Singular boundary perturbations for some eigenvalue problems

By means of two examples arising from physics we show that in contrast to a small perturbation of a regular boundary point, a small displacement of a singular boundary is singular in the sense that the expansions of the pertu rbed eigenvalues contain not only the integer powers of the small parameter involved, but also powers of the logarithm of this parameter. Examples con sidered are the Schrodinger equation for a hydrogenlike atom with nucleus o f finite small size and the linearized shallow-water equation describing wa ter waves trapped by a sloping beach. (C) 2000 American Institute of Physic s. [S0022-2488(00)04804-0].