemma 5.1.
Given x ∈ N j , equation (12) is satisfied for all a ≤ a < b < + ∞ and all f ∈ L ( N ) with compact support, if and only if for all j = 1 , . . . , n Im h w F − ,j +1 λ ( x ) F − ,jλ ( x ′ ) i = n X l,m =1 q lm ( λ ) F − ,lλ (cid:0) x ) F − ,mλ ( x ′ ) (13) for almost all x ′ ∈ N and λ ≥ a . Here again the index j has to be understood modulo n , thatis to say, if j = n , j + 1 = 1 . Proof.
If functions q l,m , l, m = 1 , . . . , n , satisfy (13) and if a ≤ a < b < + ∞ , we get by (11)( E ( a, b ) f )( x ) = b Z a n X j =1 N j ( x ) nZ N f ( x ′ ) n X l,m =1 q lm ( λ ) F − ,lλ (cid:0) x ) F − ,mλ ( x ′ ) dx ′ o dλ = Z ba n X l,m =1 q lm ( λ ) F − ,lλ ( x ) Z N F − ,mλ ( x ′ ) f ( x ′ ) dx ′ dλ, which is (12).For the converse implication, let (12) be satisfied for some x ∈ N j and all a ≤ a < b < + ∞ ,as well as all f ∈ L ( N ) with compact support. This means by (11) Z ba Z N f ( x ′ )Im h w ( λ ) F − ,j +1 λ ( x ) F − ,jλ ( x ′ ) i dx ′ dλ = Z ba n X l,m =1 q lm ( λ ) F − ,lλ ( x ) Z N F − ,mλ ( x ′ ) f ( x ′ ) dx ′ dλ. Firstly, we want to see that the integrands of the λ -integrals on both sides in fact have to beequal almost everywhere. In order to do so, we observe that they both are in L (( a , + ∞ )) anduse the following general observation: If I is an interval and g ∈ L ( I ) satisfies R J g = 0 forall intervals J ⊆ I , then g = 0 almost everywhere in I . Indeed, in this case we have for anyLebesgue point x ∈ I of g and every ε > | g ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12) ε Z x + εx − ε (cid:0) g ( x ) − g ( x ) (cid:1) dx + 12 ε Z x + εx − ε g ( x ) dx | {z } =0 (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε Z x + εx − ε (cid:12)(cid:12) g ( x ) − g ( x ) (cid:12)(cid:12) dx −→ ε → , which implies g ( x ) = 0 for almost all x ∈ I .This implies Z N f ( x ′ )Im h w ( λ ) F − ,j +1 λ ( x ) F − ,jλ ( x ′ ) i dx ′ = Z N n X l,m =1 q lm ( λ ) F − ,lλ ( x ) F − ,mλ ( x ′ ) f ( x ′ ) dx ′ for almost all λ ≥ a and all f ∈ L ( N ) with compact support. By the fundamental theorem ofcalculus this implies the assertion. (cid:3) In a next step, we explicitely write down equation (13) as a linear system for the values q lm . Lemma 5.2.
