Energy levels of a particle confined in a super-circular box

We find the energy levels of a free particle confined in a two dimensional infinite potential well having super-circular boundary (|x|n + |y|n = an where n is a rational number and a is a positive real number) by perturbing about the equivalent circle (n = 2). The ground state energies are very accurate over a wide range of n and can be improved further by introducing a phenomenological constant determined from the knowledge of exact results available for diamond (n = 1). For excited states, we find that the shape effect can cause parametric resonance which can lead to singlet-triplet crossing. (authors)