We consider a version of the stationary phase method in one dimension of A. Erdélyi, allowing the phase to have stationary points of non-integer order and the amplitude to have integrable singularities. After having completed the original proof and improved the error estimate in the case of regular amplitude, we consider a modification of the method by replacing the smooth cut-off function employed in the source by a characteristic function, leading to more precise remainder estimates.We exploit this refinement to study the time-asymptotic behaviour of the solution of the free Schrödinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. We obtain asymptotic expansions with respect to time in certain space-time cones as well as uniform and optimal estimates in curved regions which are asymptotically larger than any space-time cone. These results show the influence of the frequency band and of the singularity on the propagation and on the decay of the wave packets. Mathematics Subject Classification (2010). IntroductionThe asymptotic behaviour of oscillatory integrals with respect to a large parameter, sometimes used to study long-time asymptotics for solutions of dispersive equations, can often be described using the stationary phase method. A theorem of A. Erdélyi [11, * Université de Valenciennes et du Hainaut-Cambrésis, LAMAV, FR CNRS 2956, Le Mont Houy, 59313 Valenciennes Cedex 9, France. Email: felix.ali-mehmeti@univ-valenciennes.fr † Université Lille 1, Laboratoire Paul Painlevé, CNRS U.M.R 8524, 59655 Villeneuve d'Ascq Cedex, France. Email: florent.dewez@math.univ-lille1.fr 1 section 2.8] permits to treat oscillatory integrals with singular amplitudes and furnishes asymptotic expansions with explicit remainder estimates. The approach is specific for one integration variable and the results are interesting for applications. Unfortunately the proof is only sketched in the source [11]. In the present paper, we start by providing a complete proof and by improving the remainder estimates in the case without amplitude singularities. Then by applying the above method to a particular example, we exhibit an inherent blow-up of the expansion occurring when the endpoints of the integration interval tend to each other. In particular, we remark that the smooth cut-off function, employed in the original proof, prevents us from controlling explicitly the blow-up, restricting potentially the field of applications. This motivates an improvement of the above stationary phase method which consists in replacing the smooth cut-off function by a characteristic function, making the blow-up explicit in the applications. Finally we apply these abstract results to the solution of the free Schrödinger equation on the line for initial conditions having a singular Fourier transform with support in a compact interval. We calculate expansions to one term with respect to time in certain space-time cones, exhibiting the optimal time decay in these regions. Moreove...

Communicated by E. MeisterConsider two copies N,, N 2 of the interval [0, sc). Consider Klein-Gordon equations with (different) constant coefficients on Iw x N, ( = time x space). Assume the coincidence of the values of the solution at the boundary points of the N , for all times and a transmission condition relating its first (one-sided) space derivatives at these points.Under a symmetry condition, we extend the spatial part of the equation and the transmission conditions to a self-adjoint operator (by Fnedrichs extension) and reformulate our problem in terms of an abstract wave equation in a suitable Hilbert space. We derive an expansion of the solution in generalized eigenfunctions of this self-adjoint extension and show, that the L"-norms (in space) of the solution and its first k space derivatives at the time r decay for t -+ z at least as const. 1 -'j4, if the initial conditions satisfy a compatibility condition of order k derived in this paper. The loss of decay rate in comparison with the full line case (const. r-''', cf. [28]) is caused by the tunnel effect.Further we show that an abstract wave equation in a Hilbert space with a Friedrichs extension as spatial part can always be derived from a stationarity principle for an associated action-type functional. This yields a physical legitimation of our model by the principle of stationary action and moreover a criterion for the physical interpretability of all models created by the linear interaction concept C4.6.8, lo], in particular for the coupling of media of different dimension (alternative to 113, 161 for similar models). IntroductionWe consider two one-dimensional semi-infinite homogeneous media of damping-free wave propagation (described by Klein-Gordon equations with constant coefficients) with different dispersion relations (i.e. with different dependence of the propagation speed on the frequency or wave number). These media are related at their end points via the continuity requirement and a physically interpretable transmission condition. More precisely, find solutions u j : [0, cc) x N j --* R, N j 2 [0, cc), j = 1,2 (with finite energy, write u(t, x) with the time variable t E [0, cc) and the space variable

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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. Exploiting a spectral theoretic solution formula from a previous paper, we study the L ∞ -time decay via Hörmander's version of the stationary phase method. We analyze the coefficient c of the leading term c • t −1/2 of the asymptotic expansion of the solution with respect to time. For two branches we prove that for an initial condition in an energy band above the threshold of tunnel effect, this coefficient tends to zero on the branch with the higher potential, as the potential difference tends to infinity. At the same time the incline to the t-axis and the aperture of the cone of t −1/2 -decay in the (t, x)-plane tend to zero.