The Helmholtz resonator and the theory of operator extensions in a space with indefinite metric

To investigate the Helmholtz resonator a model is developed based on the theory of selfadjoint extensions of symmetric operators in a space with indefinite metric. In the case of a small opening compared to the wavelength, approximations of any predetermined precision are obtained for the Green functions of the Dirichlet and Neumann problems for the Helmholtz resonator. The problem of resonances is considered in the framework of the Lax–Phillips approach. Formulae to determine the resonances with any required precision are obtained and substantiated.