Semiclassical measure for the solution of the dissipative Helmholtz equation
نویسنده
چکیده
We study the semiclassical measures for the solution of a dissipative Helmholtz equation with a source term concentrated on a bounded submanifold. The potential is not assumed to be non-trapping, but trapped trajectories have to go through the region where the absorption coefficient is positive. In that case, the solution is microlocally written around any point away from the source as a sum (finite or infinite) of lagragian distributions. Moreover we prove and use the fact that the outgoing solution of the dissipative Helmholtz equation is microlocally zero in the incoming region.
منابع مشابه
Uniform resolvent estimates for a non-dissipative Helmholtz equation
We study the high frequency limit for a non-dissipative Helmholtz equation. We first prove the absence of eigenvalue on the upper half-plane and close to an energy which satisfies a weak damping assumption on trapped trajectories. Then we generalize to this setting the resolvent estimates of Robert-Tamura and prove the limiting absorption principle. We finally study the semiclassical measures o...
متن کاملCubic spline Numerov type approach for solution of Helmholtz equation
We have developed a three level implicit method for solution of the Helmholtz equation. Using the cubic spline in space and finite difference in time directions. The approach has been modied to drive Numerov type nite difference method. The method yield the tri-diagonal linear system of algebraic equations which can be solved by using a tri-diagonal solver. Stability and error estimation of the...
متن کاملAn efficient method for the numerical solution of Helmholtz type general two point boundary value problems in ODEs
In this article, we propose and analyze a computational method for numerical solution of general two point boundary value problems. Method is tested on problems to ensure the computational eciency. We have compared numerical results with results obtained by other method in literature. We conclude that propose method is computationally ecient and eective.
متن کاملSemiclassical measure for the solution of the Helmholtz equation with an unbounded source
Here V1 and V2 are smooth and real-valued potentials which go to 0 at infinity. Thus for any h ∈]0, 1] the operator Hh is a non-symmetric (unless V2 = 0) Schrödinger operator with domain H(R). The energy Eh will be chosen in such a way that for h > 0 small enough, δ > 1 2 and Sh ∈ L(R) the equation (1.1) has a unique outgoing solution uh ∈ L2,−δ(Rn). Here we denote by L(R) the weighted space L ...
متن کاملApplication of Decoupled Scaled Boundary Finite Element Method to Solve Eigenvalue Helmholtz Problems (Research Note)
A novel element with arbitrary domain shape by using decoupled scaled boundary finite element (DSBFEM) is proposed for eigenvalue analysis of 2D vibrating rods with different boundary conditions. Within the proposed element scheme, the mode shapes of vibrating rods with variable boundary conditions are modelled and results are plotted. All possible conditions for the rods ends are incorporated ...
متن کامل