The equation Im h w (cid:0) F − ,j +1 λ (cid:1) j ( x ) (cid:0) F − ,jλ (cid:1) k ( x ′ ) i = n X l,m =1 q lm ( λ ) (cid:0) F − ,lλ (cid:1) j ( x ) (cid:0) F − ,mλ (cid:1) k ( x ′ ) holds for any ( x, x ′ ) ∈ N j × N k and λ ∈ ( a p , a p +1 ) , if and only if • Case (a) : if k ≥ j > p or if j ≥ k > p q jk = 0 P l = j q lk = 0 P m = k q jm = 0 P l = j,m = k q lm = Im (cid:2) w (cid:3) , HE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 15 • Case (b) : if j < k ≤ p or if k < j ≤ p P l,m q lm = Im (cid:2) w (cid:3)P l = j,m = k q lm + s k P l = j q lk + s j P m = k q jm + s j · s k · q jk = − Im (cid:2) w (cid:3)P l = j,m = k q lm + s k P l = j q lk + P m = k q jm + s k · q jk = − i · Im (cid:2) iw (cid:3)P l = j,m = k q lm + P l = j q lk + s j P m = k q jm + s j · q jk = i · Im (cid:2) iw (cid:3) , • Case (b), j = k : if j = k ≤ p P l,m q lm = Im (cid:2) w (cid:3)P l = j,m = j q lm + s j P l = j q lj + s j P m = j q jm + s j · s j · q jj = − Im (cid:2) s j w (cid:3)P l = j,m = j q lm + s j P l = j q lj + P m = j q jm + s j · q jj = − i · Im (cid:2) iw (cid:3)P l = j,m = j q lm + P l = j q lj + s j P m = j q jm + s j · q jj = i · Im (cid:2) iw (cid:3) , • Case (c) : if j ≤ p < k q jk = 0 P l = j q lk = 0 P m = k q jm = Im (cid:2) w (cid:3)P l = j,m = k q lm + s j P m = k q jm = i · Im (cid:2) iw (cid:3) , • Case (d) : if k ≤ p < j q jk = 0 P l = j q lk = Im (cid:2) w (cid:3)P m = k q jm = 0 P l = j,m = k q lm + s k P l = j q lk = − i · Im (cid:2) iw (cid:3) . Proof.
The sum P nl,m =1 q lm ( λ ) (cid:0) F − ,lλ (cid:1) j ( x ) (cid:0) F − ,mλ (cid:1) k ( x ′ ) is explicitely written in the different cases(a), (b), (c) and (d) as it was done in Lemma 4.6. Then the linear independence of the followingfamilies of functions is used to get the above systems for the q lm ’s: • Case (a) : e Ax + Bx ′ , e Ax − Bx ′ , e − Ax + Bx ′ , e − Ax − Bx ′ , • Case (b) : cos( Ax ) cos( Bx ′ ), cos( Ax ) sin( Bx ′ ), sin( Ax ) cos( Bx ′ ), sin( Ax ) sin( Bx ′ ), • Cases (c) and (d) : e Ax cos( Bx ′ ), e Ax sin( Bx ′ ), e − Ax cos( Bx ′ ), e − Ax sin( Bx ′ ),where A and B are any fixed non-vanishing real numbers.On the other hand: if the q lm satisfy the system indicated in the lemma, on both sides of theequation of the lemma we have the same linear combination of the functions given above. Thusequality holds. (cid:3) Having the linear systems in the above lemma at hand, it remains to solve them. Indeed thisis possible and we briefly indicate the necessary steps.Firstly, if λ < a , we have ξ j ∈ i R for all j ∈ { , . . . , n } . Thus, for all j, k ∈ { , . . . , n } and all( x, x ′ ) ∈ N j × N k Im h w (cid:0) F − ,j +1 λ (cid:1) j ( x ) (cid:0) F − ,jλ (cid:1) k ( x ′ ) i = Im (cid:16) w (cid:17) e − ξ ′ j x − ξ ′ k x ′ = 0 . Now, let λ be fixed in ( a p , a p +1 ), with p ∈ { , . . . , n } , remembering the convention a n +1 = + ∞ .Due to the first equation of the corresponding system of cases (a), (c) and (d), the matrix q := ( q lm ) nl,m =1 has to be of the form q = (cid:18) Q p
