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Limiting absorption principle for the dissipative Helmholtz equation. (English) Zbl 1205.35056
The author proposes and develops an adaption of Mourre’s commutator method [E. Mourre, Commun. Math. Phys. 78, 391–408 (1981; Zbl 0489.47010)] for the dissipative setting. This allows to prove a limiting absorption principle for a class of abstract dissipative operators. Then, the general scheme applies to the high-frequency Helmholtz operator in case when trapped classical trajectories meet the region where the absorption coefficient is non-zero thus obtaining the uniform resolvent estimate for such Helmholtz equation. The resolvent estimate is given in Besov spaces, also.

MSC:
35J10 Schrödinger operator, Schrödinger equation
47A55 Perturbation theory of linear operators
47B44 Linear accretive operators, dissipative operators, etc.
47G30 Pseudodifferential operators
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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