Helmholtz equation in parabolic rotational coordinates: application to wave problems in quantum mechanics and acoustics

A method for solving exactly the Helmholtz equation in a domain bounded by two paraboloids and subject to Dirichlet, Neumann, and Dirichlet–Neumann boundary conditions is presented. It is based upon the separability of the eigenfunctions and the Frobenius power series expansion technique. Two examples of interest in wave physics are considered and analyzed quasi-analytically: the energy spectrum and wave functions of an electron in a quantum dot, and the acoustic eigenfrequencies and eigenmodes of the pressure field inside an acoustic cavity. Eigensolutions are calculated and the shape dependence of the first solution is examined. Dimensional scaling is used to factor out the size dependence of the solutions.