00 0 (cid:19) with a p × p matrix Q p .Hence, the equations P l = j q lk = 0 and P m = k q jm = 0 in case (a) are obviously fulfilled, aswell as P l = j q lk = 0 in case (c) and P m = k q jm = 0 in case (d). This means that only threeequations remain from the cases (a), (c) and (d): P l,m q lm = Im (cid:2) w (cid:3)P m = k q lm + s k P q lk = − i · Im (cid:2) iw (cid:3)P l = j q lm + s j P q jm = i · Im (cid:2) iw (cid:3) , where in all the sums, l and m belong to { , . . . , p } . Now it is important to note that thesethree equations are already contained in the following system corresponding to the case (b),i.e. the only conditions for the q lm ’s are the four following equations for a fixed ( j, k ) such that j < k ≤ p or k < j ≤ p : P l,m q lm = Im (cid:2) w (cid:3)P l = j,m = k q lm + s k P l = j q lk + s j P m = k q jm + s j · s k · q jk = − Im (cid:2) w (cid:3)P l = j,m = k q lm + s k P l = j q lk + P m = k q jm + s k · q jk = − i · Im (cid:2) iw (cid:3)P l = j,m = k q lm + P l = j q lk + s j P m = k q jm + s j · q jk = i · Im (cid:2) iw (cid:3) (14)and, for j = k ≤ p : P l,m q lm = Im (cid:2) w (cid:3)P l = j,m = j q lm + s j P l = j q lj + s j P m = j q jm + s j · s j · q jj = − Im (cid:2) s j w (cid:3)P l = j,m = j q lm + s j P l = j q lj + P m = j q jm + s j · q jj = − i · Im (cid:2) iw (cid:3)P l = j,m = j q lm + P l = j q lj + s j P m = j q jm + s j · q jj = i · Im (cid:2) iw (cid:3) , (15)where, once more, l and m belong to { , . . . , p } in all the sums.Now, we denote by Q jk := P l = j q lk and Q ′ jk := P m = k q jm . Using the fact that Im (cid:2) w (cid:3) =Re (cid:2) iw (cid:3) , the above system (14) is P l = j,m = k q lm = Im (cid:2) w (cid:3) − Q jk − Q ′ jk − q jk ( s k − Q jk + ( s j − Q ′ jk + ( s j s k − q jk = − (cid:2) iw (cid:3) ( s k − Q jk + ( s k − q jk = − i Im (cid:2) iw (cid:3) − Re (cid:2) iw (cid:3) = − iw ( s j − Q ′ jk + ( s j − q jk = i Im (cid:2) iw (cid:3) − Re (cid:2) iw (cid:3) = iw . Since s j − − iwf j , the last three equations may be rewritten as ( − if j w ) Q jk + ( if k w ) Q ′ jk + ( − if j w + if k w − | w | ) q jk = (2 f j f k )Re (cid:2) iw (cid:3) Q jk + q jk = f k | w | Q ′ jk + q jk = f j | w | and the Gauss method yields that q jk must vanish for j = k .In the case j = k , we rewrite system (15) as above and the three equations to be solved turnout to be ( − if j w ) Q jj + ( if j w ) Q ′ jj + ( − if j w + if j w − | w | ) q jj = (2 f j f j ) (cid:16) Re (cid:2) iw (cid:3) − f j (cid:17) Q jj + q jj = f j | w | Q ′ jj + q jj = f j | w | The Gauss method once more gives the only possible solution q jj = f j | w | for any j ∈ { , . . . , p } .With this candidate for a solution at hand, it is pure calculation to show that it is indeed asolution. Thus we have shown the following result. HE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 17
Theorem 5.3.
