A model of a boundary composed of the Helmholtz resonators

ABSTRACT

A model of resonators for which boundary is composed of the Helmholtz resonators is constructed. It is based on the theory of self-adjoint extensions of symmetric operators. We consider a limiting procedure when the number of resonators tends to infinity and look after the spectrum of the Neumann Laplacian. It is shown that there is a sequence of the model systems and, correspondingly, a sequence of eigenfunctions of the model operators which converge to an eigenfunction of the Laplacian with a specific boundary condition.

Keywords:

• Spectrum
• self-adjoint extension
• Helmholtz resonator

AMS Subject Classifications:

• 47B25
• 35P15

Acknowledgments

This work was partially financially supported by the Government of the Russian Federation (grant 08-08) and Russian Science Foundation (grant 16-11-10330). IYP thanks Professor A. Khrabustovskyi for useful discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Government of the Russian Federation [grant number 08-08] and Russian Science Foundation [grant number 16-11-10330].