The semiclassical limit on a star-graph with Kirchhoff conditions

We consider the dynamics of a quantum particle of mass $m$ on a $n$-edges star-graph with Hamiltonian $H_K=-(2m)^{-1}\hbar^2 \Delta$ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kre\u{\i}n's theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple $(H_K,H_{D}^{\oplus})$, where $H_{D}^{\oplus}$ is the free Hamiltonian with Dirichlet conditions in the vertex.


Introduction
Aim of this work is to provide the semiclassical dynamics and scattering for an approximate coherent state propagating freely on a star-graph, in presence of Kirchhoff conditions in the vertex.
Since the pioneering work of Kottos and Smilansky [17], having in mind applications to quantum chaos, the semiclassical limit of quantum graphs is often understood as the study of the distribution of eigenvalues (or resonances, see [18]) of self-adjoint realizations of −(2m) −1h2 ∆ on the graph.
To the best of our knowledge, a first study of the semiclassical limit for quantum dynamics on graphs is due to Barra and Gaspard [2] (see also [3], where the limiting classical model is comprehensively discussed). In this case, the semiclassical limit is understood in terms of the convergence of a Wigner-like function for graphs whenh (the reduced Planck constant) goes to zero.
Inspired by the work of Hagedorn [12], instead, we look directly at the dynamics of the wavefunction, for a class of initial states which are close to Gaussian coherent states supported on one of the edges of the graph.
Closely related to our work is a series of papers by Chernyshev and Shafarevich [6,8,9] in which the authors study theh → 0 limit of Gaussian wave packets propagating on graphs. Their main interest is the asymptotic growth (for large times) of the number of wave packets propagating on the graph. The main tool for the analysis is the complex WKB method by Maslov (see [19]). We also point out the work [7], by the same authors, in which they study the smallh asymptotics of the eigenvalues of Schrödinger operators on quantum graphs (with Kirchhoff conditions in the vertices and in the presence of potential terms).
In our previous work [5] we studied the semiclassical limit in the presence of a singular potential. Specifically, we considered the operator H α , which is the quantum Hamiltonian in L 2 (R) formally written as H α = −h 2 2m ∆ + αδ 0 , where m is the mass of the particle, δ 0 is the Dirac-delta distribution centered in x = 0, and α is a real constant measuring the strength of the potential. Given a Gaussian coherent state on the real line of the form (1.1) ψh σ,ξ (x) := with σ ∈ C, Re σ = σ 0 > 0 and ξ ≡ (q, p) ∈ R 2 , we studied the limith → 0 of e −i t h Hα ψh σ 0 ,ξ . To this aim we reasoned as follows. For fixed x ∈ R, consider the classical wave function defined by φh σ,x : R 2 → C , φh σ,x (ξ) := ψh σ,ξ (x) .
Consider the vector field X 0 (q, p) = (p/m, 0) associated to the free classical Hamiltonian h 0 (q, p) = p 2 /(2m) (q is the position and p the momentum of the classical particle of mass m), and the Liouville operator We set where ξ t is the solution of the free Hamilton equations, A t is the (free) classical action, and σ t takes into account the spreading of the wave function. If the dynamics is free (i.e., α = 0), one has the identity e −i t h H 0 ψh σ 0 ,ξ (x) = e ī h At ψh σt,ξt (x) .
In the same spirit, in the present work, we study the smallh asymptotic of e −i t h H K Ξh σ 0 ,ξ where H K is the quantum Hamiltonian obtained as a self-adjoint realization of −(2m) −1h2 ∆ on the stargraph with Kirchhoff conditions in the vertex, and Ξh σ,ξ resembles a coherent state concentrated on one edge of the graph (see Section 1.3 below for the precise definition).
In the following sections of the introduction we give the main definitions and results. Section 2 and 3 contain a detailed description of the quantum and semiclassical dynamics on the star-graph respectively. In Section 4 we give the proofs of Theorems 1.3 and 1.5. Section 5 contains some additional remarks and comments. In the appendix we give a proof of a technical result, namely an explicit formula for the wave operators for the pair (Dirichlet Laplacian, Neumann Laplacian) on the half-line.