Let A be defined as in Section 2 and the generalized eigenfunctions F − ,jλ be givenby Definition 3.4. Take f ∈ H = Q nj =1 L ( N j ) , vanishing almost everywhere outside a compactset B ⊂ N and let −∞ < a < b < + ∞ . Then for all x ∈ N ( E ( a, b ) f ) ( x ) = Z ba n X l =1 q l ( λ ) F − ,lλ ( x ) Z N F − ,lλ ( x ′ ) f ( x ′ ) dx ′ dλ, where q l ( λ ) := ( , if λ < a l , f l ( λ ) | w ( λ ) | , if a l < λ. Furthermore, for almost all λ ∈ R the matrix q l,m ( λ ) = δ lm ( a l , + ∞ ) ( λ ) f l ( λ ) / | w ( λ ) | is the uniquematrix satisfying (12) . A direct approach
We have seen in the preceding section that a matrix q ( λ ) satisfying (12) exists and is uniqueup to a null set. It is the aim of this section to deduce an alternative representation for q ( λ ),involving only n × n -matrices and not an (3 n + 1) × n system as above. Since this approach isessentially independent of the special setting, it should be more convenient for generalisations.In the following, let us consider the complex-valued generalized eigenfunctions F − ,kλ as func-tions on N and not as elements of Q nj =1 C ∞ ( N j ). We introduce the notation F λ ( x ) = F − , λ ( x )... F − ,nλ ( x ) . Denoting by e k = ( δ lk ) l =1 ,...,n = (0 , . . . , , , , . . . , T the k -th unit vector in C n , we set d ( λ ) := F λ (0) and for j = 2 , . . . , n we fix x j ∈ N j and set d j ( λ ) := F λ ( x j ). Due to the form of ourgeneralized eigenfunctions, we then have d ( λ ) = n X k =1 e k = (1 , . . . , T ,d j ( λ ) = β j e j + α j X k = j e k = ( α j , . . . , α j , β j , α j , . . . , α j ) T for j = 2 , . . . , n for suitable α j , β j ∈ C .Using these vectors, we now define D ( λ ) := n X j =1 d j ( λ ) e Tj . Since α j = β j , for every j ∈ { , . . . , n } , by construction, the matrix D is invertible for any choiceof ( x , . . . , x n ) provided that x j = 0 for all j ∈ { , . . . , n } . Indeed det D = Q nj =2 ( β j − α j ).Denoting by C ( λ ) the diagonal matrix with c = i and c jj = iα j for any j ∈ { , . . . , n } , we canformulate our theorem. Theorem 6.1.
The matrix q ( λ ) := ( q lm ( λ )) l,m =1 ,...,n satisfying (12) is given by q ( λ ) = ( D ( λ ) − ) T Im (cid:16) w ( λ ) n X j =1 e j d j ( λ ) T e j +1 e Tj D ( λ ) (cid:17) ( D ( λ ) − ) T = ( D ( λ ) − ) T Im (cid:16) − iw ( λ ) C ( λ ) D ( λ ) (cid:17) ( D ( λ ) − ) T for almost all λ > a . As previously, j has to be understood modulo n , that is to say, j + 1 = 1 if j = n .Proof. By Lemma 5.1 the function q satisfies (13). Using the matrices and vectors introducedabove, for fixed j ∈ { , . . . , n } this can be rewritten asIm (cid:16) w ( λ ) F λ ( x ) T e j +1 e Tj F λ ( x ′ ) (cid:17) = F λ ( x ) T q ( λ ) F λ ( x ′ )for x ∈ N j and x ′ ∈ N .Setting x = x j and x ′ = x k we obtain by the definition of d j Im (cid:16) w ( λ ) d j ( λ ) T e j +1 e Tj d k ( λ ) (cid:17) = d j ( λ ) T q ( λ ) d k ( λ )for all j, k ∈ { , . . . , n } and thus n X k =1 n X j =1 e j Im (cid:16) w ( λ ) d j ( λ ) T e j +1 e Tj d k ( λ ) (cid:17) e Tk = n X k =1 n X j =1 e j d j ( λ ) T q ( λ ) d k ( λ ) e Tk . Using P nk =1 d k e Tk = D, we obtainIm (cid:16) w ( λ ) n X j =1 e j d j ( λ ) T e j +1 e Tj D (cid:17) = D T q D. Since D is invertible and the relation P nj =1 e j d j ( λ ) T e j +1 e Tj = − iC ( λ ) holds true, we now getthe desired formula for q . (cid:3) A Plancherel-type Theorem
