Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation

Abstract

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 starshaped 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.

Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks

“…Proof. The proof is similar to the one of Lemma 3.13 of [3] (see also Proposition 4.5 of [6]) and is therefore omitted. The main ingredients are the use of Stone's formula, Theorem 4.12 and the limiting absorption principle Theorem 4.14 (that allows to apply the dominated convergence theorem).…”

“…Using [15], we construct N families of generalized eigenfunctions of the resulting N Schrödinger operators on IR, which we recombine on the network. This approach can be compared with the ones developed for Klein-Gordon equations in R by [5,6].…”

“…Using results of [15] for the real line case, we derive estimates showing the dependence of the generalized eigenfunctions on the potential. This enables us to prove a limiting absorption principle and then to derive an expansion of the Schrödinger group on the star in these generalized eigenfunctions (section 4) following [5,6]. The proof of the L ∞ -time decay is divided in the low frequency and high frequency part, essentialy following the lines of [16].…”

Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions

“…Such problems arise for example in the modelling of transversal vibrations of networks, gas transportation networks, traffic flow on road networks, supply chain management, water flow in open canals etc. (see [14][15][16] and the associated Klein-Gordon operator C m 2 on metric graphs play an important role in relativistic quantum theories [17,18]. Recently, there has also been a growing interest in the singularly perturbed problems on metric graphs.…”

Energy Flow Above the Threshold of Tunnel Effect

“…In Section 2 we recall the solution formula that was proved in [2] by an expansion in generalized eigenfunctions in the more general setting of a star shaped network with semi-infinite branches.…”