1.1. Quantum dynamics on the star-graph. By star-graph we mean a non-compact graph, with n edges (or leads) and one vertex. Each edge can be identified with a half-line, the origins of the half-lines coincide and identify the only vertex of the graph.
We recall that the Hilbert space associated to the star-graph is L 2 (G) ≡ ⊕ n ℓ=1 L 2 (R + ), with the natural scalar product and norm; in particular, for the L 2 -norm we use the notation If ψ ∈ L 2 (G), ψ ℓ ∈ L 2 (R + ) is its ℓ-th component with respect to the decomposition ⊕ n ℓ=1 L 2 (R + ). In a similar way one can define the associated Sobolev spaces; in particular, we set H 2 (G) ≡ ⊕ n ℓ=1 H 2 (R + ), with the natural scalar product and norm.
We are primarily interested in the semiclassical limit of the quantum dynamics generated by the Kirchhoff Laplacian on the star-graph, which is the operator here ψ ′′ denotes the element of L 2 (G) with components ψ ′′ ℓ , and ψ(0) (resp., ψ ′ (0)) the vector in C n with components ψ ℓ (0) (resp., ψ ′ ℓ (0)). Functions in dom(H K ) are said to satisfy Kirchhoff (or Neumann, or standard, or natural) boundary conditions.
In the analysis of the semiclassical limit of the wave and scattering operators, we will have to fix a reference dynamics on the star-graph. To this aim we will consider the operator H ⊕ D (see also the equivalent definition in Eqs. (2.5) -(2.6) below) we remark that H ⊕ D can be understood as the direct sum of n copies of the Dirichlet Hamiltonian on the half-line (see Section 2.2 below for further details).
We recall that the quantum wave operators and the corresponding scattering operator on L 2 (G), are defined by These operators can be computed explicitly (see Proposition 2.2 and Remark 2.3 below), and component-wisely for ℓ = 1, . . . , n they read as follows: where F s and F c are the Fourier-sine and Fourier-cosine transforms respectively (see Eqs. (2.11) and (2.12));

1.2.
Semiclassical dynamics on a star-graph. The generator of the semiclassical dynamics on the star-graph is obtained as a self-adjoint realization of the differential operator − i p m ∂ ∂q in ⊕ n ℓ=1 L 2 (R + × R), (q, p) ∈ R + × R. To recover it we will make use of the method to classify the singular perturbations of self-adjoint operators developed by one of us in [20] (see also [21]).
To do so, the first step is to identify a simple dynamics on the star-graph, more precisely its generator. We shall consider classical particles moving on the edges of the graph with elastic collision at the vertex.
We start by considering the dynamics of a classical particle on the half-line with elastic collision at the origin. We obtain its generator as a limiting case from our previous work [5] and denote it by L D . We postpone the precise definition of L D to Section 3.1. Here we just note few facts.
is self-adjoint and acts on elements of its domain as For all t ∈ R, the action of the unitary evolution group associated to it is explicitly given by The (trivial) classical dynamics of a particle on the star-graph with elastic collision at the vertex can be defined in the following way. Denote by f a function of the form . . .
If f ⊕ n ℓ=1 L 2 (R+×R) = 1, |f ℓ (q, p)| 2 dqdp can be interpreted as the probability of finding a particle on the ℓ-th edge of the graph, with position in the interval [q, q + dq] and momentum in the interval [p, p + dp].
Define the operator L ⊕ D := ⊕ n ℓ=1 L D ; the associated dynamics is generated by the unitary group e iL ⊕ D t = ⊕ n ℓ=1 e iL D t , and it is trivial in the sense that it can be fully understood in terms of the dynamics on the half-line described above.