In this section we prove a Plancherel type theorem for a Fourier type transformation V andits left inverse Z associated with the generalized eigenfunctions ( F − ,kλ ) k =1 ,...,n introduced inDefinition 3.4.Furthermore, we show that the fact that a function u belongs to the space D ( A k ) can beformulated in terms of the decay rate of V u . These results will be useful for the resolution ofevolution problems involving the spatial operator A as well as for the analysis of the propertiesof a solution. For this part, we follow Section 4 of [2]. Definition 7.1. i) Let I ⊂ R be a closed interval and let σ : I → R be a measurable,non-negative function with σ ( λ ) > λ ∈ ˚ I . Define L ( I, σ ) by(
F, G ) L ( I,σ ) := Z I σ ( λ ) F ( λ ) G ( λ ) dλ, | F | L ( I,σ ) := ( F, F ) / L ( I,σ ) L ( I, σ ) := { F : I → C measurable, | F | L ( I,σ ) < + ∞} . HE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 19 ii) Consider now the weights σ k := q k for k = 1 , . . . , n , where q k is given in Theorem 5.3.We endow L σ := Q nk =1 L ([ a k , + ∞ ) , σ k ) with the inner product( F, G ) σ := n X k =1 ( F k , G k ) L ([ a k , + ∞ ) ,σ k ) and denote | F | σ := ( F, F ) / σ . Note that L σ is a Hilbert space, since it is the product of the Hilbert spaces L ([ a k , + ∞ ) , σ k ).Now, we define the transformation V , together with its (right) inverse Z , which is later onshown to diagonalize A . Definition 7.2. i) For all f ∈ L ( N, C ) and k = 1 , . . . , n the function ( V f ) k : [ a k , + ∞ ) → C is defined by ( V f ) k ( λ ) := Z N f ( x )( F − ,kλ )( x ) dx. ii) Consider χ ∈ C ∞ ( R ) such that χ ≡ −∞ , a n + 1) and χ ≡ a n + 2 , + ∞ ). For G k ∈ C ∞ ([ a k , + ∞ ) , C ), such that χG k can be extended by zero to an element of S ( R ), k ∈ { , . . . , n } , we define Z ( G , . . . , G n ) : N → C by Z ( G , . . . , G n )( x ) := n X k =1 Z ( a k , + ∞ ) σ k ( λ ) G k ( λ )( F − ,kλ )( x ) dλ. Note that in contrast to the considerations in [2], the functions (
V f ) k here are complex-valued,due to the choice of the generalized eigenfunctions. Lemma 7.3.
Consider f ∈ L ( N ) with compact support, χ as in Definition 7.2 and for k ∈{ , . . . , n } let again G k ∈ C ∞ ([ a k , + ∞ )) be such that χG k ∈ S ( R ) . Then G = ( G , . . . , G n ) ∈ L σ , V f ∈ L σ , Z ( G ) ∈ L ( N ) = H and ( G, V f ) σ = ( Z ( G ) , f ) H . Proof.
We have(
G, V f ) σ = n X k =1 (cid:0) G k , ( V f ) k (cid:1) L ([ a k , + ∞ ) ,σ k ) = n X k =1 Z ( a k , + ∞ ) σ k ( λ ) G k ( λ )( V f ) k ( λ ) dλ = n X k =1 Z ( a k , + ∞ ) σ k ( λ ) G k ( λ ) (cid:16)Z N f ( x )( F − ,kλ )( x ) dx (cid:17) dλ = Z N (cid:16) n X k =1 Z ( a k , + ∞ ) σ k ( λ ) G k ( λ ) (cid:0) F − ,kλ (cid:1) ( x ) dλ (cid:17) f ( x ) dx = ( Z ( G ) , f ) H . (16)It only remains to make sure that the assumptions for Fubini’s Theorem are satisfied to justify(16). In fact, it is sufficient to estimate ( λ, x ) σ k ( λ )( F − ,kλ )( x ) on [ a k , a k +1 ] × N for a fixed k ∈ { , . . . , n } , since f is compactly supported and G k is rapidly decreasing.In order to do so, recall