Spectral theory for the wave equation in two adjacent wedges

Mehmeti, Félix Ali; Meister, E.; Mihalinčić, Krešo

doi:10.1002/(sici)1099-1476(199708)20:12<1015::aid-mma899>3.0.co;2-a

Cited by 6 publications

(3 citation statements)

References 4 publications

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“…In his habilitation thesis Ali Mehmeti [2] started first investigations for-the half-plane problem described above. Basing on the diploma thesis of Mihalinëié recently a 3-dimensional half-space problem for the wave equation with two adjacent wedge-shaped materials is solved by spectral theoretic means [3].…”

Section: V1 (Xyt;o) = G(t -X Cos 6 -Y Sin 9)mentioning

confidence: 99%

Initial Dirichlet Problem for Half-Plane Diffraction: Global Formulae for its Generalized Eigenfunctions, Explicit Solution by the Cagniard-de Hoop Method

Meister¹,

Rottbrand²

1999

Z. Anal. Anwend.

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This paper deals with time-dependent plane wave diffraction by a soft/soft Sommerfeld half-plane E : x > 0, y = ±0. The explicit solution is obtained as a time-convolution in two ways: The first is directly applying the Cagniard de Hoop method to the generalized Wiener-Hopf solution of the corresponding stationary problem due to Meister & Speck (1989). The second way makes use of the Laplace integral representation of the generalized eigenfunctions with respect to the spatial (Cartesian) variables derived by Ali Mehmeti in his habilitation thesis (1995) from formulae of Meister (1983). After deforming the path of.integration into the semi-infinte branch cut lines of the characteristic square root _-k2 of the Helmholtzian, he obtains representations where real wave numbers may appear. But for convergence of the integrals one has to distinguish the cases x 0 and x < 0, where the obstacle is present or not. We set the time Laplace variable s =-ik and recover the time domain functions for the diffracted field from the eigenfunctions of the stationary problem. There follows a global formula representation with polar coordinates having the diffracting edge of E as its center. The solution of the initial boundary value problem is seen to coincide with that obtained in the first way, indeed.

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Section: V1 (Xyt;o) = G(t -X Cos 6 -Y Sin 9)mentioning

confidence: 99%

Initial Dirichlet Problem for Half-Plane Diffraction: Global Formulae for its Generalized Eigenfunctions, Explicit Solution by the Cagniard-de Hoop Method

Meister¹,

Rottbrand²

1999

Z. Anal. Anwend.

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“…[4,19]. Many results have been obtained in spectral theory for elliptic operators on various types of unbounded structures for example [21,13,6,3], cf. especially the references mentioned in [3].…”

Section: Introductionmentioning

confidence: 99%

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

Mehmeti¹,

Haller-Dintelmann

Régnier³

2012

J. Evol. Equ.

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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.

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“…[4,18]. Many results have been obtained in spectral theory for elliptic operators on various types of unbounded domains for example [20,12,6,3], cf. especially the references mentioned in [3].…”

Section: Introductionmentioning

confidence: 99%

The Klein-Gordon equation with multiple tunnel effect on a star-shaped network: Expansions in generalized eigenfunctions

Mehmeti,

Haller-Dintelmann,

Régnier

2009

Preprint

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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. 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.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.

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Spectral theory for the wave equation in two adjacent wedges

Cited by 6 publications

References 4 publications

Initial Dirichlet Problem for Half-Plane Diffraction: Global Formulae for its Generalized Eigenfunctions, Explicit Solution by the Cagniard-de Hoop Method

Initial Dirichlet Problem for Half-Plane Diffraction: Global Formulae for its Generalized Eigenfunctions, Explicit Solution by the Cagniard-de Hoop Method

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

The Klein-Gordon equation with multiple tunnel effect on a star-shaped network: Expansions in generalized eigenfunctions

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