We consider the map defined on sufficiently smooth functions (we refer to Section 3 for the details). This map can be extended to a continuous one on dom(L ⊕ D ). The operator L ⊕ D ↾ ker(γ ⊕ + ) is symmetric; in Theorem 3.3, by using the approach developed in [20,21] we identify a family of self-adjoint extensions. Among those we select the one that turns out to be useful to study the semiclassical limit of exp(−iH K t/h) and denote it by L K .
We postpone the precise definition of L K to Section 3.3, see, in particular, Remark 3.4. Here we just give component-wisely the formula for the associated unitary group, for ℓ = 1, . . . , n and for all t ∈ R: We define the classical wave operators and the corresponding scattering operator on ⊕ n ℓ=1 L 2 (R + × R) by (1.10) and (1.11) S cl := (Ω + cl ) * Ω − cl . These operators can be computed explicitly (see Proposition 3.8 below), and component-wisely they read as follows for ℓ = 1, . . . , n (θ is the Heaviside step function): 1.3. Truncated coherent states on the star-graph. In general, there is no natural definition of a coherent state on a star-graph, neither there is a unique way to extend coherent states through the vertex. Since we are interested in initial states concentrated on one edge of the graph, we introduce the following class of initial states. We denote by ψh σ,ξ the non-normalized restriction of ψh σ,ξ (see Eq. (1.1)) to R + , namely, On the graph we consider the quantum states defined as Definition 1.1 (Quantum states). Let σ ∈ C, with Re σ = σ 0 > 0, and ξ = (q, p) ∈ R + × R; consider any normalized function Ξh σ,ξ ∈ L 2 (R + ), such that (1.14) Ξh σ,ξ − ψh σ,ξ L 2 (R+) ≤ C 0 e − ε q 2 h|σ| 2 for some C 0 , ε > 0 .
We are primarily interested in quantum states on the star-graph of the form Correspondingly, we will consider the family of classical states We will make use of the family of classical states on the star-graph given by

Main results.
Our first result concerns the semiclassical limit of the dynamics. Theorem 1.3. Let σ 0 > 0, ξ = (q, p) ∈ R + × R and consider any initial state of the form Ξh σ 0 ,ξ , together with its classical analogue Σh σ 0 ,x . Then, for all t ∈ R there holds In the second part of our analysis we study the semiclassical limit of the wave operators and of the scattering operator.
Theorem 1.5. Let σ 0 > 0, ξ = (q, p) ∈ R + × (R\{0}) and consider any state of the form Ξh σ 0 ,ξ , together with its classical analogue Σh σ 0 ,x . Then, there hold Identity (1.18) is an immediate consequence of Eqs. (1.7) and (1.12), and of the definitions of Ξh σ 0 ,ξ and Σh σ 0 ,x . Remark 1.6. Eq. (1.17) makes evident that Ω ± Ξh σ 0 ,ξ and (Ω ± cl Σh σ 0 ,(·) )(ξ) are exponentially close (with respect to the natural topology of L 2 (G)) in the semiclassical limithσ 2 0 /q 2 ,h/σ 2 0 p 2 → 0 + for any ξ = (q, p) with q > 0 and p = 0. As a matter of fact, it can be proved that the relation (1.17) remains valid also for p = 0 if one puts (Ω ± cl Σh σ 0 ,x )(q, 0) = Σh σ 0 ,x (q, 0); the latter position appears to be reasonable and is indeed compatible with the computations reported in the proof of Proposition 3.8. Nonetheless, since exp(− σ 2 0 p 2 /h) = 1 in this case, the resulting upper bound is of limited interest for what concerns the semi-classical limit. To say more, for p = 0 andhσ 2 0 /q 2 (or C 0 ) small enough, by a variation of the arguments described in the proof of Theorem 1.5 one can derive the lower bound This shows that, as might be expected, the classical scattering theory does not provide a good approximation for the quantum analogue when p = 0. On the contrary, notice that Eq. (1.16) ensures a significant control of the error for the dynamics at any finite time t ∈ R even for p = 0.