that c k ξ k ( λ ) = p c k ( λ − a k ) belongs to R if and only if λ ≥ a k andto i R otherwise and that for any λ > a , we have | w ( λ ) | ≥ P nl =1 c l | λ − a l | due to Lemma 3.7. Thus, putting in the expression for s k ( λ ), cf. Definition 3.4, we get for x ∈ N k (cid:12)(cid:12) σ k ( λ )( F − ,kλ )( x ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) c k ξ k ( λ ) | w ( λ ) | (cid:0) cos( ξ k ( λ ) x ) − is k ( λ ) sin( ξ k ( λ ) x ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ | c k ξ k ( λ ) | P nl =1 c l | λ − a l | (1 + | s k ( λ ) | ) ≤ | c k ξ k ( λ ) | P nl =1 c l | λ − a l | · P nl =1 c l | ξ l ( λ ) || c k ξ k ( λ ) | ≤ P nl =1 √ c l p | λ − a l | P nl =1 c l | λ − a l | . Furthermore, for x ∈ N j , j > k , we have | σ k ( λ )( F − ,kλ )( x ) | = (cid:12)(cid:12)(cid:12) c k ξ k ( λ ) | w ( λ ) | e − iξ j ( λ ) x (cid:12)(cid:12)(cid:12) ≤ √ c k p | λ − a k | exp (cid:16) − s a j − λc j x (cid:17) . Finally, for x ∈ N j , j < k , | σ k ( λ )( F − ,kλ )( x ) | = (cid:12)(cid:12)(cid:12) c k ξ k ( λ ) | w ( λ ) | e − iξ j ( λ ) x (cid:12)(cid:12)(cid:12) ≤ √ c k p | λ − a k | . For all three cases the bound is a continuous function of λ on [ a k , a k +1 ] and so, belongs to L ([ a k , a k +1 ]), which is enough for Fubini’s Theorem, due to the properties of f and G k . (cid:3) Lemma 7.4.
Consider f ∈ Q nk =1 C ∞ c ( N k ) . Then, for every k ∈ { , . . . , n } , ( V f ) k ∈ C ([ a k , + ∞ )) ∩ C ∞ ([ a k , a k +1 )) ∩ . . . ∩ C ∞ ([ a n , + ∞ )) and χσ k ( V f ) k ∈ S ( R ) with χ as in Definition 7.2.Proof. As in Section 4 of [2], (
V f ) k is a linear combination of Fourier and Laplace transformsof functions in C ∞ c ( N j ), j ∈ { , . . . , n } . (cid:3) The next step is to show that V is an isometry and Z is its right inverse. The proof of thefollowing theorem is precisely as in [2]. Theorem 7.5.
Endow Q nk =1 C ∞ c ( N k ) with the norm of H = Q nk =1 L ( N k ) . Then i) V : Q nk =1 C ∞ c ( N k ) → L σ is isometric and can therefore be extended to an isometry ˜ V : H → L σ .In particular, for all f ∈ H we have | ˜ V f | σ = ( f, f ) H . ii) Z = ˜ V − on ˜ V ( H ) . iii) Z can be extended to a continuous operator on L σ . Note that it is not clear, whether Z is injective, in contrast to the situation for n = 2, see [2].There the proof of injectivity relies on a choice of generalized eigenfunctions that are real-valued.Here, a choice of such eigenfunctions would probably destroy the feature that the matrix q , foundin Theorem 5.3, is diagonal. And it is this property that will allow us to carry over from [2] theproof of Theorem 7.7. However, the injectivity of Z is not important for the applications wehave in mind.In the sequel we shall again write V for ˜ V for simplicity.Our final aim is to show that V diagonalizes the operator, i.e. V ( A j u )( λ ) = λ j V ( u )( λ ) for all u ∈ D ( A j ). In order to formulate this precisely for a measurable function ψ : R → R we denotethe corresponding multiplication operator on L σ by M ψ , i.e. for a function F ∈ L σ we have( M ψ F ) k ( λ ) = ψ ( λ ) F k ( λ ) , λ ∈ [ a k , + ∞ ) , k ∈ { , . . . , n } . Then we have the following lemma, whose proof is again analogous to [2, Lemma 4.12].
