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A direct method for the bound states and the low energy scattering from a circular and a spherical delta shell potentials is proposed, and the results are compared with the one using the standard partial wave analysis developed for potentials with rotational symmetry. The formulation is presented in momentum space, and the scattering solutions are obtained by considering the elementary use of distributions. In this approach, the outgoing boundary conditions are imposed explicitly in contrast to the i prescription often used in quantum mechanics.

We study the hybrid type of rank one perturbations in [Formula: see text] and [Formula: see text], where the perturbation supported by a circle/sphere is considered together with the delta potential supported by a point outside of the circle/sphere. The construction of a self-adjoint Hamiltonian operator associated with formal expressions for the rank one perturbation supported by a circle and by a point is explicitly given. Bound state energies and scattering properties for each problem are also studied. Finally, we consider the rank one perturbation supported by a deformed circle/sphere and show that the first order change in bound state energies under small deformations of the circle/sphere has a simple geometric interpretation.

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We study the hybrid type of delta potentials in R 2 and R 3 , where the delta potentials supported by a circle/sphere are considered together with the delta potentials supported by a point outside of the circle/sphere. The construction of the self-adjoint Hamiltonian operator associated with the formal Hamiltonian operator for the circle and point delta potentials is explicitly given. The bound state energies and scattering properties for each problem are also studied. Finally, we consider the delta potentials supported by deformed circle/sphere and show that first order change in the bound state energies under the small deformations of circle/sphere is equal to the first order perturbative solution of the bound state energy calculated from the delta potential whose support is increased by the average of the deformation over the circle/sphere.