  We propose an analytic perturbative scheme for determining the eigenvalues of
the Helmholtz equation, $(\nabla^2 + k^2) \psi = 0$, in three dimensions with
an arbitrary boundary where $\psi$ satisfies either the Dirichlet boundary
condition ($\psi =0$ on the boundary) or the Neumann boundary condition (the
normal gradient of $\psi$, $\frac{\partial \psi}{\partial n}$ is vanishing on
the boundary). Although numerous works are available in the literature for
arbitrary boundaries in two dimensions, to best of our knowledge the
formulation in three dimensions is proposed for the first time. In this novel
prescription, we have expanded the arbitrary boundary in terms of spherical
harmonics about an equivalent sphere and obtained perturbative closed-form
solutions at each order for the problem in terms of corrections to the
equivalent spherical boundary for both the boundary conditions. This
formulation is in parallel with the standard time-independent
Rayleigh-Schr{\"o}dinger perturbation theory. The efficiency of the method is
tested by comparing the perturbative values against the numerically calculated
eigenvalues for spheroidal, superegg and superquadric shaped boundaries. It is
shown that this perturbation works quite well even for wide departure from
spherical shape and for higher excited states too. We believe this formulation
would find applications in the field of quantum dots and acoustical cavities.