HE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 21
Lemma 7.6. i) Let j ∈ N and p j : R → R be defined by p j ( x ) = x j . Then for any f ∈ D ( A j ) we have V ( A j f ) = M p j V f. ii)
Let
Ψ : [ a , ∞ ) → R be a bounded, measurable function. Then Ψ( A ) defined by thespectral Theorem is a bounded operator on H and for all f ∈ H we have V (Ψ( A ) f ) = M Ψ ( V f ) . Finally, this leads to a characterization of the spaces D ( A j ). Theorem 7.7.
For j ∈ N the following statements are equivalent: i) u ∈ D ( A j ) , ii) λ λ j ( V u )( λ ) ∈ L σ , iii) λ λ j ( V u ) k ( λ ) ∈ L σ ([ a k , + ∞ )) , k = 1 , . . . , n .Proof. Due to Theorem 7.5, it holds ( A j u, A j u ) H = | V ( A j u ) | σ . Now Lemma 7.6 implies( A j u, A j u ) H = | M p j V u | σ . (cid:3) The above results provide explicit solution formulae for evolution equations involving theoperator A . For example for the wave equation ¨ u + Au = 0 with u (0) = u and ˙ u (0) = 0a formal solution is given by u ( t ) = Z cos( √ λt ) V u . Our aim in the future will be to studyproperties of these solutions, as indicated in the introduction.8. Appendix
The book of J. Weidmann [27] describes a general approach to the spectral theory of systems ofSturm-Liouville equations, which is in principle applicable to our setting. In this point of view,our problem is seen as a system of n equations on (0 , + ∞ ) coupled by boundary conditions in0. Thus the kernel of the resolvent is a matrix-valued function K ( · , · , λ ) : (0 , + ∞ ) × (0 , + ∞ ) → C n × n . The relation to Definition 3.6 is given by K ij ( x, x ′ , λ ) = K ( x, x ′ , λ ) where ( x, x ′ ) ∈ N i × N j b = (0 , + ∞ ) × (0 , + ∞ ) . The fundamental hypothesis of Weidmann is that the kernel of the resolvent can be written inthe following form: K ( x, x ′ , λ ) = P pq =1 (cid:16) p X l =1 α lq w l ( x, λ ) (cid:17)| {z } =: m αq ( x,λ ) w q ( x ′ , λ ) T , for x ′ ≤ x, P pq =1 (cid:16) p X l =1 β lq w l ( x, λ ) (cid:17)| {z } =: m βq ( x,λ ) w q ( x ′ , λ ) T , for x ′ > x, (17)where α lq , β lq ∈ C and the w q : [0 , ∞ ) × C → C n are such that { w q ( · , λ ) : q = 1 , . . . , p } is a fundamental system of ker( A f − λI ) , the space of generalized eigenfunctions of A . Here A f : D ( A f ) → C ( R ) is the formal operator, in our case A f = A = ( − c k · ∂ x + a k ) k =1 ,...,n but D ( A f ) = Q nk =1 C ( N k ), i.e. the operator A without transmission conditions nor integrabilityconditions at ∞ . Clearly in our case dim (cid:0) ker( A f − λI ) (cid:1) = 2 n and thus p = 2 n .In contrast to the typical applications treated in [27] as for instance the Dirac system, weconsider in this paper a transmission problem, i.e. intuitively the components of the w q are functions on different domains N k (while mathematically all N k are equivalent to (0 , + ∞ )):the branches of a star. For all such applications it is highly important to use only generalizedeigenfunctions satisfying the transmission conditions ( T ) and ( T ), for example Theorem 7.7would be impossible otherwise.Supposing the ansatz (17) of Weidmann, this is not possible, what we shall show now.In Definition 3.6 we have given an explicit expression for the (unique) kernel K of the resolventof A , using only elements of ker( A T − λI ), where A T : D ( A T ) → H satisfies A T = A = ( − c k · ∂ x + a k ) k =1 ,...,n and D ( A T ) = Q