The quantum theory
2.1. Dirichlet dynamics on the half-line. Let us first consider the free quantum Hamiltonian for a quantum particle of mass m on the whole real line, defined as usual by together with the associated free unitary group U 0 t := e −i t h H 0 (t ∈ R). Correspondingly, let us recall that for any ψ ∈ L 2 (R) we have Let us further introduce the Dirichlet Hamiltonian on the half-line R + , defined as usual by and refer to the associated unitary group . As well known, the latter operator can be expressed as where, in view of the identity (2.1), we introduced the bounded operators on L 2 (R + ) defined as follows for ψ ∈ L 2 (R + ) and x ∈ R + : together with its adjoint Namely, Θ gives the extension by zero to the whole real line R of any function on R + , while Θ * is the restriction to R + of any function on R. Note that Θ is an isometry. In fact, Θ * Θ is the identity on L 2 (R + ) and Θ Θ * is an orthogonal projector (but not the identity) on L 2 (R); more precisely, we have (θ is the Heaviside step function) To proceed let us consider the parity operator Of course P is a unitary, self-adjoint involution which commutes with the free Hamiltonian H 0 , i.e., Furthermore it can be checked by direct inspection that ran(P Θ) = ker(Θ * ) Using the bounded linear maps introduced above, one can express the operators defined in Eq. (2.3) as follows: Recalling that (U 0 t ) * = U 0 −t , the above relation allow us to infer Let us finally point out that, on account of the obvious operator norms Θ = Θ * = 1, U 0 t = 1 and P = 1, from Eq. (2.4) it readily follows

2.2.
Dirichlet and Kirchhoff dynamics on the star-graph. Let us now introduce the quantum Hamiltonian on the graph G, corresponding to Dirichlet boundary conditions at the vertex. This coincides with the direct sum of n copies of the Dirichlet Hamiltonian H D on the half-line R + , namely: In view of the identity (2.2), it can be readily inferred that the corresponding unitary group where U ± t is defined as in Eq. (2.3). To proceed let us consider the Kirchhoff Hamiltonian on the graph G. This is defined as in Eqs.
In what follows we denote by S the n × n matrix with components By a slight abuse of notation we use the same symbol to denote the operator in L 2 (G) defined by By arguments similar to those given in the proof of [1, Thm. 2.1] (cf. also [11] and [15, Eq. (7.1)]) we get 2.3. The quantum wave operators and scattering operator. Let us consider the wave operators and the corresponding scattering operator on L 2 (G) respectively defined in Eqs. (1.5) and (1.6).
Since H K has purely absolutely continuous spectrum σ(H K ) = [0, ∞), we have that Ω ± are unitary on the whole Hilbert space L 2 (G), i.e., (Ω ± ) * Ω ± = 1 , which in turn ensures 1 (2.10) Let us define the unitary operators F s : L 2 (R + ) → L 2 (R + ) and F c : L 2 (R + ) → L 2 (R + ): The wave operators can be computed explicitly. To this aim one could use the results from Weder [22] (see also references therein), with some modifications, since in [22] the reference dynamics is given by the Hamiltonian with Neumann boundary conditions. For the sake of completeness, we prefer to give an explicit derivation of the result, obtained by taking the limit t → ±∞ on the unitary groups. We remark that in [22] the formulae are obtained by using the Jost functions.
We have the following explicit formulae for the wave operators: The quantum wave operators can be expressed as Proof. By Eqs. (2.7) and (2.9) we easily obtain the identity (2.14) Let us find more convenient expressions for the operators on the right-hand side. Let ψ ∈ L 2 (R + ) and define On the other hand, Recall that U D t is the unitary group generated by the Dirichlet Laplacian on the half-line; its integral kernel is given by Moreover, let U N t be the unitary group generated by the Neumann Laplacian on the half-line; its integral kernel is given by Note that, for x ∈ R + , Hence, A similar computation gives Hence To compute the wave operator we have to evaluate the limits s-lim t→±∞ U N −t U D t ; the latter give the wave operators Ω ± ND for the pair (Dirichlet Laplacian, Neumann Laplacian) on the half-line, which are computed in Proposition A. 1 [16].
