Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian: Special Cases

In a previous paper, we used a classification of the self adjoint extensions, also called self-adjoint determinations, of the differential operator -d(2)/dx(2) in order to obtain the whole list of Supersymmetric (SUSY) partners of those selfadjoint determinations for which the ground state has strictly positive energy. The existence of self adjoint determinations with a ground state of zero or even negative energy is a proved fact. In this paper, we analyze the possibility of constructing SUSY partners for those determinations. We also study those cases for which the ground state has a degeneracy, the study of their SUSY partners should be analyzed separately. So far, we have studied those determinations having an exactly solvable eigenvalue problem. On the present study, we also included some comments in relation to determinations not exactly solvable from this point of view. In addition, the use of self adjoint determinations for which the ground state wave function has nodes (zeroes) produces formal SUSY partners with a finite number of eigenvalues or even with a purely continuous spectrum. We give some worked examples of these situations.

Automated summary

In this paper, we classify the self adjoint extensions of the differential operator -d(2)/dx(2) and obtain the complete list of Supersymmetric (SUSY) partners with strictly positive energy. We also study the cases where the ground state has degeneracy and provide examples of their SUSY partners. The use of self adjoint determinations with ground state wave function nodes produces formal SUSY partners with finite eigenvalues or purely continuous spectrum.