nk =1 C ( N k ) ∩ { ( u k ) nk =1 satisfies ( T ) , ( T ) } . Note that ker( A T − λI ) isthe n -dimensional space of generalized eigenfunctions of A satisfying the transmission conditions( T ) and ( T ), but without integrability conditions at infinity.Suppose that we have a representation of K as given in (17) satisfying w q ( · , λ ) ∈ ker( A T − λI ) , q = 1 , . . . , p, λ ∈ ρ ( A )and thus we can take p = n . Let us fix x ∈ N and λ ∈ ρ ( A ). Then the m βq ( x, λ ) =: m βq ∈ C areconstants and the expression g ( x ′ , λ ) := p X q =1 m βq w q ( x ′ , λ )defines a function g ( · , λ ) : { x ′ ∈ [0 , ∞ ) : x ′ > x } → C n . Clearly, the functions [ w q ( · , λ )] j : N j → C can be uniquely extended to entire functions in x ′ because they are linear combinations of e ± iξ j ( λ ) x ′ . Thus the [ g ( · , λ )] j are entire. Our hypothesis w q ( · , λ ) ∈ ker( A T − λI ) , q = 1 , . . . , p ,implies thus g ( · , λ ) ∈ ker( A T − λI ) , q = 1 , . . . , p. (18)But comparing (17) with Definition 3.6 yields g ( x ′ , λ ) = F ± , λ ( x ) · F ± , λ ( x ′ ) for x ′ ∈ N , . . . , N n and g ( x ′ , λ ) = F ± , λ ( x ) · F ± , λ ( x ′ ) for x ′ ∈ N , which makes sense after analytic continuation. Using the assumption that g satisfies ( T ), weget from these two equalities, putting x ′ = 0 ∈ N into the first and x ′ = 0 ∈ N into the second F ± , λ ( x ) = F ± , λ ( x ) F ± , λ (0) = g (0 , λ ) = F ± , λ ( x ) F ± , λ (0) = F ± , λ ( x ) . But inspecting the definitions of the generalized eigenfunctions, bearing in mind that x ∈ N was arbitrary, this impliescos (cid:0) ξ ( λ ) x (cid:1) ± i sin (cid:0) ξ ( λ ) x (cid:1) = cos (cid:0) ξ ( λ ) x (cid:1) ± is ( λ ) sin (cid:0) ξ ( λ ) x (cid:1) and thus s ( λ ) = 1. This finally implies P nj =1 c j ξ j ( λ ) = 0, which is impossible for all real λ < a ,since then ξ j ( λ ) is purely imaginary. Furthermore, P nj =1 c j ξ j is analytic, so this equality can, ifever, only be fulfilled on a discrete set of λ ∈ C .We have thus proven: Theorem 8.1.
Representation (17) of the kernel K of the resolvent of A is not possible, usingexclusively generalized eigenfunctions w q ( · , λ ) ∈ ker( A T − λI ) , q = 1 , . . . , p. HE MULTIPLE TUNNEL EFFECT ON A STAR-SHAPED NETWORK 23
The reason for this is, shortly speaking, the rigidity of (17) caused by the use of the samelinear combinations of generalized eigenfunctions above the diagonals of all N j × N k (idembelow). This is not compatible with the fact that the kernel is non-smooth only on the maindiagonals, i.e. the diagonals of N j × N j , cf. Definition 3.6 and Figure 1.This means that the approach of J. Weidmann in [27], when applied to problems on the star-shaped domain, does not sufficiently take into account its non-manifold character. The resultingexpansion formulae would use generalized eigenfunctions which are in a sense incompatible withthe geometry of the domain. This would have undesirable consequences: for example, theimportant feature of Theorem 7.7 that the belonging of u to D ( A j ) is expressed by the decayof the components of V u would be impossible, due to artificial singularities in the expansionformula.
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Univ Lille Nord de France, F-59000 Lille, FranceUVHC, LAMAV, FR CNRS 2956, F-59313 Valenciennes, France
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