3. The semiclassical theory 3.1. Classical dynamics on the half-line with elastic collision at the origin. We start by recalling some basic definitions and results from [5]. Let X 0 (q, p) = (p/m, 0) be the vector field associated to the free (classical) Hamiltonian of a particle of mass m and consider the differential operator We denote by Moreover the action of the (free) unitary group e itL 0 (t ∈ R) is given by For any f ∈ S(R 2 ) we define the map (γf)(p) := f(0, p). For a comparison with the results in [5,Sec. 2], recall that the map γ can be equivalently defined as (γf)(p) = 1 √ 2π R dkf(k, p) wherẽ f(k, p) is the Fourier transform of f(q, p) in the variable q. By [5, Lem. 2.1], the map γ extends to a bounded operator γ : dom(L 0 ) → L 2 (R, |p| dp), where dom(L 0 ) ⊂ L 2 (R) is endowed with the graph norm. For any z ∈ C\R, we define the bounded linear map G z : L 2 (R, |p| −1 dp) → L 2 (R 2 ) , G z := (γ R 0 z ) * (here L 2 (R, |p| −1 dp) and L 2 (R, |p|dp) are considered as a dual couple with respect to the duality induced by the scalar product in L 2 (R)). An explicit calculation gives (G z u)(q, p) = θ(q p Im z) sgn(Im z) i m |p| e imzq/p u(p) .
Next we consider the classical motion of a point particle of mass m on the whole real line, with elastic collision at the origin. The generator of the dynamics, denoted by L ∞ , is obtained as a limiting case, for β → ∞, of the operator L β defined in [5]. To this aim we set Λ ∞ z : L 2 (R, |p|dp) → L 2 (R, |p| −1 dp) , (Λ ∞ z u)(p) := i sgn(Im z) 2|p| m u(p) .
Similarly, for the unitary operator describing the dynamics we have Lemma 3.2. For any t ∈ R and for any f ∈ L 2 (R 2 ), there holds Proof. First note that the identity in Eq.
which in turn implies The thesis follows upon substitution of the above identity into Eq. (3.4).
Let us now consider the natural decomposition 3 and notice that both the subspaces are left invariant by the resolvent R ∞ z , this is evident from Eq. (3.3). Taking this into account, we introduce the bounded operator By direct computations, from Eq. (3.3) (here employed with q > 0) we get We denote by L D the self-adjoint operator in L 2 (R + × R) having R D z as resolvent, so that For all (q, p) ∈ R + ×R, t ∈ R and f ∈ L 2 (R + ×R), from the above definition and from Eq.
which describes the motion of a classical particle on the half-line R + with elastic collision at q = 0. Let us also mention that, in view of the basic identity e itL 0 (0 ⊕ f) (q, p) = θ q + pt m e itL 0 (0 ⊕ f) (q, p) , (3.12) the above relation (3.11) is equivalent to Another equivalent (and more explicit) formula for the action of the unitary group e itL D is the one given in Eq. (1.8).
Finally, from Lemmata 3.1 and 3.2 (here employed with q > 0) we derive, respectively, 3.2. Classical dynamics on the graph with total reflection at the vertex. Let us now consider the "classical" Hilbert space ⊕ n ℓ=1 L 2 (R + × R) and indicate any of its elements with the vector notation Let L D be defined according to Eq. (3.10), and consider the classical dynamics on the star-graph G with total elastic collision in the vertex; this is described by the self-adjoint operator L ⊕ D := ⊕ n ℓ=1 L D . 4 Note that for q > 0 we have θ(−tqp) θ |pt| The associated resolvent and time evolution operators are respectively given by Explicitly, for ℓ = 1, . . . , n and t ∈ R, from Eq. (1.8) we derive . We use the above map to define a trace operator on the graph: . In what follows we use the technique developed by one of us in [20,21] to characterize all the self-adjoint extensions of the symmetric operator L ⊕ D ↾ ker(γ ⊕ + ) (see Theorem 3.3 below). Among those we select the one that turns out to be useful to study the semiclassical limit of exp(−iH K t/h), see Remark 3.4.
In the sequel, we proceed to determine the unitary evolution associated to the above choices by means of arguments analogous to those described in the proof of [5, Prop. 2.4].
Proposition 3.5. For all f ∈ ⊕ n ℓ=1 L 2 (R + × R) and for all t ∈ R there holds where S := 1−2 Π was already defined in relation with the quantum scattering operator, see Eq. (2.8).
Proof. Throughout the whole proof we work component-wisely, denoting with ℓ ∈ {1, ..., n} a fixed index. Let us first remark that the resolvent (3.18), with B, Π as in Eq. (3.20), acts on any element f ∈ ⊕ n ℓ=1 L 2 (R + × R) according to From the above relation we derive the following, recalling the explicit expressions for g z (q, p) and m ∞ z (p) given in Eqs. (3.16) and (3.17), as well as Eq. (3.15) for γ + R D z f =G + z f: We now proceed to compute the unitary operator e −itL K (t ∈ R) by inverse Laplace transform, using the above representation for the resolvent R K z . Let us first assume t > 0; then, for any c > 0 and f ∈ dom(L K ), we get (see [10,Ch. III,Cor. 5.15]) On the one hand, recalling Eq.
On the other hand, noting that θ p Im(k+ic) = θ(pc) = θ(p) and sgn Im(k+ic) = sgn(c) = +1 for c > 0, by computations similar to those reported in the proof of [ Summing up, the above relations imply For t < 0 one can perform similar computations, starting from the following identity where c > 0: We omit the related details for brevity. In the end, one obtains exactly the same expression as in Eq. (3.22), which with the trivial replacement t → −t proves Eq. (3.21).
Remark 3.6. By Eq. (3.12) we infer that the action of the unitary group e itL K is explicitly given, component-wisely, by Eq. (1.9).
Remark 3.7. Recalling the explicit form of Π (see Eq. (3.20)) we obtain In particular, for a star-graph with three edges (n = 3) we have whence S = − M with respect to the notation used in [1]. 5 Especially, recall the following basic identity regarding the unitary Fourier transform F and its inverse F −1 : which holds true whenever e c· |a| h(·/a) ∈ L 2 (R). In addition, keep in mind the relation written in Eq. (3.12).

3.4.
The semiclassical wave operators and scattering operator. Let us consider the wave operators and the corresponding scattering operator on ⊕ n ℓ=1 L 2 (R + × R), respectively defined by Eqs. (1.10) and (1.11). The following proposition provides explicit expressions for these operators.
To proceed let us point out the forthcoming lemma which characterizes a large class of functions satisfying the condition in Definition 1.1.
On one hand, recalling the definition (1.13) of ψh σ,ξ and that ψh σ,ξ L 2 (R) = 1, using the basic inequality (a − b) 2 ≤ |a 2 − b 2 | for a, b > 0 we get Recalling the hypothesis (4.2), the explicit expression (1.1) for ψh σ,ξ , and that we are assuming q > 0, from the above results we derive On the other hand, by arguments similar to those employed above we get Summing up, the previous results and the basic relation which suffices to infer the thesis on account of the uniform boundedness of χ q,η .
Again, the assumptions of Lemma 4.2 are certainly verified and the related functions Ξh σ,ξ have compact support in R + , besides satisfying the bound (1.14).
Proof of Theorem 1.3. Let Ψh σ 0 ,ξ and Φh σ 0 ,(·) (ξ) be, respectively, as in Eqs. (4.4) and (4.5), and note that from the triangular inequality it follows  Regarding the first term on the right-hand side of Eq. (4.6), by the unitarity of e −i t h H K and the condition (1.14) we infer As for the second term in Eq. (4.6), note that Eqs. (2.9) and (3.21) give Since e itL 0 (0 ⊕ φh σt,x ) (±ξ) = φh σt,x (±ξ t ) = ψh σt,±ξt (x) for x ∈ R + , from the above identity and from Lemma 4.1 we deduce 6 Let us finally consider the third term in Eq. (4.6). Recalling again the identity (3.21), we obtain From the above identity, by arguments similar to those employed previously we get e itL K Φh σt,(·) (ξ) − e itL K Σh σt,(·) (ξ) ( 6 Note also that, on account of Eq. Additionally, recalling the expression of Ω ± cl given in Eq. (3.24), we get In view of these results together with the identity (1.15) we derive By the bound (1 − 2θ(∓p))F c ∓ F s ≤ |1 − 2θ(∓p)| F c + F s ≤ 2 and by Eq. (1.14), we infer In what follows we prove the following upper bound which concludes the proof of the theorem We start with the identity Considering separately the cases p > 0 and p < 0 for the two possible choices of the signs, it is easy to convince oneself that Recall that the Fourier transform of ψh σ 0 ,ξ is given by Let us assume p > 0, we have the chain of inequalities/identities 2 π ∞ 0 dk ∞ 0 dx e ikx ψh σ 0 ,ξ (x) Reasoning like for the bound in Eq. (4.1), we obtain which conclude the proof of the bound (4.7) for p > 0. The proof of the bound for p < 0 is identical and we omit it. Identity (1.18) follows immediately from Eqs. (1.7) and (1.12).

5.1.
A comparison with the standard approach to the definition of a classical dynamics on the graph. There is no trajectory of a classical particle which is the semiclassical limit of e −i t h H K Ξh σ 0 ,ξ . As a consequence, the semiclassical dynamics is not described by the Hamilton equations. One way to overcome this difficulty is to assign a probability to every possible path on the graph. Typically the probability of a certain path is postulated, and given in terms of the square modulus of the quantum transition (or stability) amplitudes (see, e.g., [ [17] and in several other works, see, e.g., [3], the review [14], and the monograph [4]. We have already noted that, up to a sign, the coefficients 2 n − δ ℓ,ℓ ′ coincide with the elements of the matrix S identifying both the classical and quantum scattering operators.
In our paper we followed a different train of thought. We wanted to recover the limiting classical dynamics on the star-graph starting from the trivial dynamics of classical particle on the half-line with elastic collision in the origin. To do so we made use of a Kreȋn's formula to find and classify singular perturbations of self-adjoint operators, see [20] and [21]. We remark that, in a similar way, one can reconstruct the Hamiltonian H K starting from the free Hamiltonian of a quantum particle on the half-line with Dirichlet conditions in the origin.
We defined the generator of the trivial dynamics on the half-line through Eq. (1.8). Note that if f ∈ dom(L D ), then f t = e −itL D f satisfies the Liouville equation but the action of the group can be extended in a natural way to any bounded function.
Since the evolution is unitary in as a density in the phase space R + × R. Setting ρ(q, p) := |f(q, p)| 2 , for all t ∈ R one has that and it satisfies the equation here V is the interaction potential and the initial datum is (q 0 , p 0 ) = (q, p); σ t andσ t are given by and A t is given by In our notation, one can think to a classical function given by φh(σ,σ, x; q, p) = ψh(σ,σ, q, p; x). With this identification ψh(σ t ,σ t , q t , p t ; x) = φh(σ t ,σ t , x; q t , p t ) = e iL V t φh(σ t ,σ t , x)(q, p), with L V the Liouville operator associated to the vector field of the classical Hamiltonian p 2 2m + V(q). This is analogous to e itL K Σh σt,x (ξ).

√ 2π
∞ 0 dk (i sin(kx) ∓ cos(kx)) e −ik 2 t (F s ϕ)(k) We have obtained the following explicit formula for the quantity we are interested in We note that, to prove the statement for W ND + we have to study the limit t → +∞ of while, to prove the statement for W ND − we have to study the limit t → −∞ of (U N −t U D t + F * c F s )ϕ 2 L 2 (R+) = 2 π ∞ 0 dx ∞ 0 dk e ikx+ik 2 |t| (F s ϕ)(k) 2 .